Discretization error estimates for discontinuous Galerkin isogeometric analysis

Abstract

Isogeometric analysis is a spline-based discretization method to partial differential equations which show the approximation power of a high-order method. The number of degrees of freedom, however, is as small as the number of degrees of freedom of a low-order method. This does not come for free as the original formulation of isogeometric analysis requires a global geometry function. Since this is too restrictive for many kinds of applications, the domain is usually decomposed into patches, where each patch is parameterized with its own geometry function. In simpler cases, the patches can be combined in a conforming way. However, for non-matching discretizations or for varying coefficients, a non-conforming discretization is desired. An symmetric interior penalty discontinuous Galerkin method for isogeometric analysis has been previously introduced. In the present paper, we give error estimates that are explicit in the spline degree. This opens the door towards the construction and the analysis of fast linear solvers, particularly multigrid solvers for non-conforming multipatch isogeometric analysis.

Keywords:

• Isogeometric analysis
• multipatch domains
• symmetric interior penalty
• discontinuous Galerkin

2010 Mathematics Subject Classifications:

• 65N15
• 41A15

1. Introduction

The original design goal of isogeometric analysis (IgA) [1] was to unite the world of computer aided design (CAD) and the world of finite element (FEM) simulation. In IgA, both the computational domain and the solution of the partial differential equation are represented by spline functions, like tensor-product B-splines or non-uniform rational B-splines (NURBS). This follows the design goal since such spline functions are also used in standard CAD systems to represent the geometric objects of interest.

The parameterization of the computational domain using just one tensor-product spline function is possible only in simple cases. A necessary condition for this to be possible is that the computational domain is topologically equivalent to the unit square or the unit cube. This might not be the case for more complicated computational domains. Such domains are typically decomposed into subdomains, in IgA called patches, where each of them is parameterized by its own geometry function. The standard approach is to set up a conforming discretization. For a standard Poisson problem, this means that the overall discretization needs to be continuous. For higher order problems, like the biharmonic problem, even more regularity is required, conforming discretizations in this case are rather hard to construct, cf. Ref. [2] and references therein.

Even for the Poisson problem, a conforming discretization requires the discretizations to agree on the interfaces. This excludes many cases of practical interest, like having different grid sizes or different spline degrees on the patches. Since such cases might be of interest, alternatives to conforming discretizations are of interest. One promising alternative is discontinuous Galerkin approaches, cf. Refs. [3, 4], particularly the symmetric interior penalty discontinuous Galerkin (SIPG) method [5]. The idea of applying this technique to couple patches in IgA has been previously discussed in Refs. [6–8].

Concerning the approximation error, in early IgA literature, only its dependence on the grid size has been studied, cf. Refs. [1, 9]. In recent publications [10–13] also the dependence on the spline degree has been investigated. These error estimates are restricted to the single-patch case. In Ref. [14], the results from Ref. [13] on approximation errors for B-splines of maximum smoothness have been extended to the conforming multipatch case.

For the case of discontinuous Galerkin discretizations, only error estimates in the grid size are known, cf. Ref. [8]. The goal of the present paper is to present an analysis that is explicit both in the grid size h and the spline degree p. We show that the penalty parameter has to grow like $p^2$ for the SIPG method to be well posed. If the solution is sufficiently smooth, the error in the energy norm decays like $h^p$. For the analysis of linear solvers, we need also estimates for less smooth functions. We give a bound that degrades only poly-logarithmically with p, cf. (21(21) $\parallel u-u_h{\parallel }_{Q_h}\lesssim \underset{v_h\in V_h}{inf}\parallel u-v_h{\parallel }_{Q_h^{+}}\lesssim {\sigma }_0^{1/2}(\ln p{)}^{2}h|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)},$(21) ). These error estimates have recently been used to analyze multigrid solvers for discontinuous Galerkin multipatch discretizations, see Ref. [15]. They might also be helpful for the analysis of Finite Element Tearing and Interconnecting (FETI) solvers, cf. Refs. [6, 16], Balancing Domain Decomposition by Constraints (BDDC) solvers, cf. Refs. [17–19] and references therein, Schwarz type solvers, cf. Ref. [20] and references therein, and similar methods.

The remainder of the paper is organized as follows. In Section 2, we introduce the model problem and give a detailed description of its discretization. A discussion of the existence of a unique solution and the discretization and the approximation error is provided in Section 3. The proof of the approximation error estimate is given in Section 4. In Section 5, we provide numerical experiments that depict our estimates. In Section 6, we summarize the findings and discuss possible extensions.

2. The model problem and its discretization

We consider the following Poisson model problem. Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be an open and simply connected Lipschitz domain. For any given source function $f\in L_2\left(\mathrm{\Omega }\right)$, we are interested in the function $u\in H^{1,\circ }\left(\mathrm{\Omega }\right):=H^1\left(\mathrm{\Omega }\right)\cap L_2^{\circ }\left(\mathrm{\Omega }\right)$ solving (1) $(\mathrm{\nabla }u,\mathrm{\nabla }v{)}_{L_2\left(\mathrm{\Omega }\right)}=(f,v{)}_{L_2\left(\mathrm{\Omega }\right)}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in H^{1,\circ }\left(\mathrm{\Omega }\right).$(1) Here and in what follows, for any $r\in \mathbb{N}:=\{1,2,3,\dots \}$, $L_2\left(\mathrm{\Omega }\right)$ and $H^r\left(\mathrm{\Omega }\right)$ are the standard Lebesgue and Sobolev spaces with standard scalar products $(\cdot ,\cdot {)}_{L_2\left(\mathrm{\Omega }\right)}$, $(\cdot ,\cdot {)}_{H^r\left(\mathrm{\Omega }\right)}:=({\mathrm{\nabla }}^r\cdot ,{\mathrm{\nabla }}^r\cdot {)}_{L_2\left(\mathrm{\Omega }\right)}$, norms $\parallel \cdot {\parallel }_{L_2\left(\mathrm{\Omega }\right)}$ and $\parallel \cdot {\parallel }_{H^r\left(\mathrm{\Omega }\right)}$, and seminorms $|\cdot {|}_{H^r\left(\mathrm{\Omega }\right)}$. The Lebesgue space of function with zero mean is given by $L_2^{\circ }\left(\mathrm{\Omega }\right):=\{v\in L_2(\mathrm{\Omega }):(v,1{)}_{L_2\left(\mathrm{\Omega }\right)}=0\}$.

The computational domain Ω is the union of K non-overlapping open patches ${\mathrm{\Omega }}_k$, i.e. (2) $\overline{\mathrm{\Omega }}=\bigcup _{k=1}^K\overline{{\mathrm{\Omega }}_k}\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{\Omega }}_k\cap {\mathrm{\Omega }}_l=set\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{y}k\ne l$(2) holds, where $\overline{T}$ denotes the closure of T. Each patch ${\mathrm{\Omega }}_k$ is represented by a bijective geometry function $G_k:\hat{\mathrm{\Omega }}:=(0,1{)}^2\to {\mathrm{\Omega }}_k:=G_k(\hat{\mathrm{\Omega }})\subset {\mathbb{R}}^2,$which can be continuously extended to the closure of $\hat{\mathrm{\Omega }}$ such that $G_k\left(\overline{\hat{\mathrm{\Omega }}}\right)=\overline{{\mathrm{\Omega }}_k}$. We use the notation $v_k:=v{|}_{{\mathrm{\Omega }}_k}\mathrm{a}\mathrm{n}\mathrm{d}{\hat{v}}_k:=v_k\circ G_k$for any function v on Ω. If $v\in H^1\left(\mathrm{\Omega }\right)$, we can use standard trace theorems to extend $v_k$ to $\overline{{\mathrm{\Omega }}_k}$ and to extend ${\hat{v}}_k$ to $\overline{\hat{\mathrm{\Omega }}}$.

We assume that the mesh induced by the interfaces between the patches does not have any T-junctions, i.e. we assume as follows.

Assumption 2.1

For any two patches ${\mathrm{\Omega }}_k$ and ${\mathrm{\Omega }}_l$ with $k\ne l$, the intersection $\mathrm{\partial }{\mathrm{\Omega }}_k\cap \mathrm{\partial }{\mathrm{\Omega }}_l$ is either (a) empty, (b) a common vertex or (c) a common edge $I_{k,l}=I_{l,k}$ such that (3) ${\hat{I}}_{k,l}:=I_{k,l}\circ G_k\in \hat{\mathcal{I}}:=\left\{\right\{0\}\times (0,1),\{1\}\times (0,1),(0,1)\times \{0\},(0,1)\times \{1\left\}\right\}.$(3)

Note that the pre-images ${\hat{I}}_{k,l}$ and ${\hat{I}}_{l,k}$ do not necessarily agree. We define $\mathcal{N}\left(k\right):=\{l\in \{1,\dots ,K\}:{\mathrm{\Omega }}_k\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{\Omega }}_l\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}\},$$\mathcal{N}:=\left\{\right(k,l):k<l\mathrm{a}\mathrm{n}\mathrm{d}l\in \mathcal{N}(k\left)\right\},$ and ${\mathcal{N}}^{\ast }:=\left\{\right(k,l):k>l\mathrm{a}\mathrm{n}\mathrm{d}l\in \mathcal{N}(k\left)\right\},$ and the parameterization ${\gamma }_{k,l}:(0,1)\to {\hat{I}}_{k,l}$ via (4) ${\gamma }_{k,l}\left(t\right):=\left\{\begin{array}{cc}(t,s)& \mathrm{i}\mathrm{f}{\hat{I}}_{k,l}=(0,1)\times \left\{s\right\},s\in \{0,1\},\\ (s,t)& \mathrm{i}\mathrm{f}{\hat{I}}_{k,l}=\left\{s\right\}\times (0,1),s\in \{0,1\}.\end{array}\right.$(4) We assume that the geometry functions agree on the interface (up to the orientation); this does not require any smoothness of the overall geometry function normal to the interface.

Assumption 2.2

For all $(k,l)\in {\mathcal{N}}^{\ast }$ and $t\in (0,1)$, we have ${\gamma }_{k,l}\left(t\right)=G_k^{-1}\circ G_l\circ {\gamma }_{l,k}\left(t\right)\mathrm{o}\mathrm{r}{\gamma }_{k,l}\left(t\right)=G_k^{-1}\circ G_l\circ {\gamma }_{l,k}(1-t).$

Remark 2.1

For any domain satisfying Assumption 2.1, we can reparameterize each patch such that this condition is satisfied. Assume to have two patches ${\mathrm{\Omega }}_k$ and ${\mathrm{\Omega }}_l$, sharing the patch $I_{k,l}=G_k\left(\right(0,1)\times \{0\left\}\right)=G_l\left(\right(0,1)\times \{0\left\}\right)$. Using ${\tilde{G}}_k(x,y):=G_k(yx+(1-y\left)\rho \right(x),y),$where $\left(\rho \right(t),0):=\left\{\begin{array}{ll}{G}_k^{-1}\circ G_l(t,0)& \mathrm{i}\mathrm{f}{G}_k(s,0)=G_l(s,0)\mathrm{f}\mathrm{o}\mathrm{r}s\in \{0,1\},\\ G_k^{-1}\circ G_l(1-t,0)& \mathrm{i}\mathrm{f}{G}_k(s,0)=G_l(1-s,0)\mathrm{f}\mathrm{o}\mathrm{r}s\in \{0,1\},\end{array}\right.$we obtain a reparameterization of $G_k$, which (a) matches the parameterization of ${\mathrm{\Omega }}_l$ at the interface, (b) is unchanged on the other interfaces, and (c) keeps the patch ${\mathrm{\Omega }}_k$ unchanged. By iteratively applying this approach to all patches, we obtain a discretization satisfying Assumption 2.2.

We assume that the geometry function is sufficiently smooth such that the following assumption holds.

Assumption 2.3

There is a constant $C_G>0$ such that the geometry functions $G_k$ satisfy the estimates (5) $\underset{x\in \hat{\mathrm{\Omega }}}{sup}\parallel {\mathrm{\nabla }}^r{G}_k\left(x\right){\parallel }_{{\ell }_2}\le C_G\mathrm{a}\mathrm{n}\mathrm{d}\underset{x\in \hat{\mathrm{\Omega }}}{sup}\parallel \left({\mathrm{\nabla }}^r{G}_k\right(x){)}^{-1}{\parallel }_{{\ell }_2}\le C_G$(5) for r = 1, 2.

We assume full elliptic regularity.

Assumption 2.4

The solution u of the model problem (1(1) $(\mathrm{\nabla }u,\mathrm{\nabla }v{)}_{L_2\left(\mathrm{\Omega }\right)}=(f,v{)}_{L_2\left(\mathrm{\Omega }\right)}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in H^{1,\circ }\left(\mathrm{\Omega }\right).$(1) ) satisfies $u_k\in H^2\left({\mathrm{\Omega }}_k\right)$for all $k=1,\dots ,K$.

For domains Ω with a sufficiently smooth boundary, cf. Ref. [21], and for convex polygonal domains Ω, cf. Refs. [22, 23], we have $u\in H^2\left(\mathrm{\Omega }\right)$ and thus also Assumption 2.4. In case of varying diffusion conditions (which are uniform on each patch), we might have $u\notin H^2\left(\mathrm{\Omega }\right)$, but Assumption 2.4 might still be satisfied, cf. Refs. [24, 25] and others. The theory of this paper can be extended to cases where we only know $u_k\in H^{3/2+ϵ}\left({\mathrm{\Omega }}_k\right)$ for some $ϵ>0$. For simplicity, we restrict ourselves to the case of full elliptic regularity, i.e. Assumption 2.4.

Having a representation of the domain, we introduce the isogeometric function space. Following Refs. [7, 8], we use a conforming isogeometric discretization for each patch and couple the contributions for the patches using a SIPG method, cf. Ref. [5], as follows.

For the univariate case, the space of spline functions of degree $p\in \mathbb{N}$ over a grid (vector of breakpoints) $Z:=({\zeta }_0,{\zeta }_1,\dots ,{\zeta }_{n-1},{\zeta }_n)$ with $\zeta =0$ and ${\zeta }_n=1$ and size $h:=\underset{i=0,\dots ,n-1}{max}|{\zeta }_{i+1}-{\zeta }_i|$ is given by $S_{p,\mathrm{\Xi }}(0,1):=\left\{v\in C^{p-1}(0,1):v{|}_{({\zeta }_i,{\zeta }_{i+1}]}\in {\mathbb{P}}^p\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i=1,\dots ,n-1\right\},$where ${\mathbb{P}}^p$ is the space of polynomials of degree p.

On the parameter domain $\hat{\mathrm{\Omega }}:=(0,1{)}^2$, we introduce tensor-product B-spline functions ${\hat{V}}_k:=S_{p_k,Z_{k,1}}(0,1)\otimes S_{p_k,Z_{k,2}}(0,1).$The multipatch function space $V_h$ is given by (6) $V_h:=\{u_h\in L_2^{\circ }(\mathrm{\Omega }):u_h\circ G_k\in {\hat{V}}_k\mathrm{f}\mathrm{o}\mathrm{r}k=1,\dots ,K,\}.$(6) Note that the grid sizes $h_k$ and the spline degrees $p_k$ can be different for each of the patches. We define $p:=\underset{k\in \{1,\dots ,K\}}{max}{p}_k,p_{min}:=\underset{k\in \{1,\dots ,K\}}{min}{p}_k,h:=\underset{\delta \in \{1,2\}}{max}\underset{k\in \{1,\dots ,K\}}{max}{h}_{k,\delta }$to be the largest spline degree, the smallest spline degree, and the grid size. We assume (7) $p_{min}\ge 2\mathrm{a}\mathrm{n}\mathrm{d}\underset{\delta \in \{1,2\}}{min}\underset{k\in \{1,\dots ,K\}}{min}{h}_{k,\delta ,min}\ge C_{h}h$(7) for some constant $C_h>0$, where $h_{k,\delta ,min}$ refers to the smallest knot span.

