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Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 5
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Research Article

Discretization error estimates for discontinuous Galerkin isogeometric analysis

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Pages 1439-1462 | Received 03 Feb 2021, Accepted 19 Sep 2021, Published online: 01 Oct 2021

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.

2010 Mathematics Subject Classifications:

1. Introduction

The original design goal of isogeometric analysis (IgA) [Citation1] 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. [Citation2] 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. [Citation3, Citation4], particularly the symmetric interior penalty discontinuous Galerkin (SIPG) method [Citation5]. The idea of applying this technique to couple patches in IgA has been previously discussed in Refs. [Citation6–8].

Concerning the approximation error, in early IgA literature, only its dependence on the grid size has been studied, cf. Refs. [Citation1, Citation9]. In recent publications [Citation10–13] also the dependence on the spline degree has been investigated. These error estimates are restricted to the single-patch case. In Ref. [Citation14], the results from Ref. [Citation13] 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. [Citation8]. 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 p2 for the SIPG method to be well posed. If the solution is sufficiently smooth, the error in the energy norm decays like hp. 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. (Equation21). These error estimates have recently been used to analyze multigrid solvers for discontinuous Galerkin multipatch discretizations, see Ref. [Citation15]. They might also be helpful for the analysis of Finite Element Tearing and Interconnecting (FETI) solvers, cf. Refs. [Citation6, Citation16], Balancing Domain Decomposition by Constraints (BDDC) solvers, cf. Refs. [Citation17–19] and references therein, Schwarz type solvers, cf. Ref. [Citation20] 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 ΩR2 be an open and simply connected Lipschitz domain. For any given source function fL2(Ω), we are interested in the function uH1,(Ω):=H1(Ω)L2(Ω) solving (1) (u,v)L2(Ω)=(f,v)L2(Ω)for all vH1,(Ω).(1) Here and in what follows, for any rN:={1,2,3,}, L2(Ω) and Hr(Ω) are the standard Lebesgue and Sobolev spaces with standard scalar products (,)L2(Ω), (,)Hr(Ω):=(r,r)L2(Ω), norms L2(Ω) and Hr(Ω), and seminorms ||Hr(Ω). The Lebesgue space of function with zero mean is given by L2(Ω):={vL2(Ω):(v,1)L2(Ω)=0}.

The computational domain Ω is the union of K non-overlapping open patches Ωk, i.e. (2) Ω¯=k=1KΩk¯andΩkΩl=set for any kl(2) holds, where T¯ denotes the closure of T. Each patch Ωk is represented by a bijective geometry function Gk:Ωˆ:=(0,1)2Ωk:=Gk(Ωˆ)R2,which can be continuously extended to the closure of Ωˆ such that Gk(Ωˆ¯)=Ωk¯. We use the notation vk:=v|Ωkandvˆk:=vkGkfor any function v on Ω. If vH1(Ω), we can use standard trace theorems to extend vk to Ωk¯ and to extend vˆk to Ωˆ¯.

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 Ωk and Ωl with kl, the intersection ΩkΩl is either (a) empty, (b) a common vertex or (c) a common edge Ik,l=Il,k such that (3) Iˆk,l:=Ik,lGkIˆ:={{0}×(0,1),{1}×(0,1),(0,1)×{0},(0,1)×{1}}.(3)

Note that the pre-images Iˆk,l and Iˆl,k do not necessarily agree. We define N(k):={l{1,,K} : Ωk and Ωl have common edge},N:={(k,l):k<l and lN(k)}, and N:={(k,l):k>l and lN(k)}, and the parameterization γk,l:(0,1)Iˆk,l via (4) γk,l(t):=(t,s)if Iˆk,l=(0,1)×{s},s{0,1},(s,t)if Iˆk,l={s}×(0,1),s{0,1}.(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)N and t(0,1), we have γk,l(t)=Gk1Glγl,k(t)orγk,l(t)=Gk1Glγl,k(1t).

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 Ωk and Ωl, sharing the patch Ik,l=Gk((0,1)×{0})=Gl((0,1)×{0}). Using G~k(x,y):=Gk(yx+(1y)ρ(x),y),where (ρ(t),0):=Gk1Gl(t,0)if Gk(s,0)=Gl(s,0) for s{0,1},Gk1Gl(1t,0)if Gk(s,0)=Gl(1s,0) for s{0,1},we obtain a reparameterization of Gk, which (a) matches the parameterization of Ωl at the interface, (b) is unchanged on the other interfaces, and (c) keeps the patch Ωk 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 CG>0 such that the geometry functions Gk satisfy the estimates (5) supxΩˆrGk(x)2CGandsupxΩˆ(rGk(x))12CG(5) for r = 1, 2.