Following the assumption that $u_h$ is a patchwise function, we define for each $r\in \mathbb{N}$, a broken Sobolev space ${\mathcal{H}}^r\left(\mathrm{\Omega }\right):=\{v\in L_2(\mathrm{\Omega }):v_k\in H^r(\mathrm{\Omega }\left)\right\}$with associated norms and scalar products $\parallel v{\parallel }_{{\mathcal{H}}^r\left(\mathrm{\Omega }\right)}:=(v,v{)}_{{\mathcal{H}}^r\left(\mathrm{\Omega }\right)}^{1/2}\mathrm{a}\mathrm{n}\mathrm{d}(u,v{)}_{{\mathcal{H}}^r\left(\mathrm{\Omega }\right)}:=\sum _{k=1}^K(u,v{)}_{H^r\left({\mathrm{\Omega }}_k\right)}.$For each patch, we define on its boundary $\mathrm{\partial }{\mathrm{\Omega }}_k$, the outer normal vector ${\mathbf{\text{n}}}_k$. On each interface $I_{k,l}$, we define the jump operator $\text{〚}\cdot \text{〛}$ by $\text{〚}v\text{〛}:=v_k-v_l\mathrm{o}\mathrm{n}{I}_{k,l}=I_{l,k}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}(k,l)\in \mathcal{N}$and the average operator $\{\cdot \}$ by $\left\{v\right\}:=\frac{1}{2}(v_k+v_l)\mathrm{o}\mathrm{n}{I}_{k,l}=I_{l,k}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}(k,l)\in \mathcal{N}.$The discretization of the variational problem using the SIPG method reads as follows. Find $u_h\in V_h$ such that (8) $(u_h,v_h{)}_{A_h}=(f,v_h{)}_{L_2\left(\mathrm{\Omega }\right)}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}{v}_h\in V_h,$(8) where $\begin{array}{rl}(u,v{)}_{A_h}& :=(u,v{)}_{{\mathcal{H}}^1\left(\mathrm{\Omega }\right)}-(u,v{)}_{B_h}-(v,u{)}_{B_h}+(u,v{)}_{C_h},\\ (u,v{)}_{B_h}& :=\sum _{(k,l)\in \mathcal{N}}(\text{〚}u\text{〛},\{\mathrm{\nabla }v\}\cdot {\mathbf{\text{n}}}_k{)}_{L_2\left(I_{k,l}\right)},\\ (u,v{)}_{C_h}& :=\frac{\sigma }{h}\sum _{(k,l)\in \mathcal{N}}(\text{〚}u\text{〛},\text{〚}v\text{〛}{)}_{L_2\left(I_{k,l}\right)}\end{array}$for all $u,v\in {\mathcal{H}}^{2,\circ }\left(\mathrm{\Omega }\right)$, where the penalty parameter (9) $\sigma ={\sigma }_0{p}^2>0$(9) is chosen sufficiently large.

Using a basis for the space $V_h$, we obtain a standard matrix-vector problem: find ${\underset{\_}{u}}_h\in {\mathbb{R}}^N$ such that (10) $A_h{\underset{\_}{u}}_h={\underset{\_}{f}}_h.$(10) Here and in what follows, ${\underset{\_}{u}}_h=[u_i{]}_{i=1}^N$ is the coefficient vector representing $u_h$ with respect to the chosen basis, i.e. $u_h=\sum _{i=1}^N{u}_i{\phi }_i$, and ${\underset{\_}{f}}_h=\left[\right(f,{\phi }_i{)}_{L_2\left(\mathrm{\Omega }\right)}{]}_{i=1}^N$ is the coefficient vector obtained by testing the right-hand-side functional with the basis functions.

As the dependence on the geometry function is not in the focus of this paper, unspecified constants might depend on $C_G$, $C_I$, and $C_h$. Before we proceed, we introduce a convenient notation.

Definition 2.5

Any generic constant c>0 used within this paper is understood to be independent of the grid size h, the spline degree p, and the number of patches K, but it might depend on the constants $C_G$, $C_I$, and $C_h$.

We use the notation $a\lesssim b$ if there is a generic constant c>0 such that $a\le cb$ and the notation $a\eqsim b$ if $a\lesssim b$ and $b\lesssim a$.

For symmetric positive definite matrices A and B, we write $A\le B\mathrm{i}\mathrm{f}{\underset{\_}{v}}_h^{\mathrm{\top }}A{\underset{\_}{v}}_h\le {\underset{\_}{v}}_h^{\mathrm{\top }}B{\underset{\_}{v}}_h\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}{\underset{\_}{v}}_h.$The notations $A\lesssim B$ and $A\eqsim B$ are defined analogously.

3. A discretization error estimate

In Ref. [7], it has been shown that the bilinear form $(\cdot ,\cdot {)}_{A_h}$ is coercive and bounded in the dG-norm. For our further analysis, it is vital to know these conditions to be satisfied with constants that are independent of the spline degree p. Thus, we define the dG-norm via (11) $\parallel u{\parallel }_{Q_h}^2:=(u,u{)}_{Q_h}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}(u,v{)}_{Q_h}:=(u,v{)}_{{\mathcal{H}}^1\left(\mathrm{\Omega }\right)}+(u,v{)}_{C_h}$(11) for all $u,v\in {\mathcal{H}}^{2,\circ }\left(\mathrm{\Omega }\right)$. Note that we define the norm differently to Ref. [7], where the dG-norm was independent of p.

Before we proceed, we give some estimates on the geometry functions.

Lemma 3.1

We have (12) $\parallel v\circ G_k^{-1}{\parallel }_{H^r\left({\mathrm{\Omega }}_k\right)}\eqsim \parallel v{\parallel }_{H^r\left(\hat{\mathrm{\Omega }}\right)}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in H^r\left(\hat{\mathrm{\Omega }}\right),$(12) where r = 0, 1, 2. For ease of notation, here and in what follows, we define $H^0:=L_2$. If (5(5) $\underset{x\in \hat{\mathrm{\Omega }}}{sup}\parallel {\mathrm{\nabla }}^r{G}_k\left(x\right){\parallel }_{{\ell }_2}\le C_G\mathrm{a}\mathrm{n}\mathrm{d}\underset{x\in \hat{\mathrm{\Omega }}}{sup}\parallel \left({\mathrm{\nabla }}^r{G}_k\right(x){)}^{-1}{\parallel }_{{\ell }_2}\le C_G$(5) ) holds for $r=1,2,\dots ,s$ with some s>r, then (12(12) $\parallel v\circ G_k^{-1}{\parallel }_{H^r\left({\mathrm{\Omega }}_k\right)}\eqsim \parallel v{\parallel }_{H^r\left(\hat{\mathrm{\Omega }}\right)}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in H^r\left(\hat{\mathrm{\Omega }}\right),$(12) ) also holds for those choices of r. Moreover, we have $\parallel v\circ G_k^{-1}{\parallel }_{L_2\left(I_{k,l}\right)}\eqsim \parallel v{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in H^1\left(\hat{\mathrm{\Omega }}\right).$

Proof.

The statements follow directly from the chain rule for differentiation, the substitution rule for integration, and Assumption 2.3.

Lemma 3.2

We have $\parallel (\mathrm{\nabla }v\circ G_k^{-1})\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left(I_{k,l}\right)}\lesssim \parallel \mathrm{\nabla }v{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}$ for all $v\in H^2\left(\hat{\mathrm{\Omega }}\right)$.

Proof.

We have $\parallel (\mathrm{\nabla }v\circ G_k^{-1})\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left(I_{k,l}\right)}\le \parallel \mathrm{\nabla }v\circ G_k^{-1}{\parallel }_{L_2\left(I_{k,l}\right)}\parallel {\mathbf{\text{n}}}_k{\parallel }_{L_{\mathrm{\infty }}\left(I_{k,l}\right)},$ where certainly $\parallel {\mathbf{\text{n}}}_k{\parallel }_{L_{\mathrm{\infty }}\left(I_{k,l}\right)}=1$ because the length of ${\mathbf{\text{n}}}_k$ is always 1. The estimate $\parallel \mathrm{\nabla }v\circ G_k^{-1}{\parallel }_{L_2\left(I_{k,l}\right)}\lesssim \parallel \mathrm{\nabla }v{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}$ follows directly from the chain rule for differentiation, the substitution rule for integration, and Assumption 2.3.

For σ sufficiently large, the symmetric bilinear form $(\cdot ,\cdot {)}_{A_h}$ is coercive and bounded, i.e. a scalar product.

Theorem 3.3

Coercivity and boundedness

There is some ${\sigma }_0>0$ that only depends on $C_G$, $C_I$, and $C_h$ such that $(u_h,u_h{)}_{A_h}⪆\parallel u_h{\parallel }_{Q_h}^2\mathrm{a}\mathrm{n}\mathrm{d}(u_h,v_h{)}_{A_h}\lesssim \parallel u_h{\parallel }_{Q_h}\parallel v_h{\parallel }_{Q_h}$hold for all $u_h,v_h\in V_h$ and all $\sigma \ge p^2{\sigma }_0$.

Proof.

Note that $(u_h,v_h{)}_{A_h}=(u_h,v_h{)}_{Q_h}-(u_h,v_h{)}_{B_h}-(v_h,u_h{)}_{B_h}.$ Using Lemma 3.2, [14, Lemma 4.4], [26, Corollary 3.94], and Lemma 3.1, we obtain (13) $\begin{array}{rl}\parallel \mathrm{\nabla }{v}_h\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left(I_{k,l}\right)}^2& \lesssim \parallel \mathrm{\nabla }(v_h\circ G_k)\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}^2\lesssim \parallel v_h\circ G_k{\parallel }_{H^2\left(\hat{\mathrm{\Omega }}\right)}\parallel v_h\circ G_k{\parallel }_{H^1\left(\hat{\mathrm{\Omega }}\right)}\\ & \lesssim \frac{p^2}{h}\parallel v_h\circ G_k{\parallel }_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2\lesssim \frac{p^2}{h}\parallel v_h{\parallel }_{H^1\left({\mathrm{\Omega }}_k\right)}^2\end{array}$(13) for all $v_h\in V_h$, $k=1,\dots ,K$, and $l\in \mathcal{N}\left(k\right)$. As $V_h\subset L_2^{\circ }\left(\mathrm{\Omega }\right)$, the Poincaré inequality (see, e.g. Ref. [26, Theorem A.25]) yields also $\parallel \mathrm{\nabla }{v}_h\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left(I_{k,l}\right)}^2\lesssim \frac{p^2}{h}|v_h{|}_{H^1\left({\mathrm{\Omega }}_k\right)}^2.$The Cauchy–Schwarz inequality, the triangle inequality, (13(13) $\begin{array}{rl}\parallel \mathrm{\nabla }{v}_h\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left(I_{k,l}\right)}^2& \lesssim \parallel \mathrm{\nabla }(v_h\circ G_k)\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}^2\lesssim \parallel v_h\circ G_k{\parallel }_{H^2\left(\hat{\mathrm{\Omega }}\right)}\parallel v_h\circ G_k{\parallel }_{H^1\left(\hat{\mathrm{\Omega }}\right)}\\ & \lesssim \frac{p^2}{h}\parallel v_h\circ G_k{\parallel }_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2\lesssim \frac{p^2}{h}\parallel v_h{\parallel }_{H^1\left({\mathrm{\Omega }}_k\right)}^2\end{array}$(13) ), and $|\mathcal{N}\left(k\right)|\le 4$ yield (14) $\begin{array}{rl}|(u_h,v_h{)}_{B_h}|& \le {\left(\sum _{(k,l)\in \mathcal{N}}\parallel \text{〚}{u}_h\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}{\left(\sum _{(k,l)\in \mathcal{N}}\parallel \left\{\mathrm{\nabla }{v}_h\right\}\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}\\ & \lesssim {\left(\sum _{(k,l)\in \mathcal{N}}\parallel \text{〚}{u}_h\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}{\left(\sum _{k=1}^K\sum _{j\in \mathcal{N}\left(k\right)}\parallel \mathrm{\nabla }{v}_h\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}\\ & \lesssim {\left(\frac{p^2}{h}\right)}^{1/2}{\left(\sum _{(k,l)\in \mathcal{N}}\parallel \text{〚}{u}_h\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}{\left(\sum _{k=1}^K|v_h{|}_{H^1\left({\mathrm{\Omega }}_k\right)}^2\right)}^{1/2}\\ & \le p{\sigma }^{-1/2}\parallel u_h{\parallel }_{Q_h}\parallel v_h{\parallel }_{Q_h}\end{array}$(14) for all $u_h,v_h\in V_h$. Let $c_0\eqsim 1$ be the hidden constant, i.e. such that (15) $|(u_h,v_h{)}_{B_h}|\le c_{0}p{\sigma }^{-1/2}\parallel u_h{\parallel }_{Q_h}\parallel v_h{\parallel }_{Q_h}.$(15) For $\sigma \ge 16c_0{p}^2$, we obtain $(u_h,u_h{)}_{A_h}=\parallel u_h{\parallel }_{Q_h}^2-2(u_h,u_h{)}_{B_h}\ge \frac{1}{2}\parallel u_h{\parallel }_{Q_h}^2,$i.e. coercivity. Using (14(14) $\begin{array}{rl}|(u_h,v_h{)}_{B_h}|& \le {\left(\sum _{(k,l)\in \mathcal{N}}\parallel \text{〚}{u}_h\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}{\left(\sum _{(k,l)\in \mathcal{N}}\parallel \left\{\mathrm{\nabla }{v}_h\right\}\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}\\ & \lesssim {\left(\sum _{(k,l)\in \mathcal{N}}\parallel \text{〚}{u}_h\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}{\left(\sum _{k=1}^K\sum _{j\in \mathcal{N}\left(k\right)}\parallel \mathrm{\nabla }{v}_h\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}\\ & \lesssim {\left(\frac{p^2}{h}\right)}^{1/2}{\left(\sum _{(k,l)\in \mathcal{N}}\parallel \text{〚}{u}_h\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}{\left(\sum _{k=1}^K|v_h{|}_{H^1\left({\mathrm{\Omega }}_k\right)}^2\right)}^{1/2}\\ & \le p{\sigma }^{-1/2}\parallel u_h{\parallel }_{Q_h}\parallel v_h{\parallel }_{Q_h}\end{array}$(14) ) and the Cauchy–Schwarz inequality, we obtain further $(u_h,v_h{)}_{A_h}=(u_h,v_h{)}_{Q_h}-(u_h,v_h{)}_{B_h}-(v_h,u_h{)}_{B_h}\le \frac{3}{2}\parallel u_h{\parallel }_{Q_h}\parallel v_h{\parallel }_{Q_h},$i.e. boundedness.

As we have boundedness and coercivity (Theorem 3.3), the Lax Milgram theorem (see, e.g. Ref. [26, Theorem 1.24]) yields states existence and uniqueness of a solution, i.e. the following statement.

Theorem 3.4

Existence and uniqueness

If σ is chosen as in Theorem 3.3, the problem (8(8) $(u_h,v_h{)}_{A_h}=(f,v_h{)}_{L_2\left(\mathrm{\Omega }\right)}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}{v}_h\in V_h,$(8) ) has exactly one solution $u_h\in V_h$.

The following theorem shows that the solution of the original problem also satisfies the discretized bilinear form.

Theorem 3.5

Consistency

The solution $u\in H^{1,\circ }\left(\mathrm{\Omega }\right)\cap {\mathcal{H}}^2\left(\mathrm{\Omega }\right)$ of the original problem (1(1) $(\mathrm{\nabla }u,\mathrm{\nabla }v{)}_{L_2\left(\mathrm{\Omega }\right)}=(f,v{)}_{L_2\left(\mathrm{\Omega }\right)}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in H^{1,\circ }\left(\mathrm{\Omega }\right).$(1) ) satisfies $(u,v_h{)}_{A_h}=(f,v_h{)}_{L_2\left(\mathrm{\Omega }\right)}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}{v}_h\in V_h.$

For a proof, see, e.g. Ref. [4, Proposition 2.9]; the proof requires elliptic regularity (cf. Assumption 2.4).

If boundedness of the bilinear form $(\cdot ,\cdot {)}_{A_h}$ was also satisfied for $u\in {\mathcal{H}}^{2,\circ }\left(\mathrm{\Omega }\right)$, Ceá's Lemma (see, e.g. Ref. [26, Theorem 2.19.iii]) would allow to bound the discretization error. However, the bilinear form is not bounded in the norm $\parallel \cdot {\parallel }_{Q_h}$, but only in the stronger norm $\parallel \cdot {\parallel }_{Q_h^{+}}$, given by (16) $\parallel u{\parallel }_{Q_h^{+}}^2:=\parallel u{\parallel }_{Q_h}^2+\frac{h^2}{{\sigma }^2}|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2.$(16)

Theorem 3.6

If σ is chosen as in Theorem 3.3, $(u,v_h{)}_{A_h}\lesssim \parallel u{\parallel }_{Q_h^{+}}\parallel v_h{\parallel }_{Q_h}$holds for all $u\in {\mathcal{H}}^{2,\circ }\left(\mathrm{\Omega }\right)$ and all $v_h\in V_h$.

Proof.

Let $u\in {\mathcal{H}}^{2,\circ }\left(\mathrm{\Omega }\right)$ and $v_h\in V_h$ be arbitrarily but fixed. Note that the arguments from (14(14) $\begin{array}{rl}|(u_h,v_h{)}_{B_h}|& \le {\left(\sum _{(k,l)\in \mathcal{N}}\parallel \text{〚}{u}_h\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}{\left(\sum _{(k,l)\in \mathcal{N}}\parallel \left\{\mathrm{\nabla }{v}_h\right\}\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}\\ & \lesssim {\left(\sum _{(k,l)\in \mathcal{N}}\parallel \text{〚}{u}_h\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}{\left(\sum _{k=1}^K\sum _{j\in \mathcal{N}\left(k\right)}\parallel \mathrm{\nabla }{v}_h\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}\\ & \lesssim {\left(\frac{p^2}{h}\right)}^{1/2}{\left(\sum _{(k,l)\in \mathcal{N}}\parallel \text{〚}{u}_h\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}{\left(\sum _{k=1}^K|v_h{|}_{H^1\left({\mathrm{\Omega }}_k\right)}^2\right)}^{1/2}\\ & \le p{\sigma }^{-1/2}\parallel u_h{\parallel }_{Q_h}\parallel v_h{\parallel }_{Q_h}\end{array}$(14) ) also hold if the first parameter of the bilinear form $(\cdot ,\cdot {)}_{B_h}$ is not in $V_h$. So, we obtain $|(u,v_h{)}_{B_h}|\le \frac{1}{4}\parallel u{\parallel }_{Q_h}\parallel v_h{\parallel }_{Q_h}.$Using Lemma 3.2, [14, Lemma 4.4], Lemma 3.1, and the Poincaré inequality, we obtain (17) $\begin{array}{rl}\parallel \mathrm{\nabla }v\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left(I_{k,l}\right)}^2& \lesssim \parallel \mathrm{\nabla }(v\circ G_k)\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}^2\lesssim \parallel v\circ G_k{\parallel }_{H^2\left(\hat{\mathrm{\Omega }}\right)}\parallel v\circ G_k{\parallel }_{H^1\left(\hat{\mathrm{\Omega }}\right)}\\ & \lesssim \parallel v{\parallel }_{H^2\left({\mathrm{\Omega }}_k\right)}\parallel v{\parallel }_{H^1\left({\mathrm{\Omega }}_k\right)}\le \frac{1}{\text{β}}\parallel v{\parallel }_{H^2\left({\mathrm{\Omega }}_k\right)}^2+\text{β}\parallel v{\parallel }_{H^1\left({\mathrm{\Omega }}_k\right)}^2\le \frac{1}{\text{β}}|v{|}_{H^2\left({\mathrm{\Omega }}_k\right)}^2+\text{β}|v{|}_{H^1\left({\mathrm{\Omega }}_k\right)}^2\end{array}$(17) for all $v\in H^2\left({\mathrm{\Omega }}_k\right)$, all $k=1,\dots ,K$, all $l\in \mathcal{N}\left(k\right)$, and all $\text{β}>1$. Using this estimate, and $|\mathcal{N}\left(k\right)|\le 4$, we obtain for $\text{β}:=h^{-2}\sigma$ $\begin{array}{rl}|(v_h,u{)}_{B_h}|& \le {\left(\frac{\sigma }{h}\sum _{(k,l)\in \mathcal{N}}\parallel \text{〚}{v}_h\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}{\left(\frac{h}{\sigma }\sum _{k=1}^K\sum _{l\in \mathcal{N}\left(k\right)}\parallel \mathrm{\nabla }u\cdot {\mathbf{\text{n}}}_k{\parallel }_{L_2\left(I_{k,l}\right)}^2\right)}^{1/2}\\ & \le \parallel v_h{\parallel }_{Q_h}{\left(\sum _{k=1}^K|u{|}_{H^1\left({\mathrm{\Omega }}_k\right)}^2+\frac{h^2}{{\sigma }^2}\sum _{k=1}^K|u{|}_{H^2\left({\mathrm{\Omega }}_k\right)}^2\right)}^{1/2}\lesssim \parallel v_h{\parallel }_{Q_h}\parallel u{\parallel }_{Q_h^{+}}.\end{array}$Using these estimates, we obtain $(u,v_h{)}_{A_h}=(u,v_h{)}_{Q_h}-(u,v_h{)}_{B_h}-(v_h,u{)}_{B_h}\lesssim \parallel u{\parallel }_{Q_h^{+}}\parallel v_h{\parallel }_{Q_h},$which finishes the proof.

Using consistency (Theorem 3.5), coercivity, and boundedness (Theorems 3.3 and 3.6), we can bound the discretization error using the approximation error.

Theorem 3.7

Discretization error estimate

Provided the assumptions of Theorems 3.3 and 3.5, the estimate $\parallel u-u_h{\parallel }_{Q_h}\lesssim \underset{v_h\in V_h}{inf}\parallel u-v_h{\parallel }_{Q_h^{+}}$holds, where u is the solution of the original problem (1(1) $(\mathrm{\nabla }u,\mathrm{\nabla }v{)}_{L_2\left(\mathrm{\Omega }\right)}=(f,v{)}_{L_2\left(\mathrm{\Omega }\right)}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in H^{1,\circ }\left(\mathrm{\Omega }\right).$(1) ) and $u_h$ is the solution of the discrete problem (8(8) $(u_h,v_h{)}_{A_h}=(f,v_h{)}_{L_2\left(\mathrm{\Omega }\right)}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}{v}_h\in V_h,$(8) ).

Proof.

For any $v_h\in V_h$, the triangle inequality yields (18) $\parallel u-u_h{\parallel }_{Q_h}\le \parallel u-v_h{\parallel }_{Q_h}+\parallel v_h-u_h{\parallel }_{Q_h}.$(18) Theorem 3.5 and Galerkin orthogonality yield $(u-u_h,w_h{)}_{A_h}=0$ for all $w_h\in V_h$. So, we obtain using Theorems 3.3 and 3.6 that $\parallel v_h-u_h{\parallel }_{Q_h}^2\lesssim (v_h-u_h,v_h-u_h{)}_{A_h}=(v_h-u,v_h-u_h{)}_{A_h}\lesssim \parallel v_h-u{\parallel }_{Q_h^{+}}\parallel v_h-u_h{\parallel }_{Q_h},$which shows $\parallel v_h-u_h{\parallel }_{Q_h}\lesssim \parallel u-v_h{\parallel }_{Q_h^{+}}$. Together with (18(18) $\parallel u-u_h{\parallel }_{Q_h}\le \parallel u-v_h{\parallel }_{Q_h}+\parallel v_h-u_h{\parallel }_{Q_h}.$(18) ), this shows $\parallel u-u_h{\parallel }_{Q_h}\lesssim \parallel u-v_h{\parallel }_{Q_h^{+}}$. Since this holds for all $v_h\in V_h$, this finishes the proof.

Theorem 3.8

Approximation error estimate

Let $q\in \{1,\dots ,p_{min}\}$. Provided that σ is as in Theorem 3.3 and that $\parallel u_k{\parallel }_{H^{q+1}\left({\mathrm{\Omega }}_k\right)}\eqsim \parallel {\hat{u}}_k{\parallel }_{H^{q+1}\left(\hat{\mathrm{\Omega }}\right)}$ for $k=1,\dots ,K$ (cf. Lemma 3.1), then (19) $\underset{v_h\in V_h}{inf}\parallel u-v_h{\parallel }_{Q_h^{+}}\lesssim {\sigma }^{1/2}{\pi }^{-q}{h}^q\parallel u{\parallel }_{{\mathcal{H}}^{q+1}\left(\mathrm{\Omega }\right)}$(19) holds for all $u\in H^{1,\circ }\left(\mathrm{\Omega }\right)\cap {\mathcal{H}}^{q+1}\left(\mathrm{\Omega }\right)$.

A proof of this theorem is given at the end of the next section.

Assuming $p\eqsim p_{min}$, then we have for the case $q=p_{min}$ that $\parallel u-u_h{\parallel }_{Q_h}\lesssim \underset{v_h\in V_h}{inf}\parallel u-v_h{\parallel }_{Q_h^{+}}\lesssim {\sigma }_0^{1/2}\underset{{\displaystyle <1}}{\underbrace{q^2{\pi }^{-q}}}{h}^q\parallel u{\parallel }_{{\mathcal{H}}^{q+1}\left(\mathrm{\Omega }\right)}.$For the analysis of linear solvers, like multigrid solvers [15], we also need low-order approximation error estimates. In the convergence proofs, we usually have to estimate errors of the iterative scheme. Even if we know that the true solution satisfies certain regularity assumptions, this does not extend to the errors. For them, we can only rely on the regularity statements arising from the domain, which means that $H^2$-regularity is usually the best we can hope for. For this case, we obtain $\underset{v_h\in V_h}{inf}\parallel u-v_h{\parallel }_{Q_h^{+}}\lesssim {\sigma }_0^{1/2}{p}^{2}h\parallel u{\parallel }_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}.$This means that we obtain a quadratic increase in the spline degree p. Using a refined analysis, we obtain as follows.

Theorem 3.9

Low-order approximation error estimate

Provided that σ is as in Theorem 3.3, then the estimate (20) $\underset{v_h\in V_h}{inf}\parallel u-v_h{\parallel }_{Q_h^{+}}\lesssim (\ln \sigma {)}^2{\sigma }^{1/(2p_{min}-1)}h|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}$(20) holds for all $u\in H^{1,\circ }\left(\mathrm{\Omega }\right)\cap {\mathcal{H}}^2\left(\mathrm{\Omega }\right)$.

The proof is given at the end of the next section. Assuming again $p\eqsim p_{min}\ge 2$, we obtain (21) $\parallel u-u_h{\parallel }_{Q_h}\lesssim \underset{v_h\in V_h}{inf}\parallel u-v_h{\parallel }_{Q_h^{+}}\lesssim {\sigma }_0^{1/2}(\ln p{)}^{2}h|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)},$(21) i.e. an only poly-logarithmic increase in the spline degree p.

4. Proof of the approximation error estimates

Before we can give the proof, we give some auxiliary results. This section is organized as follows. In Section 4.1, we give patchwise projectors and estimates for them. We introduce a mollifying operator and give estimates for that operator in Section 4.2. Finally, in Section 4.3, we give the proof for the approximation error estimate.

4.1. Patchwise projectors

As first step, we recall the projection operators from Ref. [14, Sections 3.1 and 3.2]. Let ${\mathrm{\Pi }}_{p,Z}$ be the $H_D^1(0,1)$-orthogonal projection into $S_{p,Z}(0,1)$, where $(u,v{)}_{H_D^1(0,1)}=(u^{\prime },v^{\prime }{)}_{L_2(0,1)}+u\left(0\right)v\left(0\right).$In what follows, we also write $S_{p,h}(0,1)$ and ${\mathrm{\Pi }}_{p,h}$ if we refer to a uniform grid of size h. Takacs [14, Lemma 3.1] states that $\left({\mathrm{\Pi }}_{p,Z}u\right)\left(0\right)=u\left(0\right)$ and $\left({\mathrm{\Pi }}_{p,Z}u\right)\left(1\right)=u\left(1\right)$. Using $0=(u-{\mathrm{\Pi }}_{p,Z}u,x^2{)}_{H_D^1(0,1)}=2((u-{\mathrm{\Pi }}_{p,Z}u{)}^{\prime },x{)}_{L_2(0,1)}=-2(u-{\mathrm{\Pi }}_{p,Z}u,1{)}_{L_2(0,1)}+u\left(1\right)-\left({\mathrm{\Pi }}_{p,Z}u\right)\left(1\right)$, we obtain for $p\ge 2$ and $u\in H^1(0,1)$ that (22) $(u-{\mathrm{\Pi }}_{p,Z}u,1{)}_{L_2(0,1)}=0.$(22) The next step is to consider the multivariate case, more precisely the parameter domain $\hat{\mathrm{\Omega }}=(0,1{)}^2$. Let ${\mathrm{\Pi }}_k^x:H^2\left(\hat{\mathrm{\Omega }}\right)\to H^2\left(\hat{\mathrm{\Omega }}\right)$ and ${\mathrm{\Pi }}_k^y:H^2\left(\hat{\mathrm{\Omega }}\right)\to H^2\left(\hat{\mathrm{\Omega }}\right)$ be given by $\left({\mathrm{\Pi }}_k^{x}u\right)(x,y)=\left({\mathrm{\Pi }}_{p_k,Z_{k,1}}u\right(\cdot ,y\left)\right)\left(x\right)\mathrm{a}\mathrm{n}\mathrm{d}\left({\mathrm{\Pi }}_k^{y}u\right)(x,y)=\left({\mathrm{\Pi }}_{p_k,Z_{k,2}}u\right(x,\cdot \left)\right)\left(y\right)$and let ${\hat{\mathrm{\Pi }}}_k:H^2\left(\hat{\mathrm{\Omega }}\right)\to S_{p_k,Z_{k,1}}\left(\hat{\mathrm{\Omega }}\right)$ be such that (23) ${\hat{\mathrm{\Pi }}}_k={\mathrm{\Pi }}_k^x{\mathrm{\Pi }}_k^y.$(23) For the physical domain, define $\mathrm{\Pi }:H^{1,\circ }\left(\mathrm{\Omega }\right)\cap {\mathcal{H}}^2\left(\mathrm{\Omega }\right)\to V_h$ to be such that $\left(\mathrm{\Pi }v\right){|}_{{\mathrm{\Omega }}_k}=\left({\hat{\mathrm{\Pi }}}_k\right(v\circ G_k\left)\right)\circ G_k^{-1}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in {\mathcal{H}}^2\left(\mathrm{\Omega }\right)\mathrm{a}\mathrm{n}\mathrm{d}k=1,\dots ,K.$Observe that we obtain using (22(22) $(u-{\mathrm{\Pi }}_{p,Z}u,1{)}_{L_2(0,1)}=0.$(22) ) that (24) $(u-{\hat{\mathrm{\Pi }}}_{k}u,1{)}_{L_2\left(\hat{\mathrm{\Omega }}\right)}=0,{\hat{\mathrm{\Pi }}}_{k}c=c,\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\Pi }c=c$(24) for all $c\in \mathbb{R}$.

The projectors ${\hat{\mathrm{\Pi }}}_k$ satisfy robust error estimates and are almost stable in $H^2$.

Lemma 4.1

Let Z be a grid of size h, $p\in \{2,3,\dots \}$, and $r\in \{1,2,\dots ,p\}$. Then, $\begin{array}{rl}\parallel (I-{\mathrm{\Pi }}_{p,Z})u{\parallel }_{L_2(0,1)}& \le {\pi }^{-r-1}{h}^{r+1}|u{|}_{H^{r+1}(0,1)}and\\ |(I-{\mathrm{\Pi }}_{p,Z})u{|}_{H^1(0,1)}& \le {\pi }^{-r}{h}^r|u{|}_{H^{r+1}(0,1)}\end{array}$hold for all $u\in H^{r+1}(0,1)$.

Proof.

The identity (22(22) $(u-{\mathrm{\Pi }}_{p,Z}u,1{)}_{L_2(0,1)}=0.$(22) ) implies that the projector ${\mathrm{\Pi }}_{p,Z}$ coincides with the projector $Q_p^1$ from Ref. [12, Equations (3.8) and (3.9)]. Thus, the desired result follows from Ref. [12, Theorem 3.1].

Lemma 4.2

Let Z be a quasi-uniform grid of size h, $p\in \{2,3,\dots \}$, and $r\in \{1,2,\dots ,p\}$. Then, $|(I-{\mathrm{\Pi }}_{p,Z})u{|}_{H^2(0,1)}\lesssim p^2{\pi }^{-r+1}{h}^{r-1}|u{|}_{H^{r+1}(0,1)}$holds for all $u\in H^{r+1}(0,1)$.

Proof.

The proof is analogous to the proof of Ref. [27, Theorem 4]. Let $Q_p^2$ be the $H^2$-orthogonal projector into $S_{p,Z}(0,1)$ as introduced in Ref. [12]. Using the triangle inequality and a standard inverse estimate [26, Corollary 3.94], we obtain $\begin{array}{rl}|(I-{\mathrm{\Pi }}_{p,Z})u{|}_{H^2(0,1)}& \le |(I-Q_p^2)u{|}_{H^2(0,1)}+|(Q_p^2-{\mathrm{\Pi }}_{p,Z})u{|}_{H^2(0,1)}\\ & \lesssim |(I-Q_p^2)u{|}_{H^2(0,1)}+h^{-1}{p}^2|(Q_p^2-{\mathrm{\Pi }}_{p,Z})u{|}_{H^1(0,1)}\\ & \le |(I-Q_p^2)u{|}_{H^2(0,1)}+h^{-1}{p}^2|(I-Q_p^2)u{|}_{H^1(0,1)}+h^{-1}{p}^2|(I-{\mathrm{\Pi }}_{p,Z})u{|}_{H^1(0,1)}.\end{array}$Thus, the desired result follows from Ref. [12, Theorem 3.1].

Lemma 4.3

Let $r\in \{1,2,\dots ,p_{min}\}$. The estimates $\begin{array}{rl}\parallel (I-{\hat{\mathrm{\Pi }}}_k)u{\parallel }_{L_2\left(\hat{\mathrm{\Omega }}\right)}& \lesssim {\pi }^{-r-1}{h}_k^{r+1}|u{|}_{H^{r+1}\left(\hat{\mathrm{\Omega }}\right)},\\ |(I-{\hat{\mathrm{\Pi }}}_k)u{|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}& \lesssim {\pi }^{-r}{h}_k^r|u{|}_{H^{r+1}\left(\hat{\mathrm{\Omega }}\right)}and\\ |(I-{\hat{\mathrm{\Pi }}}_k)u{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}& \lesssim p^2{\pi }^{-r+1}{h}_k^{r-1}|u{|}_{H^{r+1}\left(\hat{\mathrm{\Omega }}\right)}\end{array}$hold for all $u\in H^{r+1}\left(\hat{\mathrm{\Omega }}\right)$.

Proof.

The proof is based on the univariate estimates given in Lemmas 4.1 and 4.2 ($p_{min}\ge 2$ and the quasi-uniformity of the grids have been required in (7(7) $p_{min}\ge 2\mathrm{a}\mathrm{n}\mathrm{d}\underset{\delta \in \{1,2\}}{min}\underset{k\in \{1,\dots ,K\}}{min}{h}_{k,\delta ,min}\ge C_{h}h$(7) )) and follows the standard construction that can be found, e.g. in the proof of Ref. [14, Theorem 3.3].

On the interfaces, we have the following approximation error estimate.

Lemma 4.4

Let $r\in \{1,2,\dots ,p_{min}\}$ and $(k,l)\in \mathcal{N}\cup {\mathcal{N}}^{\ast }$. Then, the estimate $\parallel (I-{\hat{\mathrm{\Pi }}}_k)u{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}\lesssim {\pi }^{-r}{h}^r|u{|}_{H^r\left({\hat{I}}_{k,l}\right)}$ holds for all $u\in H^2\left(\hat{\mathrm{\Omega }}\right)\cap H^r\left({\hat{I}}_{k,l}\right)$.

Proof.

Without loss of generality, we assume ${\hat{I}}_{k,l}=(0,1)\times \left\{0\right\}$. Because of Ref. [14, Theorem 3.4], we know that $\left(\right(I-{\hat{\mathrm{\Pi }}}_k\left)u\right)(\cdot ,0)=(I-{\mathrm{\Pi }}_{p_k,Z_{k,1}})\left(u\right(\cdot ,0\left)\right)$. So, we have $\parallel (I-{\hat{\mathrm{\Pi }}}_k)u{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}^2=\parallel (I-{\mathrm{\Pi }}_{p_k,Z_{k,1}})u(\cdot ,0){\parallel }_{L_2(0,1)}^2.$Using Refs. [14, Equation (3.4)], [28, Lemma 8] and that ${\mathrm{\Pi }}_{p_k,Z_{k,1}}$ minimizes the $H^1$-seminorm, we further obtain $\parallel (I-{\hat{\mathrm{\Pi }}}_k)u{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}^2\lesssim h^2\underset{v_h\in S_{p,h}(0,1)}{inf}|u(\cdot ,0)-v_h{|}_{H^1(0,1)}^2.$Sande et al. [12, Theorem 3.1] provide the desired result.

4.2. A mollifying operator

A second step of the proof is the introduction of a particular mollification operator for the interfaces.

For $(k,l)\in \mathcal{N}$, let ${\mathrm{{\rm Y}}}_{k,l}$ be given by ${\mathrm{{\rm Y}}}_{k,l}v:=v\circ {\gamma }_{k,l}$. For $(k,l)\in {\mathcal{N}}^{\ast }$, we define ${\mathrm{{\rm Y}}}_{k,l}v:=v\circ G_k^{-1}\circ G_l\circ {\gamma }_{l,k}$, i.e. we have $\left({\mathrm{{\rm Y}}}_{k,l}v\right)\left(t\right)=v\left({\gamma }_{l,k}\right(t\left)\right)$ or $\left({\mathrm{{\rm Y}}}_{k,l}v\right)\left(t\right)=v\left({\gamma }_{l,k}\right(1-t\left)\right)$, cf. Assumption 2.2. For all cases, ${\mathrm{{\rm Y}}}_{k,l}$ is a bijective function $H^s(0,1)\to H^s\left({\hat{I}}_{k,l}\right)$ and (25) $|u{|}_{H^s(0,1)}\eqsim |{\mathrm{{\rm Y}}}_{k,l}u{|}_{H^s\left({\hat{I}}_{k,l}\right)}$(25) holds for all s. For $v\in H^s\left(\hat{\mathrm{\Omega }}\right)$, we define the abbreviated notation ${\mathrm{{\rm Y}}}_{k,l}^{-1}v:={\mathrm{{\rm Y}}}_{k,l}^{-1}\left(v{|}_{{\hat{I}}_{k,l}}\right)$ and observe ${\mathrm{{\rm Y}}}_{k,l}^{-1}u\in H^{3/2}(0,1)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}u\in H^2\left(\hat{\mathrm{\Omega }}\right).$For $(k,l)\in \mathcal{N}\cup {\mathcal{N}}^{\ast }$, we define extension operators ${\mathrm{\Xi }}_{k,l}:H^s\left({\hat{I}}_{k,l}\right)\to H^s\left(\hat{\mathrm{\Omega }}\right)$ by $\left({\mathrm{\Xi }}_{k,l}w\right)(x,y):=\left\{\begin{array}{ll}\varphi \left(x\right)w(0,y)& \mathrm{i}\mathrm{f}{\hat{I}}_{k,l}=\left\{0\right\}\times (0,1),\\ \varphi (1-x)w(1,y)& \mathrm{i}\mathrm{f}{\hat{I}}_{k,l}=\left\{1\right\}\times (0,1),\\ \varphi \left(y\right)w(x,0)& \mathrm{i}\mathrm{f}{\hat{I}}_{k,l}=(0,1)\times \left\{0\right\},\\ \varphi (1-y)w(x,1)& \mathrm{i}\mathrm{f}{\hat{I}}_{k,l}=(0,1)\times \left\{1\right\},\end{array}\right.$where (26) $\varphi \left(x\right):=max\{0,1-{\eta }^{-1}x\}\mathrm{a}\mathrm{n}\mathrm{d}\eta \in (0,1).$(26) Now, define for each patch ${\mathrm{\Omega }}_k$, a mollifying operator ${\hat{\mathcal{M}}}_k$ by (27) ${\hat{\mathcal{M}}}_k:=I-\sum _{l\in \mathcal{N}\left(k\right)}{\mathrm{\Xi }}_{k,l}{\mathrm{{\rm Y}}}_{k,l}(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}.$(27) The combination of the patch local operators yields a global operator $\mathcal{M}$: (28) $\left(\mathcal{M}u\right){|}_{{\mathrm{\Omega }}_k}:=\left({\hat{\mathcal{M}}}_k\right(u\circ G_k\left)\right)\circ G_k^{-1}.$(28) Observe that $\mathcal{M}$ preserves constants, i.e. (29) $\mathcal{M}c=c\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}c\in \mathbb{R}.$(29)

Lemma 4.5

For all $(k,l)\in \mathcal{N}\cup {\mathcal{N}}^{\ast }$ and all $u\in H_0^1\left({\hat{I}}_{k,l}\right):=\{u\in H^1({\hat{I}}_{k,l}):u=0\mathrm{o}\mathrm{n}\mathrm{\partial }{\hat{I}}_{k,l}\}$, we have ${\mathrm{\Xi }}_{k,l}u=0\mathrm{o}\mathrm{n}\mathrm{\partial }\hat{\mathrm{\Omega }}\mathrm{\setminus }{\hat{I}}_{k,l}\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{\Xi }}_{k,l}u=u\mathrm{o}\mathrm{n}{\hat{I}}_{k,l}.$

Proof.

Assume without loss of generality that ${\hat{I}}_{k,l}=\left\{0\right\}\times (0,1)$. For this case, we have $\left({\mathrm{\Xi }}_{k,l}u\right)(x,y)=\varphi \left(x\right)u(0,y).$As $u\in H_0^1\left({\hat{I}}_{k,l}\right)$, we obtain $u(0,0)=u(0,1)=0$. This shows the first statement for the two boundary segments adjacent to ${\hat{I}}_{k,l}$, i.e. $[0,1]\times \left\{0\right\}$ and $[0,1]\times \left\{1\right\}$. Since $\eta <1$ yields $\varphi \left(1\right)=0$, we also have the first statement for the boundary segment $\left\{1\right\}\times (0,1)$. This finishes the proof for the first statement. The proof for the second statement follows directly from $\varphi \left(0\right)=1$.

Lemma 4.6

${\mathrm{{\rm Y}}}_{k,l}^{-1}{\hat{\mathcal{M}}}_k={\mathrm{\Pi }}_{r,\eta }{\mathrm{{\rm Y}}}_{k,l}^{-1}$ holds for all $(k,l)\in \mathcal{N}\cup {\mathcal{N}}^{\ast }$.

Proof.

Equation (27(27) ${\hat{\mathcal{M}}}_k:=I-\sum _{l\in \mathcal{N}\left(k\right)}{\mathrm{\Xi }}_{k,l}{\mathrm{{\rm Y}}}_{k,l}(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}.$(27) ) implies ${\mathrm{{\rm Y}}}_{k,l}^{-1}{\hat{\mathcal{M}}}_k={\mathrm{{\rm Y}}}_{k,l}^{-1}-\sum _{j\in \mathcal{N}\left(k\right)}{\mathrm{{\rm Y}}}_{k,l}^{-1}{\mathrm{\Xi }}_{k,j}{\mathrm{{\rm Y}}}_{k,j}(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,j}^{-1}$. Observe that the projector ${\mathrm{\Pi }}_{r,\eta }$ is interpolatory on the boundary [14, Lemma 3.1]. So, $(I-{\mathrm{\Pi }}_{r,\eta })$ maps into $H_0^1(0,1)$ and ${\mathrm{{\rm Y}}}_{k,j}(I-{\mathrm{\Pi }}_{r,\eta })$ maps into $H_0^1\left({\hat{I}}_{k,j}\right)$. Therefore, Lemma 4.5 yields ${\mathrm{{\rm Y}}}_{k,l}^{-1}{\hat{M}}_k={\mathrm{{\rm Y}}}_{k,l}^{-1}-{\mathrm{{\rm Y}}}_{k,l}^{-1}{\mathrm{{\rm Y}}}_{k,l}(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}$, which immediately implies the desired result.

Before we proceed, we give a certain trace like estimate.

Lemma 4.7

The estimate $\mathrm{\Psi }\left(u\right):=\underset{v\in H^1\left({\hat{I}}_{k,l}\right)}{inf}\parallel u-v{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}^2+{\theta }^2|v{|}_{H^1\left({\hat{I}}_{k,l}\right)}^2\lesssim \theta |u{|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2$holds for all $u\in H^1\left(\hat{\mathrm{\Omega }}\right)$ and $(k,l)\in \mathcal{N}\cup {\mathcal{N}}^{\ast }$ and all $\theta >0$.

Proof.

A trace theorem [14, Lemma 4.4] yields (30) $\mathrm{\Psi }\left(u\right)\lesssim \underset{v\in H^2\left(\hat{\mathrm{\Omega }}\right)}{inf}\parallel u-v{\parallel }_{L_2\left(\hat{\mathrm{\Omega }}\right)}|u-v{|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}+{\theta }^2|v{|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}|v{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}.$(30) Case 1. Assume $\theta <1$. In this case, we choose v to be the $H^1$-orthogonal projection of u into $S_{3,\lceil {\theta }^{-1}{\rceil }^{-1}}\left(\hat{\mathrm{\Omega }}\right)$. Since the spline degree of that space is fixed, we obtain using a standard inverse inequality [26, Corollary 3.94] and a standard approximation error estimate (like from Ref. [13]) that $\mathrm{\Psi }\left(u\right)\lesssim (\lceil {\theta }^{-1}{\rceil }^{-1}+{\theta }^2\lceil {\theta }^{-1}\rceil )|v{|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2\lesssim \theta |v{|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2.$Case 2. Assume $\theta \ge 1$. In this case, we choose $v:=(u,1{)}_{L_2\left(\mathrm{\Omega }\right)}$ and obtain from (30(30) $\mathrm{\Psi }\left(u\right)\lesssim \underset{v\in H^2\left(\hat{\mathrm{\Omega }}\right)}{inf}\parallel u-v{\parallel }_{L_2\left(\hat{\mathrm{\Omega }}\right)}|u-v{|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}+{\theta }^2|v{|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}|v{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}.$(30) ) directly $\mathrm{\Psi }\left(u\right)\lesssim \parallel u-v{\parallel }_{H^1\left(\hat{\mathrm{\Omega }}\right)}|u{|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}.$In this case, the Poincaré inequality finishes the proof.

As a next step, we show that the mollifier constructs functions that are very smooth on the interfaces.

Lemma 4.8

The estimate $|{\hat{\mathcal{M}}}_k{\hat{u}}_k{|}_{H^r\left({\hat{I}}_{k,l}\right)}^2\lesssim (2\sqrt{3}{r}^2{\eta }^{-1}{)}^{2r-3}{r}^2|{\hat{u}}_k{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2$ holds for all ${\hat{u}}_k\in H^r\left(\hat{\mathrm{\Omega }}\right)$ and all $(k,l)\in \mathcal{N}\cup {\mathcal{N}}^{\ast }$.

Proof.

We have using (25(25) $|u{|}_{H^s(0,1)}\eqsim |{\mathrm{{\rm Y}}}_{k,l}u{|}_{H^s\left({\hat{I}}_{k,l}\right)}$(25) ) and Lemma 4.6 $|{\hat{\mathcal{M}}}_k{\hat{u}}_k{|}_{H^r\left({\hat{I}}_{k,l}\right)}^2\eqsim |{\mathrm{{\rm Y}}}_{k,l}^{-1}{\hat{\mathcal{M}}}_k{\hat{u}}_k{|}_{H^r(0,1)}^2=|{\mathrm{\Pi }}_{r,\eta }{\mathrm{{\rm Y}}}_{k,l}^{-1}{\hat{u}}_k{|}_{H^r(0,1)}^2.$Now, a standard inverse estimate [26, Corollary 3.94] yields $|{\hat{\mathcal{M}}}_k{\hat{u}}_k{|}_{H^r\left({\hat{I}}_{k,l}\right)}^2\lesssim {\psi }^{2(r-s)}|{\mathrm{\Pi }}_{r,\eta }{\mathrm{{\rm Y}}}_{k,l}^{-1}{\hat{u}}_k{|}_{H^s(0,1)}^2\mathrm{f}\mathrm{o}\mathrm{r}s\in \{1,2\},$where $\psi :=2\sqrt{3}{r}^2{\eta }^{-1}$. Lemma 4.2 and the $H^1$-stability of ${\mathrm{\Pi }}_{r,\eta }$ yield $|{\mathrm{\Pi }}_{r,\eta }w{|}_{H^2(0,1)}^2\lesssim r^4|w{|}_{H^2(0,1)}^2\mathrm{a}\mathrm{n}\mathrm{d}|{\mathrm{\Pi }}_{r,\eta }w{|}_{H^1(0,1)}^2\le |w{|}_{H^1(0,1)}^2,$so we obtain $|{\hat{\mathcal{M}}}_k{\hat{u}}_k{|}_{H^r\left({\hat{I}}_{k,l}\right)}^2\lesssim {\psi }^{2r-2}\underset{v\in H^2(0,1)}{inf}\left(|{\mathrm{{\rm Y}}}_{k,l}^{-1}{\hat{u}}_k-v{|}_{H^1(0,1)}^2+{\psi }^{-2}{r}^4|v{|}_{H^2(0,1)}^2\right).$Using (25(25) $|u{|}_{H^s(0,1)}\eqsim |{\mathrm{{\rm Y}}}_{k,l}u{|}_{H^s\left({\hat{I}}_{k,l}\right)}$(25) ), we obtain $|{\hat{\mathcal{M}}}_k{\hat{u}}_k{|}_{H^r\left({\hat{I}}_{k,l}\right)}^2\lesssim {\psi }^{2r-2}\underset{v\in H^2\left({\hat{I}}_{k,l}\right)}{inf}\left(|{\hat{u}}_k-v{|}_{H^1\left({\hat{I}}_{k,l}\right)}^2+{\psi }^{-2}{r}^4|v{|}_{H^2\left({\hat{I}}_{k,l}\right)}^2\right).$By applying Lemma 4.7 to the derivative of ${\hat{u}}_k$, we obtain the desired result.

Lemma 4.9

$\parallel \text{〚}(I-\mathcal{M})u\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}=0$ holds for all $u\in H^{1,\circ }\left(\mathrm{\Omega }\right)\cap {\mathcal{H}}^2\left(\mathrm{\Omega }\right)$ and $(k,l)\in \mathcal{N}$.

Proof.

Let $u\in H^{1,\circ }\left(\mathrm{\Omega }\right)\cap {\mathcal{H}}^2\left(\mathrm{\Omega }\right)$ be arbitrary but fixed.

We obtain using the definition of ${\mathrm{{\rm Y}}}_{k,l}$ and ${\mathrm{{\rm Y}}}_{l,k}$ and Lemma 3.1 that (31) $\begin{array}{rl}\parallel \text{〚}w\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}& =\parallel w_k-w_l{\parallel }_{L_2\left(I_{k,l}\right)}\eqsim \parallel {\hat{w}}_k-{\hat{w}}_l\circ G_l^{-1}\circ G_k{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}\\ & =\parallel {\mathrm{{\rm Y}}}_{k,l}^{-1}({\hat{w}}_k-{\hat{w}}_l\circ G_l^{-1}\circ G_k){\parallel }_{L_2(0,1)}\\ & =\parallel {\mathrm{{\rm Y}}}_{k,l}^{-1}{\hat{w}}_k-{\mathrm{{\rm Y}}}_{l,k}^{-1}{\hat{w}}_l{\parallel }_{L_2(0,1)}\end{array}$(31) holds, where ${\hat{w}}_k:=w_k\circ G_k$ and ${\hat{w}}_l:=w_l\circ G_l$. Since $u\in H^1\left(\mathrm{\Omega }\right)$, a standard trace theorem yields (32) ${\mathrm{{\rm Y}}}_{k,l}^{-1}{\hat{u}}_k-{\mathrm{{\rm Y}}}_{l,k}^{-1}{\hat{u}}_l=0.$(32) Thus, (31(31) $\begin{array}{rl}\parallel \text{〚}w\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}& =\parallel w_k-w_l{\parallel }_{L_2\left(I_{k,l}\right)}\eqsim \parallel {\hat{w}}_k-{\hat{w}}_l\circ G_l^{-1}\circ G_k{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}\\ & =\parallel {\mathrm{{\rm Y}}}_{k,l}^{-1}({\hat{w}}_k-{\hat{w}}_l\circ G_l^{-1}\circ G_k){\parallel }_{L_2(0,1)}\\ & =\parallel {\mathrm{{\rm Y}}}_{k,l}^{-1}{\hat{w}}_k-{\mathrm{{\rm Y}}}_{l,k}^{-1}{\hat{w}}_l{\parallel }_{L_2(0,1)}\end{array}$(31) ) implies $\parallel \text{〚}u\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}=0$. By plugging $\mathcal{M}u$ into (31(31) $\begin{array}{rl}\parallel \text{〚}w\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}& =\parallel w_k-w_l{\parallel }_{L_2\left(I_{k,l}\right)}\eqsim \parallel {\hat{w}}_k-{\hat{w}}_l\circ G_l^{-1}\circ G_k{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}\\ & =\parallel {\mathrm{{\rm Y}}}_{k,l}^{-1}({\hat{w}}_k-{\hat{w}}_l\circ G_l^{-1}\circ G_k){\parallel }_{L_2(0,1)}\\ & =\parallel {\mathrm{{\rm Y}}}_{k,l}^{-1}{\hat{w}}_k-{\mathrm{{\rm Y}}}_{l,k}^{-1}{\hat{w}}_l{\parallel }_{L_2(0,1)}\end{array}$(31) ), we obtain using Lemma 4.6 $\parallel \text{〚}\mathcal{M}u\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}\eqsim \parallel {\mathrm{\Pi }}_{r,\eta }({\mathrm{{\rm Y}}}_{k,l}^{-1}{\hat{u}}_k-{\mathrm{{\rm Y}}}_{l,k}^{-1}{\hat{u}}_l){\parallel }_{L_2(0,1)}.$Using (32(32) ${\mathrm{{\rm Y}}}_{k,l}^{-1}{\hat{u}}_k-{\mathrm{{\rm Y}}}_{l,k}^{-1}{\hat{u}}_l=0.$(32) ) and ${\mathrm{\Pi }}_{r,\eta }0=0$, we obtain $\parallel \text{〚}\mathcal{M}u\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}=0$ and consequently also $\parallel \text{〚}(I-\mathcal{M})u\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}=0$.

Lemma 4.10

The estimate $\parallel {\hat{\mathrm{\Pi }}}_k(I-{\hat{\mathcal{M}}}_k)u{\parallel }_{H^{1,\circ }\left(\hat{\mathrm{\Omega }}\right)}^2\lesssim (1+{\eta }^2{h}^{-2})h^2|u{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2$ holds for all $u\in H^2\left(\hat{\mathrm{\Omega }}\right)$ and $k=1,\dots ,K$.

Proof.

Using the definition of ${\hat{\mathcal{M}}}_k$ and of the $H^{1,\circ }$-norm, we obtain (33) $\begin{array}{rl}\parallel {\hat{\mathrm{\Pi }}}_k(I-{\hat{\mathcal{M}}}_k)u{\parallel }_{H^{1,\circ }\left(\hat{\mathrm{\Omega }}\right)}& \le \sum _{l\in \mathcal{N}\left(k\right)}\parallel {\hat{\mathrm{\Pi }}}_k{\mathrm{\Xi }}_{k,l}{\mathrm{{\rm Y}}}_{k,l}(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}u{\parallel }_{H^{1,\circ }\left(\hat{\mathrm{\Omega }}\right)}\\ & \lesssim \sum _{l\in \mathcal{N}\left(k\right)}({\mathrm{\Psi }}_{x,l}+{\mathrm{\Psi }}_{y,l}+{\mathrm{\Psi }}_{\circ ,l}),\end{array}$(33) where ${\mathrm{\Psi }}_{◻,l}:=\parallel \frac{\mathrm{\partial }}{\mathrm{\partial }◻}{\hat{\mathrm{\Pi }}}_k{\mathrm{\Xi }}_{k,l}{\mathrm{{\rm Y}}}_{k,l}(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}u{\parallel }_{L_2\left(\hat{\mathrm{\Omega }}\right)}$ for $◻\in \{x,y\}$ and ${\mathrm{\Psi }}_{\circ ,l}:=\left({\hat{\mathrm{\Pi }}}_k{\mathrm{\Xi }}_{k,l}{\mathrm{{\rm Y}}}_{k,l}\right(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}u,1{)}_{L_2\left(\hat{\mathrm{\Omega }}\right)}.$ We estimate the terms ${\mathrm{\Psi }}_{x,l}$, ${\mathrm{\Psi }}_{y,l}$, and ${\mathrm{\Psi }}_{\circ ,l}$ separately. Let without loss of generality ${\hat{I}}_{k,l}=\left\{0\right\}\times (0,1)$.

Step 1. Using (23(23) ${\hat{\mathrm{\Pi }}}_k={\mathrm{\Pi }}_k^x{\mathrm{\Pi }}_k^y.$(23) ) and the $H^1$-stability of the $H^{1,D}$-orthogonal projection, and $w:={\mathrm{{\rm Y}}}_{k,l}(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}u$, we obtain $\begin{array}{rl}{\mathrm{\Psi }}_{x,l}^2& =\parallel \frac{\mathrm{\partial }}{\mathrm{\partial }x}{\mathrm{\Pi }}_k^x{\mathrm{\Pi }}_k^y{\mathrm{\Xi }}_{k,l}w{\parallel }_{L_2\left(\hat{\mathrm{\Omega }}\right)}^2\le \parallel \frac{\mathrm{\partial }}{\mathrm{\partial }x}{\mathrm{\Pi }}_k^y{\mathrm{\Xi }}_{k,l}w{\parallel }_{L_2\left(\hat{\mathrm{\Omega }}\right)}^2\\ & ={\int }_0^1{\int }_0^1{\left({\varphi }^{\prime }\left(x\right)\left({\mathrm{\Pi }}_{p_k,Z_{k,2}}w\right(0,\cdot \left)\right)\left(y\right)\right)}^2\text{d}x\text{d}y\\ & =|\varphi {|}_{H^1(0,1)}^2\parallel {\mathrm{\Pi }}_{p_k,Z_{k,2}}\left(w\right(0,\cdot \left)\right){\parallel }_{L_2(0,1)}^2\eqsim {\hat{\mathrm{\Psi }}}_{x,l}^2:={\eta }^{-1}\parallel {\mathrm{\Pi }}_{p_k,Z_{k,2}}\left(w\right(0,\cdot \left)\right){\parallel }_{L_2(0,1)}^2,\end{array}$where we use $|\varphi {|}_{H^1(0,1)}^2\eqsim {\eta }^{-1}$. The triangle inequality yields ${\mathrm{\Psi }}_{x,l}^2\le {\hat{\mathrm{\Psi }}}_{x,l}^2\lesssim {\eta }^{-1}\parallel w(0,\cdot ){\parallel }_{L_2(0,1)}^2+{\eta }^{-1}\parallel (I-{\mathrm{\Pi }}_{p_k,Z_{k,2}})\left(w\right(0,\cdot \left)\right){\parallel }_{L_2(0,1)}^2.$Lemma 4.1 yields ${\mathrm{\Psi }}_{x,l}^2\le {\hat{\mathrm{\Psi }}}_{x,l}^2\lesssim {\eta }^{-1}\parallel w(0,\cdot ){\parallel }_{L_2(0,1)}^2+{\eta }^{-1}{h}^2|w(0,\cdot ){|}_{H^1(0,1)}^2.$The definition of w and (25(25) $|u{|}_{H^s(0,1)}\eqsim |{\mathrm{{\rm Y}}}_{k,l}u{|}_{H^s\left({\hat{I}}_{k,l}\right)}$(25) ) yield ${\mathrm{\Psi }}_{x,l}^2\le {\hat{\mathrm{\Psi }}}_{x,l}^2\lesssim {\eta }^{-1}\parallel (I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}u{\parallel }_{L_2(0,1)}^2+{\eta }^{-1}{h}^2|(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}u{|}_{H^1(0,1)}^2.$Lemma 4.1 yields ${\mathrm{\Psi }}_{x,l}^2\le {\hat{\mathrm{\Psi }}}_{x,l}^2\lesssim (\eta +{\eta }^{-1}{h}^2)\left(\underset{v\in H^2(0,1)}{inf}|{\mathrm{{\rm Y}}}_{k,l}^{-1}u-v{|}_{H^1(0,1)}^2+{\eta }^2|v{|}_{H^2(0,1)}^2\right).$ Equation (25(25) $|u{|}_{H^s(0,1)}\eqsim |{\mathrm{{\rm Y}}}_{k,l}u{|}_{H^s\left({\hat{I}}_{k,l}\right)}$(25) ) yields further ${\mathrm{\Psi }}_{x,l}^2\le {\hat{\mathrm{\Psi }}}_{x,l}^2\lesssim (\eta +{\eta }^{-1}{h}^2)\left(\underset{v\in H^2\left({\hat{I}}_{k,l}\right)}{inf}|u-v{|}_{H^1\left({\hat{I}}_{k,l}\right)}^2+{\eta }^2|v{|}_{H^2\left({\hat{I}}_{k,l}\right)}^2\right).$Now, Lemma 4.7 applied to the derivative of u yields (34) ${\mathrm{\Psi }}_{x,l}^2\le {\hat{\mathrm{\Psi }}}_{x,l}^2\lesssim (\eta +{\eta }^{-1}{h}^2)\eta |u{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2=(1+{\eta }^2{h}^{-2})h^2|u{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2.$(34) Step 2. Using (23(23) ${\hat{\mathrm{\Pi }}}_k={\mathrm{\Pi }}_k^x{\mathrm{\Pi }}_k^y.$(23) ) and the $H^1$-stability of the $H^{1,D}$-orthogonal projection and $w:={\mathrm{{\rm Y}}}_{k,l}(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}u$, we obtain $\begin{array}{rl}{\mathrm{\Psi }}_{y,l}^2& =\parallel \frac{\mathrm{\partial }}{\mathrm{\partial }y}{\mathrm{\Pi }}_k^x{\mathrm{\Pi }}_k^y{\mathrm{\Xi }}_{k,l}w{\parallel }_{L_2\left(\hat{\mathrm{\Omega }}\right)}^2\le \parallel \frac{\mathrm{\partial }}{\mathrm{\partial }y}{\mathrm{\Pi }}_k^x{\mathrm{\Xi }}_{k,l}w{\parallel }_{L_2\left(\hat{\mathrm{\Omega }}\right)}^2\\ & ={\int }_0^1{\int }_0^1{\left(\left({\mathrm{\Pi }}_{p_k,Z_{k,1}}\varphi \right)\left(x\right)\frac{\mathrm{\partial }}{\mathrm{\partial }y}w(0,y)\right)}^2\text{d}x\text{d}y\eqsim \parallel {\mathrm{\Pi }}_{p_k,Z_{k,1}}\varphi {\parallel }_{L_2(0,1)}^2|w{|}_{H^1\left({\hat{I}}_{k,l}\right)}^2.\end{array}$Using $\parallel {\mathrm{\Pi }}_{p_k,Z_{k,1}}\varphi {\parallel }_{L_2(0,1)}^2\lesssim \parallel \varphi {\parallel }_{L_2(0,1)}^2+\parallel (I-{\mathrm{\Pi }}_{p_k,Z_{k,1}})\varphi {\parallel }_{L_2(0,1)}^2\lesssim \parallel \varphi {\parallel }_{L_2(0,1)}^2+h^2|\varphi {|}_{H^1(0,1)}^2\eqsim \eta +h^2{\eta }^{-1}=({\eta }^2{h}^{-2}+1)h^2{\eta }^{-1}$ and the definition of w, we obtain ${\mathrm{\Psi }}_{y,l}^2\lesssim ({\eta }^2{h}^{-2}+1)h^2{\eta }^{-1}|{\mathrm{{\rm Y}}}_{k,l}(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}u{|}_{H^1\left({\hat{I}}_{k,l}\right)}^2.$Using (25(25) $|u{|}_{H^s(0,1)}\eqsim |{\mathrm{{\rm Y}}}_{k,l}u{|}_{H^s\left({\hat{I}}_{k,l}\right)}$(25) ), we obtain further ${\mathrm{\Psi }}_{y,l}^2\lesssim ({\eta }^2{h}^{-2}+1)h^2{\eta }^{-1}|(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}u{|}_{H^1(0,1)}^2.$Using the $H^1$-stability of ${\mathrm{\Pi }}_{r,\eta }$ and the approximation error estimate [14, Theorem 3.1], we obtain ${\mathrm{\Psi }}_{y,l}^2\lesssim ({\eta }^2{h}^{-2}+1)h^2\left(\underset{v\in H^2(0,1)}{inf}{\eta }^{-1}|{\mathrm{{\rm Y}}}_{k,l}^{-1}u-v{|}_{H^1(0,1)}^2+\eta |v{|}_{H^2(0,1)}^2\right).$Using (25(25) $|u{|}_{H^s(0,1)}\eqsim |{\mathrm{{\rm Y}}}_{k,l}u{|}_{H^s\left({\hat{I}}_{k,l}\right)}$(25) ) and Lemma 4.7 applied to the derivative of ${\mathrm{{\rm Y}}}_{k,l}^{-1}u$, we obtain (35) $\begin{array}{rl}{\mathrm{\Psi }}_{y,l}^2& \lesssim ({\eta }^2{h}^{-2}+1)h^2\left(\underset{v\in H^2\left({\hat{I}}_{k,l}\right)}{inf}{\eta }^{-1}|u-v{|}_{H^1\left({\hat{I}}_{k,l}\right)}^2+\eta |v{|}_{H^2\left({\hat{I}}_{k,l}\right)}^2\right)\\ & \lesssim ({\eta }^2{h}^{-2}+1)h^2|u{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2.\end{array}$(35) Step 3. Using $w:={\mathrm{{\rm Y}}}_{k,l}(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}u$ and (22(22) $(u-{\mathrm{\Pi }}_{p,Z}u,1{)}_{L_2(0,1)}=0.$(22) ), we obtain $\begin{array}{rl}{\mathrm{\Psi }}_{\circ ,l}^2& =({\mathrm{\Pi }}_k^x{\mathrm{\Pi }}_k^y{\mathrm{\Xi }}_{k,l}w,1{)}_{L_2\left(\hat{\mathrm{\Omega }}\right)}^2={\int }_0^1{\int }_0^1({\mathrm{\Pi }}_{p_k,Z_{k,1}}\varphi \left)\right(x\left)\right({\mathrm{\Pi }}_{p_k,Z_{k,2}}w(0,\cdot )\left)\right(y)\text{d}x\text{d}y\\ & ={\int }_0^1\left({\mathrm{\Pi }}_{p_k,Z_{k,1}}\varphi \right)\left(x\right)\text{d}x{\int }_0^1\left({\mathrm{\Pi }}_{p_k,Z_{k,2}}w\right(0,\cdot \left)\right)\left(y\right)\text{d}y\\ & ={\int }_0^1\varphi \left(x\right)\text{d}x{\int }_0^1\left({\mathrm{\Pi }}_{p_k,Z_{k,2}}w\right(0,\cdot \left)\right)\left(y\right)\text{d}y\le \frac{\eta }{2}\parallel {\mathrm{\Pi }}_{p_k,Z_{k,2}}w(0,\cdot ){\parallel }_{L_2(0,1)}^2=\frac{{\eta }^2}{2}{\hat{\mathrm{\Psi }}}_{x,l}^2.\end{array}$Using (34(34) ${\mathrm{\Psi }}_{x,l}^2\le {\hat{\mathrm{\Psi }}}_{x,l}^2\lesssim (\eta +{\eta }^{-1}{h}^2)\eta |u{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2=(1+{\eta }^2{h}^{-2})h^2|u{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2.$(34) ), and using $\eta \le 1$, we further obtain (36) ${\mathrm{\Psi }}_{\circ ,l}^2\lesssim (1+{\eta }^2{h}^{-2})h^2|u{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2.$(36) Concluding step. The combination of (33(33) $\begin{array}{rl}\parallel {\hat{\mathrm{\Pi }}}_k(I-{\hat{\mathcal{M}}}_k)u{\parallel }_{H^{1,\circ }\left(\hat{\mathrm{\Omega }}\right)}& \le \sum _{l\in \mathcal{N}\left(k\right)}\parallel {\hat{\mathrm{\Pi }}}_k{\mathrm{\Xi }}_{k,l}{\mathrm{{\rm Y}}}_{k,l}(I-{\mathrm{\Pi }}_{r,\eta }){\mathrm{{\rm Y}}}_{k,l}^{-1}u{\parallel }_{H^{1,\circ }\left(\hat{\mathrm{\Omega }}\right)}\\ & \lesssim \sum _{l\in \mathcal{N}\left(k\right)}({\mathrm{\Psi }}_{x,l}+{\mathrm{\Psi }}_{y,l}+{\mathrm{\Psi }}_{\circ ,l}),\end{array}$(33) )–(36(36) ${\mathrm{\Psi }}_{\circ ,l}^2\lesssim (1+{\eta }^2{h}^{-2})h^2|u{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2.$(36) ) yields the desired result.

4.3. The approximation error estimate

The following three lemmas give approximation error estimates (20(20) $\underset{v_h\in V_h}{inf}\parallel u-v_h{\parallel }_{Q_h^{+}}\lesssim (\ln \sigma {)}^2{\sigma }^{1/(2p_{min}-1)}h|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}$(20) ) for the choice $u_h:=\mathrm{\Pi }\mathcal{M}u$ separately for the individual parts of $\parallel \cdot {\parallel }_{Q_h^{+}}$.

Lemma 4.11

$|(I-\mathrm{\Pi }\mathcal{M})u{|}_{{\mathcal{H}}^1\left(\mathrm{\Omega }\right)}^2\le (1+{\eta }^2{h}^{-2})h^2|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2$ holds for all $u\in H^{1,\circ }\left(\mathrm{\Omega }\right)\cap {\mathcal{H}}^2\left(\mathrm{\Omega }\right)$.

Proof.

First note that the Poincaré inequality yields (37) $\parallel \cdot {\parallel }_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2\eqsim \parallel \cdot {\parallel }_{H^{1,\circ }\left(\hat{\mathrm{\Omega }}\right)}^2:=|\cdot {|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2+(\cdot ,1{)}_{L_2\left(\hat{\mathrm{\Omega }}\right)}^2.$(37) Let $u\in H^{1,\circ }\left(\mathrm{\Omega }\right)\cap {\mathcal{H}}^2\left(\mathrm{\Omega }\right)$ be arbitrary but fixed and let ${\hat{u}}_k:=u\circ G_k$. Using Lemma 3.1, the triangle inequality, (37(37) $\parallel \cdot {\parallel }_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2\eqsim \parallel \cdot {\parallel }_{H^{1,\circ }\left(\hat{\mathrm{\Omega }}\right)}^2:=|\cdot {|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2+(\cdot ,1{)}_{L_2\left(\hat{\mathrm{\Omega }}\right)}^2.$(37) ), and (24(24) $(u-{\hat{\mathrm{\Pi }}}_{k}u,1{)}_{L_2\left(\hat{\mathrm{\Omega }}\right)}=0,{\hat{\mathrm{\Pi }}}_{k}c=c,\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\Pi }c=c$(24) ), we obtain $|(I-\mathrm{\Pi }\mathcal{M})u{|}_{{\mathcal{H}}^1\left(\mathrm{\Omega }\right)}^2\lesssim \sum _{k=1}^K|(I-{\hat{\mathrm{\Pi }}}_k){\hat{u}}_k{|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2+\sum _{k=1}^K\parallel {\hat{\mathrm{\Pi }}}_k(I-{\hat{\mathcal{M}}}_k){\hat{u}}_k{\parallel }_{H^{1,\circ }\left(\hat{\mathrm{\Omega }}\right)}^2.$We further obtain using Lemmas 4.3 and 4.10, $|(I-\mathrm{\Pi }\mathcal{M})u{|}_{{\mathcal{H}}^1\left(\mathrm{\Omega }\right)}^2\lesssim h^2\sum _{k=1}^K|{\hat{u}}_k{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2+\sum _{k=1}^K(1+{\eta }^2{h}^{-2})h^2\parallel {\hat{u}}_k{\parallel }_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2.$Lemma 3.1, (24(24) $(u-{\hat{\mathrm{\Pi }}}_{k}u,1{)}_{L_2\left(\hat{\mathrm{\Omega }}\right)}=0,{\hat{\mathrm{\Pi }}}_{k}c=c,\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\Pi }c=c$(24) ), (29(29) $\mathcal{M}c=c\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}c\in \mathbb{R}.$(29) ), and the Poincaré inequality finish the proof.

Lemma 4.12

$|(I-\mathrm{\Pi }\mathcal{M})u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2\le (1+{\eta }^2{h}^{-2})p^4|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2$ holds for all $u\in H^{1,\circ }\left(\mathrm{\Omega }\right)\cap {\mathcal{H}}^2\left(\mathrm{\Omega }\right)$.

Proof.

Using Lemma 3.1, the triangle inequality, (37(37) $\parallel \cdot {\parallel }_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2\eqsim \parallel \cdot {\parallel }_{H^{1,\circ }\left(\hat{\mathrm{\Omega }}\right)}^2:=|\cdot {|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2+(\cdot ,1{)}_{L_2\left(\hat{\mathrm{\Omega }}\right)}^2.$(37) ), (24(24) $(u-{\hat{\mathrm{\Pi }}}_{k}u,1{)}_{L_2\left(\hat{\mathrm{\Omega }}\right)}=0,{\hat{\mathrm{\Pi }}}_{k}c=c,\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\Pi }c=c$(24) ), and a standard inverse inequality ([26, Corollary 3.94]), we obtain $\begin{array}{rl}|(I-\mathrm{\Pi }\mathcal{M})u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2& \lesssim \sum _{k=1}^K\parallel (I-{\hat{\mathrm{\Pi }}}_k{\hat{\mathcal{M}}}_k){\hat{u}}_k{\parallel }_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2\\ & \lesssim \sum _{k=1}^K\parallel (I-{\hat{\mathrm{\Pi }}}_k){\hat{u}}_k{\parallel }_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2+\sum _{k=1}^K\parallel {\hat{\mathrm{\Pi }}}_k(I-{\hat{\mathcal{M}}}_k){\hat{u}}_k{\parallel }_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2\\ & \lesssim \sum _{k=1}^K|(I-{\hat{\mathrm{\Pi }}}_k){\hat{u}}_k{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2+p^4{h}^{-2}\sum _{k=1}^K\parallel {\hat{\mathrm{\Pi }}}_k(I-{\hat{\mathcal{M}}}_k){\hat{u}}_k{\parallel }_{H^{1,\circ }\left(\hat{\mathrm{\Omega }}\right)}^2.\end{array}$By again applying Lemmas 4.3 and 4.10, we obtain $\begin{array}{rl}|(I-\mathrm{\Pi }\mathcal{M})u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2& \lesssim p^2\sum _{k=1}^K|{\hat{u}}_k{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2+p^4(1+{\eta }^2{h}^{-2})\sum _{k=1}^K\parallel {\hat{u}}_k{\parallel }_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2\\ & \lesssim p^4(1+{\eta }^2{h}^{-2})\sum _{k=1}^K\parallel {\hat{u}}_k{\parallel }_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2.\end{array}$Lemma 3.1, (24(24) $(u-{\hat{\mathrm{\Pi }}}_{k}u,1{)}_{L_2\left(\hat{\mathrm{\Omega }}\right)}=0,{\hat{\mathrm{\Pi }}}_{k}c=c,\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\Pi }c=c$(24) ), (29(29) $\mathcal{M}c=c\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}c\in \mathbb{R}.$(29) ) and the Poincaré inequality finish the proof.

Lemma 4.13

$\sum _{(k,l)\in \mathcal{I}}\parallel \text{〚}(I-\mathrm{\Pi }\mathcal{M})u\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\le {\pi }^{-2r}{h}^{2r}(2\sqrt{3}{r}^2{\eta }^{-1}{)}^{2r-3}{r}^2|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2$ holds for all $u\in H^{1,\circ }\left(\mathrm{\Omega }\right)\cap {\mathcal{H}}^2\left(\mathrm{\Omega }\right)$.

Proof.

Let $u\in H^{1,\circ }\left(\mathrm{\Omega }\right)\cap {\mathcal{H}}^2\left(\mathrm{\Omega }\right)$ be arbitrary but fixed and let ${\hat{u}}_k:=u\circ G_k$. Observe that the triangle inequality, Lemmas 4.9 and 3.1 yield $\begin{array}{rl}\sum _{(k,l)\in \mathcal{N}}\parallel \text{〚}(I-\mathrm{\Pi }\mathcal{M})u\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2& \lesssim \sum _{(k,l)\in \mathcal{N}}\parallel \text{〚}(I-\mathrm{\Pi })\mathcal{M}u\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\\ & \lesssim \sum _{(k,l)\in \mathcal{N}\cup {\mathcal{N}}^{\ast }}\parallel \left(\right(I-\mathrm{\Pi }\left)\mathcal{M}u\right){|}_{{\mathrm{\Omega }}_k}{\parallel }_{L_2\left(I_{k,l}\right)}^2\\ & \lesssim \sum _{(k,l)\in \mathcal{N}\cup {\mathcal{N}}^{\ast }}\parallel (I-{\hat{\mathrm{\Pi }}}_k){\hat{\mathcal{M}}}_k{\hat{u}}_k{\parallel }_{L_2\left({\hat{I}}_{k,l}\right)}^2.\end{array}$Lemma 4.4 yields $\sum _{(k,l)\in \mathcal{I}}\parallel \text{〚}(I-\mathrm{\Pi }\mathcal{M})u\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\lesssim {\pi }^{-2r}{h}^{2r}\sum _{(k,l)\in \mathcal{N}\cup {\mathcal{N}}^{\ast }}|{\hat{\mathcal{M}}}_k{\hat{u}}_k{|}_{H^r\left({\hat{I}}_{k,l}\right)}^2.$Since Assumption 2.2 yields $|{\hat{\mathcal{M}}}_k{\hat{u}}_k{|}_{H^r\left({\hat{I}}_{k,l}\right)}^2\eqsim |{\hat{\mathcal{M}}}_l{\hat{u}}_l{|}_{H^r\left({\hat{I}}_{l,k}\right)}^2$, we obtain $|{\hat{\mathcal{M}}}_k{\hat{u}}_k{|}_{H^r\left({\hat{I}}_{k,l}\right)}^2+|{\hat{\mathcal{M}}}_l{\hat{u}}_l{|}_{H^r\left({\hat{I}}_{l,k}\right)}^2\lesssim |{\hat{\mathcal{M}}}_k{\hat{u}}_k{|}_{H^r\left({\hat{I}}_{k,l}\right)}^2+|{\hat{\mathcal{M}}}_l{\hat{u}}_l{|}_{H^r\left({\hat{I}}_{l,k}\right)}^2$and therefore also $\sum _{(k,l)\in \mathcal{I}}\parallel \text{〚}(I-\mathrm{\Pi }\mathcal{M})u\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\lesssim {\pi }^{-2r}{h}^{2r}\sum _{(k,l)\in \mathcal{N}\cup {\mathcal{N}}^{\ast }}|{\hat{\mathcal{M}}}_k{\hat{u}}_k{|}_{H^r\left({\hat{I}}_{k,l}\right)}^2.$Now, Lemma 4.8 yields $\sum _{(k,l)\in \mathcal{I}}\parallel \text{〚}(I-\mathrm{\Pi }\mathcal{M})u\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2\lesssim {\pi }^{-2r}{h}^{2r}(2\sqrt{3}{r}^2{\eta }^{-1}{)}^{2r-3}{r}^2\sum _{k=1}^K|{\hat{u}}_k{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2.$Lemma 3.1 finishes the proof.

Finally, we can show Theorems 3.8 and 3.9.

Proof of Theorem 3.9.

Proof of Theorem 3.9.

Let $u\in {\mathcal{H}}^{2,\circ }\left(\mathrm{\Omega }\right)\cap H^1\left(\mathrm{\Omega }\right)$ be arbitrary but fixed and define $u_h:=\mathrm{\Pi }\mathcal{M}u$.

First, we show that (38) $\parallel u-u_h{\parallel }_{Q_h^{+}}^2\lesssim \underset{{\displaystyle \mathrm{\Psi }:=}}{\underbrace{\left(1+{\eta }_0^2{h}^{-2}+\sigma (3r^{2}h{\eta }_0^{-1}{)}^{2r-3}{r}^2\right)}}{h}^2|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2$(38) holds for any $r\in \{2,\dots ,p_{min}\}$ and all ${\eta }_0>0$.

Case 1. Assume ${\eta }_0\le 1$. In this case, we define $\eta :=\lceil {\eta }_0^{-1}{\rceil }^{-1}$ and observe $\frac{1}{2}{\eta }_0\le \eta \le {\eta }_0.$Equation (16(16) $\parallel u{\parallel }_{Q_h^{+}}^2:=\parallel u{\parallel }_{Q_h}^2+\frac{h^2}{{\sigma }^2}|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2.$(16) ), Lemmas 4.11–4.13, and $\sigma ⪆p^2$ yield $\parallel u-u_h{\parallel }_{Q_h^{+}}^2\lesssim \left(1+{\eta }^2{h}^{-2}+\frac{\sigma }{h^3}{\pi }^{-2r}{h}^{2r}(2\sqrt{3}{r}^2{\eta }^{-1}{)}^{2r-3}{r}^2\right)h^2|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2$and since $4\sqrt{3}{\pi }^{-1}\le 3$ further (38(38) $\parallel u-u_h{\parallel }_{Q_h^{+}}^2\lesssim \underset{{\displaystyle \mathrm{\Psi }:=}}{\underbrace{\left(1+{\eta }_0^2{h}^{-2}+\sigma (3r^{2}h{\eta }_0^{-1}{)}^{2r-3}{r}^2\right)}}{h}^2|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2$(38) ).

Case 2. Assume ${\eta }_0>1$. Define $W:=\{u\in H^1(\mathrm{\Omega }):u\circ G_k\in S_{1,1}(\hat{\mathrm{\Omega }})\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}k=1,\dots ,K\},$i.e. the set of all globally continuous functions which are locally just linear. Observe that $W\subseteq V_h$. Using u and w being continuous, we obtain $\parallel u-w{\parallel }_{Q_h^{+}}^2=|u-w{|}_{{\mathcal{H}}^1\left(\mathrm{\Omega }\right)}^2+\frac{h^2}{{\sigma }^2}|u-w{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2.$For the choice $w\in H^1\left(\mathrm{\Omega }\right)\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}w{|}_{{\mathrm{\Omega }}_k}:=w_k={\hat{w}}_k\circ G_k^{-1},$where ${\hat{w}}_k(x,y):=\sum _{i=0}^1\sum _{j=0}^1{\hat{\varphi }}_i\left(x\right){\varphi }_j\left(y\right){\hat{u}}_k(i,j)\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{\hat{\varphi }}_0\left(t\right):=1-t\mathrm{a}\mathrm{n}\mathrm{d}{\hat{\varphi }}_1\left(t\right):=t,$we further obtain using standard approximation error estimates and Lemma 3.1 $\underset{v_h\in V_h}{inf}\parallel u-v_h{\parallel }_{Q_h^{+}}^2\le \parallel u-w+c{\parallel }_{Q_h^{+}}^2=\parallel u-w{\parallel }_{Q_h^{+}}^2\lesssim (1+h^2{\sigma }^{-2})|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2,$where $c:=(w,1{)}_{L_2\left(\mathrm{\Omega }\right)}/(1,1{)}_{L_2\left(\mathrm{\Omega }\right)}$. Using ${\sigma }^{-2}\le p^{-4}\le 1/16$, $h\lesssim 1$, and ${\eta }_0>1$, we have $\underset{v_h\in V_h}{inf}\parallel u-v_h{\parallel }_{Q_h^{+}}^2\lesssim \left({\eta }_0^2+\frac{h^2}{16}\right)|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2\le (1+{\eta }_0^2{h}^{-2})h^2|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2,$which shows (38(38) $\parallel u-u_h{\parallel }_{Q_h^{+}}^2\lesssim \underset{{\displaystyle \mathrm{\Psi }:=}}{\underbrace{\left(1+{\eta }_0^2{h}^{-2}+\sigma (3r^{2}h{\eta }_0^{-1}{)}^{2r-3}{r}^2\right)}}{h}^2|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2$(38) ) also for the second case.

Finally, we show that Ψ is such that the desired bound (20(20) $\underset{v_h\in V_h}{inf}\parallel u-v_h{\parallel }_{Q_h^{+}}\lesssim (\ln \sigma {)}^2{\sigma }^{1/(2p_{min}-1)}h|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}$(20) ) follows. We again consider two cases.

Case 1. Assume $\ln \left({\sigma }^{1/2}\right)\le p_{min}$. In this case, we choose $r:=max\{2,\lceil \ln ({\sigma }^{1/2})\rceil \}\mathrm{a}\mathrm{n}\mathrm{d}{\eta }_0:=3\mathbf{e}{r}^{2}h,$where $\mathbf{e}$ is Euler's number ($\ln \mathbf{e}=1$), and obtain $\mathrm{\Psi }\lesssim r^4+\sigma {\mathbf{e}}^{-2r}\eqsim r^4\lesssim {\left(\ln \sigma \right)}^4\lesssim {\left(\ln \sigma \right)}^4{\sigma }^{2/(2p_{min}-1)},$which finishes the proof for Case 1.

Case 2. Assume $\ln \left({\sigma }^{1/2}\right)\ge p_{min}$. In this case, we choose $r:=p_{min}\mathrm{a}\mathrm{n}\mathrm{d}{\eta }_0:=3{\sigma }^{1/(2r-1)}{r}^{2}h$and obtain immediately $\mathrm{\Psi }=1+3^2{\sigma }^{2/(2r-1)}{r}^4+{\sigma }^{2/(2r-1)}{r}^2\lesssim {\sigma }^{2/(2p_{min}-1)}(\ln \sigma {)}^4,$which finishes the proof for Case 2.

Proof of Theorem 3.8.

Proof of Theorem 3.8.

Let $u\in {\mathcal{H}}^{2,\circ }\left(\mathrm{\Omega }\right)\cap H^1\left(\mathrm{\Omega }\right)$ be arbitrary but fixed and define $u_h:=\mathrm{\Pi }u$. The definition of the $Q_h^{+}$-norm and the $Q_h$-norm yield $\parallel u-v_h{\parallel }_{Q_h^{+}}^2=\parallel (I-\mathrm{\Pi })u{\parallel }_{{\mathcal{H}}^1}^2+\frac{\sigma }{h}\sum _{(k,l)\in \mathcal{N}}\parallel \text{〚}(I-\mathrm{\Pi })u\text{〛}{\parallel }_{L_2\left(I_{k,l}\right)}^2+\frac{h^2}{{\sigma }^2}|(I-\mathrm{\Pi })u{|}_{{\mathcal{H}}^2}^2.$Using the triangle inequality and Lemma 3.1, we have further $\parallel u-v_h{\parallel }_{Q_h^{+}}^2\lesssim \sum _{k=1}^K\left(\parallel {\hat{w}}_k{\parallel }_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2+\frac{\sigma }{h}\parallel {\hat{w}}_k{\parallel }_{L_2\left(\mathrm{\partial }\hat{\mathrm{\Omega }}\right)}^2+\frac{h^2}{{\sigma }^2}|{\hat{w}}_k{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2\right),$where ${\hat{w}}_k:=(I-{\hat{\mathrm{\Pi }}}_k){\hat{u}}_k$. Using a trace theorem [14, Lemma 4.4] and $AB\le \gamma A^2+{\gamma }^{-1}{B}^2$, we obtain $\parallel {\hat{w}}_k{\parallel }_{L_2\left(\mathrm{\partial }\hat{\mathrm{\Omega }}\right)}^2\lesssim \parallel {\hat{w}}_k{\parallel }_{L_2\left(\hat{\mathrm{\Omega }}\right)}\parallel {\hat{w}}_k{\parallel }_{H^1\left(\hat{\mathrm{\Omega }}\right)}\le h^{-1}\parallel {\hat{w}}_k{\parallel }_{L_2\left(\hat{\mathrm{\Omega }}\right)}^2+h\parallel {\hat{w}}_k{\parallel }_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2.$Using this, $h\lesssim 1$, and $2\le p^2\lesssim \sigma$, we obtain $\parallel u-v_h{\parallel }_{Q_h^{+}}^2\lesssim \sum _{k=1}^K\left(\frac{\sigma }{h^2}\parallel {\hat{w}}_k{\parallel }_{L_2\left(\hat{\mathrm{\Omega }}\right)}^2+\sigma |{\hat{w}}_k{|}_{H^1\left(\hat{\mathrm{\Omega }}\right)}^2+\frac{h^2}{p^2}|{\hat{w}}_k{|}_{H^2\left(\hat{\mathrm{\Omega }}\right)}^2\right).$The desired result is then a consequence of Lemma 4.3 and the assumed equivalence of the norms on the physical domain and the parameter domain.

5. Numerical experiments

We depict the results of this paper with numerical results.

For the first experiments, we choose a spline approximation of the quarter annulus $\mathrm{\Omega }:=\left\{\right(x,y):x,y>0,1<x^2+y^2<4\}$. This domain is uniformly split into $4\times 4$ patches in the obvious way. We solve the Poisson equation $-\mathrm{\Delta }u=200{\pi }^2\sin \left(10\pi x\right)\sin \left(10\pi y\right)\mathrm{i}\mathrm{n}\mathrm{\Omega }\mathrm{a}\mathrm{n}\mathrm{d}u=u^{\ast }\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },$where $u^{\ast }(x,y):=\sin \left(10\pi x\right)\sin \left(10\pi y\right)$ is the exact solution. On each patch, we introduce a coarse discretization space $V_{h_0}$ for $\ell =0$, which only consists of global polynomials of degree p. We then refine all grids $\ell =1,2,\dots$ times uniformly. Next, we modify the discretization spaces in order to obtain non-matching discretizations at the interfaces (since fully matching discretizations would be a special case that would allow a conforming discretization that would not be of interest in a discontinuous Galerkin setting): we refine the grid one additional time for one-third of the patches and we increase the spline degree to p + 1 for another third of the patches.

In Table 1 and Figures 1 and 2, we depict the discretization errors $e_{\ell ,p}$ in the ${\mathcal{H}}^1$-norm relative to that of the solution and the rates $r_{\ell ,p}$, given by $e_{\ell ,p}:=\frac{|u_{h_{\ell },p}-u^{\ast }{|}_{{\mathcal{H}}^1}}{|u^{\ast }{|}_{{\mathcal{H}}^1}}\mathrm{a}\mathrm{n}\mathrm{d}{r}_{\ell ,p}:=\frac{|u_{h_{\ell -1},p}-u^{\ast }{|}_{{\mathcal{H}}^1}}{|u_{h_{\ell },p}-u^{\ast }{|}_{{\mathcal{H}}^1}}$for the case that the smoothness is $C^{p-1}$ on the patches with spline degree p and $C^p$ on the patches with spline degree p + 1 (maximum smoothness). The numerical experiments show that the error decreases like $h_{\ell }\eqsim 2^{-\ell }$ or even faster. Whether or not the error bound depends on $p^2$ or $\ln p$ cannot be seen in this experiment. The almost p-robust convergence of multigrid solvers whose analysis follows from the presented results can be seen in Ref. [15].

Figure 1. Discretization errors for splines of maximum smoothness as function in ℓ.

Figure 2. Discretization errors for splines of maximum smoothness as function in p.

Table 1. Discretization errors in case of maximum smoothness.

	p = 2			p = 3		p = 4		p = 5
ℓ	dofs	$e_{\ell ,p}$	$r_{\ell ,p}$	$e_{\ell ,p}$	$r_{\ell ,p}$	$e_{\ell ,p}$	$r_{\ell ,p}$	dofs	$e_{\ell ,p}$	$r_{\ell ,p}$
2	911	0.7115		0.6170		0.5513		1757	0.5104
3	2855	0.3651	1.9	0.3375	1.8	0.2736	2.0	4229	0.2669	1.9
4	10007	0.0650	5.6	0.0246	13.7	0.0096	28.5	12437	0.0037	72.0
5	37028	0.0112	5.8	0.0016	15.1	2.6e−4	36.3	41909	4.4e−5	84.1
6	144311	0.0026	4.3	1.8e−4	9.3	1.3e−5	20.3	153077	9.8e−7	44.9
7	567095	6.4e−4	4.1	2.1e−5	8.3	7.6e−7	17.1	584309	2.8e−8	35.4
Expected:			4		8		16			32

In Table 2 and Figure 3, we choose $C^0$ smoothness on all patches (minimum smoothness). The grid sizes and the degrees are chosen as for the first experiment. The configurations for which the computation could not be completed due to a lack of memory are indicated by ‘OoM '. We obtain qualitatively the same behavior as for splines of maximum smoothness. We observe that the number of degrees of freedom in case of minimum smoothness is much larger than in the case of minimum smoothness, which outweighs the fact that for the minimally smooth approximation, slightly smaller errors are obtained.

Figure 3. Discretization errors for splines of minimum smoothness as function in ℓ.

Table 2. Discretization errors in case of minimum smoothness.

	p = 2			p = 3		p = 4		p = 5
ℓ	dofs	$e_{\ell ,p}$	$r_{\ell ,p}$	$e_{\ell ,p}$	$r_{\ell ,p}$	$e_{\ell ,p}$	$r_{\ell ,p}$	dofs	$e_{\ell ,p}$	$r_{\ell ,p}$
2	2788	0.4242		0.2292		0.1087		14980	0.0408
3	10724	0.1439	1.9	0.0424	5.4	0.0094	11.6	58916	0.0018	22.7
4	42052	0.0398	5.6	0.0057	7.7	6.5e−4	14.5	233668	6.0e−5	30.2
5	166532	0.0101	5.8	7.3e−4	7.8	4.1e−5	15.7	930692	1.9e−6	31.5
6	662788	0.0025	4.3	9.2e−5	8.0	2.6e−6	15.9	3714820	5.9e−8	31.9
7	2644484	6.4e−4	4.1	1.1e−5	8.0	OoM		14843396	OoM
Expected:			4		8		16			32

In Figure 4, one can observe the dependence of the discretization error on the number of patches. Here, we consider a decomposition of the initial domain into $2^{\iota }\times 2^{\iota }$ patches, where $\iota =1,2,\dots ,6$. We again consider a heterogeneous discretization (different spline degrees, different grid sizes), where we apply $\ell =7-\iota$ uniform refinement steps. So, the grid sizes are constant. We observe that also the discretization errors are rather constant.

Figure 4. Errors for splines of maximum smoothness with $4^{\iota }$ patches and $\ell =7-\iota$.

Finally, we present an experiment that goes beyond the presented theory. Here, we consider the Fichera corner as computational domain. This domain is composed of the patches ${\mathrm{\Omega }}_1:=(0,1{)}^3$, ${\mathrm{\Omega }}_2:=(1,2)\times (0,1)\times (0,1)$, ${\mathrm{\Omega }}_3:=(0,1)\times (1,2)\times (0,1)$, and ${\mathrm{\Omega }}_4:=(0,1)\times (0,1)\times (1,2)$. Each of the patches is parameterized with the appropriately shifted identity function. We solve (39) $-\mathrm{\nabla }\cdot \left(\alpha \mathrm{\nabla }u\right)=27{\pi }^2\alpha \sin \left(3\pi x\right)\sin \left(3\pi y\right)\sin \left(3\pi z\right)\mathrm{i}\mathrm{n}\mathrm{\Omega }\mathrm{a}\mathrm{n}\mathrm{d}u=u^{\ast }\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },$(39) where $u^{\ast }(x,y,z)=\sin \left(3\pi x\right)\sin \left(3\pi y\right)\sin \left(3\pi z\right)$ is the exact solution. We choose $\alpha =100$ on the patch ${\mathrm{\Omega }}_1:=(0,1{)}^3$ and $\alpha =1$ on the remaining patches. The SIPG formulation reads as follows. Find $u\in V_{h,\ast }:=\{v\in V_h:v{|}_{\mathrm{\partial }\mathrm{\Omega }}=u^{\ast }\}$ such that (40) $(u,v{)}_{{\mathcal{H}}^1}-(u,v{)}_{B_h}-(v,u{)}_{B_h}+(u,v{)}_{C_h}=(f,v{)}_{L_2\left(\mathrm{\Omega }\right)}$(40) holds for all $v\in V_{h,0}:=\{v\in V_h:v{|}_{\mathrm{\partial }\mathrm{\Omega }}=0\}$, where (41) $\begin{array}{rl}& (u,v{)}_{{\mathcal{H}}^r}:=\sum _{k=1}^K{\alpha }_k({\mathrm{\nabla }}^{r}u,{\mathrm{\nabla }}^{r}v{)}_{L_2\left({\mathrm{\Omega }}_k\right)}\mathrm{f}\mathrm{o}\mathrm{r}r=1,2,\dots ,\\ & (u,v{)}_{B_h}:=\sum _{(k,l)\in \mathcal{N}}(\text{〚}u\text{〛},\left\{\alpha \mathrm{\nabla }u\right\}\cdot {\mathbf{n}}_k{)}_{L_2\left(I_{k,l}\right)},\mathrm{a}\mathrm{n}\mathrm{d}\\ & (u,v{)}_{C_h}:=\sum _{(k,l)\in \mathcal{N}}\frac{({\alpha }_k+{\alpha }_l)\delta }{2h}(\text{〚}u\text{〛},\text{〚}v\text{〛}{)}_{L_2\left(I_{k,l}\right)}.\end{array}$(41) The discretization spaces on the patches are obtained by $\ell =0,1,2,\dots$ uniform refinement steps, where we use splines of maximum smoothness with degree p + 1 on the patch ${\mathrm{\Omega }}_1$ and with degree p on the remaining patches. This means that, again, for each interface, the intersection of the traces of the local function spaces of the involved spaces only contains global polynomials, so this is again a non-conforming discretization. We present the corresponding results in Table 3. As for the first experiment, we obtain the convergence rates that are slightly better than the expected ones.

Table 3. Discretization errors in case of maximum smoothness in 3D.

	p = 2			p = 3		p = 4		p = 5
ℓ	dofs	$e_{\ell ,p}$	$r_{\ell ,p}$	$e_{\ell ,p}$	$r_{\ell ,p}$	$e_{\ell ,p}$	$r_{\ell ,p}$	dofs	$e_{\ell ,p}$	$r_{\ell ,p}$
0	33	1.0210		1.0180		0.8095		456	0.3298
1	100	0.8180	1.2	0.8530	1.2	0.1746	4.6	793	0.2096	1.6
2	456	0.3623	2.3	0.2310	3.7	0.0602	2.9	1905	0.0415	5.0
3	2728	0.0558	6.5	0.0117	19.7	0.0029	21.1	6553	6.1e−4	68.6
4	18888	0.0118	4.7	0.0011	10.8	1.1e−4	25.5	30921	1.1e−5	55.1
Expected:			4		8		16			32

6. Conclusions and possible extensions

In this paper, we have introduced an SIPG discretization of the Poisson equation in two dimensions which is well posed in the dG-norm, see Theorem 3.3. We have seen that the well-posedness is robust in the grid size and the spline degree. The approximation error estimate presented in Theorem 3.8 shows that the proposed method satisfies the expected approximation order. The approximation error estimate presented in Theorem 3.9 gives an error estimate in the ${\mathcal{H}}^2$-norm, which only grows logarithmically in the spline degree and which is of particular interest for the analysis of iterative solvers.

Remark 6.1

Splines of reduced smoothness

The proofs of Theorems 3.3 and 3.6 (coercivity and boundedness) do not use that the considered spline space is of maximum smoothness. Thus, Theorems 3.4 (existence and uniqueness), 3.5 ( consistency), and 3.7 (discretization error estimate) also apply if splines of reduced smoothness are used. Since the space of splines of maximum smoothness forms a subspace of the spaces of splines of reduced smoothness, the approximation error estimates given in Theorems 3.8 and 3.9 are also valid for spline spaces of reduced smoothness.

Remark 6.2

Three-dimensional domains

The extension of the proposed discretization technique to three-dimensional domains is completely straight-forward. Here, the interfaces $I_{k,l}$ are two-dimensional faces between patches, so the set $\mathcal{N}$ contains the indices of patches that share such faces. The proofs of Lemmas 3.1 and 3.2 and Theorems 3.3 and 3.6 (coercivity and boundedness) apply almost verbatim also to three-dimensional domains. Thus, Theorems 3.4 (existence and uniqueness), 3.5 ( consistency), and 3.7 (discretization error estimate) apply as well. The extension of the approximation error estimates to three dimensions is more involved since the extension of (23(23) ${\hat{\mathrm{\Pi }}}_k={\mathrm{\Pi }}_k^x{\mathrm{\Pi }}_k^y.$(23) ) to ${\hat{\mathrm{\Pi }}}_k:={\mathrm{\Pi }}_k^x{\mathrm{\Pi }}_k^y{\mathrm{\Pi }}_k^z$ yields a projector that is well defined only in $H^3$. Based on this observation, Theorem 3.8 (approximation error estimate) can be extended to three dimensions, provided that $q\ge 2$. For the extension of Theorem 3.8 (low-order approximation error estimate), one would need to construct an $H^2$-stable projectors and appropriate mollifiers, which goes beyond the scope of this paper.

Remark 6.3

Other differential equations

The theory presented in this paper can be extended to other elliptic differential equations whose variational formulation lives in the Sobolev space $H^1$. If the differential equation is parameter-dependent, like (39(39) $-\mathrm{\nabla }\cdot \left(\alpha \mathrm{\nabla }u\right)=27{\pi }^2\alpha \sin \left(3\pi x\right)\sin \left(3\pi y\right)\sin \left(3\pi z\right)\mathrm{i}\mathrm{n}\mathrm{\Omega }\mathrm{a}\mathrm{n}\mathrm{d}u=u^{\ast }\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },$(39) ), a parameter-dependent SIPG formulation can be introduced, cf. (40(40) $(u,v{)}_{{\mathcal{H}}^1}-(u,v{)}_{B_h}-(v,u{)}_{B_h}+(u,v{)}_{C_h}=(f,v{)}_{L_2\left(\mathrm{\Omega }\right)}$(40) )–(41(41) $\begin{array}{rl}& (u,v{)}_{{\mathcal{H}}^r}:=\sum _{k=1}^K{\alpha }_k({\mathrm{\nabla }}^{r}u,{\mathrm{\nabla }}^{r}v{)}_{L_2\left({\mathrm{\Omega }}_k\right)}\mathrm{f}\mathrm{o}\mathrm{r}r=1,2,\dots ,\\ & (u,v{)}_{B_h}:=\sum _{(k,l)\in \mathcal{N}}(\text{〚}u\text{〛},\left\{\alpha \mathrm{\nabla }u\right\}\cdot {\mathbf{n}}_k{)}_{L_2\left(I_{k,l}\right)},\mathrm{a}\mathrm{n}\mathrm{d}\\ & (u,v{)}_{C_h}:=\sum _{(k,l)\in \mathcal{N}}\frac{({\alpha }_k+{\alpha }_l)\delta }{2h}(\text{〚}u\text{〛},\text{〚}v\text{〛}{)}_{L_2\left(I_{k,l}\right)}.\end{array}$(41) ). For the problem (40(40) $(u,v{)}_{{\mathcal{H}}^1}-(u,v{)}_{B_h}-(v,u{)}_{B_h}+(u,v{)}_{C_h}=(f,v{)}_{L_2\left(\mathrm{\Omega }\right)}$(40) )–(41(41) $\begin{array}{rl}& (u,v{)}_{{\mathcal{H}}^r}:=\sum _{k=1}^K{\alpha }_k({\mathrm{\nabla }}^{r}u,{\mathrm{\nabla }}^{r}v{)}_{L_2\left({\mathrm{\Omega }}_k\right)}\mathrm{f}\mathrm{o}\mathrm{r}r=1,2,\dots ,\\ & (u,v{)}_{B_h}:=\sum _{(k,l)\in \mathcal{N}}(\text{〚}u\text{〛},\left\{\alpha \mathrm{\nabla }u\right\}\cdot {\mathbf{n}}_k{)}_{L_2\left(I_{k,l}\right)},\mathrm{a}\mathrm{n}\mathrm{d}\\ & (u,v{)}_{C_h}:=\sum _{(k,l)\in \mathcal{N}}\frac{({\alpha }_k+{\alpha }_l)\delta }{2h}(\text{〚}u\text{〛},\text{〚}v\text{〛}{)}_{L_2\left(I_{k,l}\right)}.\end{array}$(41) ), analogous versions to Theorems 3.3 and 3.6 ( coercivity and boundedness) can be shown for the corresponding parameter-dependent dG-norms, i.e. the norms are given by the combination of (11(11) $\parallel u{\parallel }_{Q_h}^2:=(u,u{)}_{Q_h}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}(u,v{)}_{Q_h}:=(u,v{)}_{{\mathcal{H}}^1\left(\mathrm{\Omega }\right)}+(u,v{)}_{C_h}$(11) ), (16(16) $\parallel u{\parallel }_{Q_h^{+}}^2:=\parallel u{\parallel }_{Q_h}^2+\frac{h^2}{{\sigma }^2}|u{|}_{{\mathcal{H}}^2\left(\mathrm{\Omega }\right)}^2.$(16) ), and (41(41) $\begin{array}{rl}& (u,v{)}_{{\mathcal{H}}^r}:=\sum _{k=1}^K{\alpha }_k({\mathrm{\nabla }}^{r}u,{\mathrm{\nabla }}^{r}v{)}_{L_2\left({\mathrm{\Omega }}_k\right)}\mathrm{f}\mathrm{o}\mathrm{r}r=1,2,\dots ,\\ & (u,v{)}_{B_h}:=\sum _{(k,l)\in \mathcal{N}}(\text{〚}u\text{〛},\left\{\alpha \mathrm{\nabla }u\right\}\cdot {\mathbf{n}}_k{)}_{L_2\left(I_{k,l}\right)},\mathrm{a}\mathrm{n}\mathrm{d}\\ & (u,v{)}_{C_h}:=\sum _{(k,l)\in \mathcal{N}}\frac{({\alpha }_k+{\alpha }_l)\delta }{2h}(\text{〚}u\text{〛},\text{〚}v\text{〛}{)}_{L_2\left(I_{k,l}\right)}.\end{array}$(41) ). Thus, Theorems 3.4 (existence and uniqueness), 3.5 ( consistency), and 3.7 (discretization error estimate) apply as well. Approximation error estimates analogous to Theorems 3.8 ( approximation error estimate) and Theorem 3.8 (low-order approximation error estimate) hold, where the constants depend on $\underset{(k,l)\in \mathcal{N}}{max}{\alpha }_k/{\alpha }_l$ and the ${\mathcal{H}}^r$-norms are parameter-dependent norms as defined in (41(41) $\begin{array}{rl}& (u,v{)}_{{\mathcal{H}}^r}:=\sum _{k=1}^K{\alpha }_k({\mathrm{\nabla }}^{r}u,{\mathrm{\nabla }}^{r}v{)}_{L_2\left({\mathrm{\Omega }}_k\right)}\mathrm{f}\mathrm{o}\mathrm{r}r=1,2,\dots ,\\ & (u,v{)}_{B_h}:=\sum _{(k,l)\in \mathcal{N}}(\text{〚}u\text{〛},\left\{\alpha \mathrm{\nabla }u\right\}\cdot {\mathbf{n}}_k{)}_{L_2\left(I_{k,l}\right)},\mathrm{a}\mathrm{n}\mathrm{d}\\ & (u,v{)}_{C_h}:=\sum _{(k,l)\in \mathcal{N}}\frac{({\alpha }_k+{\alpha }_l)\delta }{2h}(\text{〚}u\text{〛},\text{〚}v\text{〛}{)}_{L_2\left(I_{k,l}\right)}.\end{array}$(41) ).

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No potential conflict of interest was reported by the author.

Additional information

Funding

The author was supported by the Austrian Science Fund (FWF) [S117 and P31048] and by the bilateral project DNTS-Austria 01/3/2017 (WTZ BG 03/2017), funded by Bulgarian National Science Fund and OeAD-GmbH (Austria).