We assume full elliptic regularity.

Assumption 2.4

The solution u of the model problem (Equation1) satisfies ukH2(Ωk)for all k=1,,K.

For domains Ω with a sufficiently smooth boundary, cf. Ref. [Citation21], and for convex polygonal domains Ω, cf. Refs. [Citation22, Citation23], we have uH2(Ω) and thus also Assumption 2.4. In case of varying diffusion conditions (which are uniform on each patch), we might have uH2(Ω), but Assumption 2.4 might still be satisfied, cf. Refs. [Citation24, Citation25] and others. The theory of this paper can be extended to cases where we only know ukH3/2+ϵ(Ωk) 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. [Citation7, Citation8], we use a conforming isogeometric discretization for each patch and couple the contributions for the patches using a SIPG method, cf. Ref. [Citation5], as follows.

For the univariate case, the space of spline functions of degree pN over a grid (vector of breakpoints) Z:=(ζ0,ζ1,,ζn1,ζn) with ζ=0 and ζn=1 and size h:=maxi=0,,n1|ζi+1ζi| is given by Sp,Ξ(0,1):=vCp1(0,1):v|(ζi,ζi+1]Pp for all i=1,,n1,where Pp is the space of polynomials of degree p.

On the parameter domain Ωˆ:=(0,1)2, we introduce tensor-product B-spline functions Vˆk:=Spk,Zk,1(0,1)Spk,Zk,2(0,1).The multipatch function space Vh is given by (6) Vh:={uhL2(Ω) : uhGkVˆk for k=1,,K,}.(6) Note that the grid sizes hk and the spline degrees pk can be different for each of the patches. We define p:=maxk{1,,K}pk,pmin:=mink{1,,K}pk,h:=maxδ{1,2}maxk{1,,K}hk,δto be the largest spline degree, the smallest spline degree, and the grid size. We assume (7) pmin2andminδ{1,2}mink{1,,K}hk,δ,minChh(7) for some constant Ch>0, where hk,δ,min refers to the smallest knot span.

Following the assumption that uh is a patchwise function, we define for each rN, a broken Sobolev space Hr(Ω):={vL2(Ω):vkHr(Ω)}with associated norms and scalar products vHr(Ω):=(v,v)Hr(Ω)1/2and(u,v)Hr(Ω):=k=1K(u,v)Hr(Ωk).For each patch, we define on its boundary Ωk, the outer normal vector nk. On each interface Ik,l, we define the jump operator by v:=vkvlon Ik,l=Il,k where (k,l)Nand the average operator {} by {v}:=12(vk+vl)on Ik,l=Il,k where (k,l)N.The discretization of the variational problem using the SIPG method reads as follows. Find uhVh such that (8) (uh,vh)Ah=(f,vh)L2(Ω)for all vhVh,(8) where (u,v)Ah:=(u,v)H1(Ω)(u,v)Bh(v,u)Bh+(u,v)Ch,(u,v)Bh:=(k,l)N(u,{v}nk)L2(Ik,l),(u,v)Ch:=σh(k,l)N(u,v)L2(Ik,l)for all u,vH2,(Ω), where the penalty parameter (9) σ=σ0p2>0(9) is chosen sufficiently large.

Using a basis for the space Vh, we obtain a standard matrix-vector problem: find u_hRN such that (10) Ahu_h=f_h.(10) Here and in what follows, u_h=[ui]i=1N is the coefficient vector representing uh with respect to the chosen basis, i.e. uh=i=1Nuiφi, and f_h=[(f,φi)L2(Ω)]i=1N 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 CG, CI, and Ch. 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 CG, CI, and Ch.

We use the notation ab if there is a generic constant c>0 such that acb and the notation ab if ab and ba.

For symmetric positive definite matrices A and B, we write ABifv_hAv_hv_hBv_hfor all vectors v_h.The notations AB and AB are defined analogously.

3. A discretization error estimate

In Ref. [Citation7], it has been shown that the bilinear form (,)Ah 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) uQh2:=(u,u)Qhwhere(u,v)Qh:=(u,v)H1(Ω)+(u,v)Ch(11) for all u,vH2,(Ω). Note that we define the norm differently to Ref. [Citation7], 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) vGk1Hr(Ωk)vHr(Ωˆ)for all vHr(Ωˆ),(12) where r = 0, 1, 2. For ease of notation, here and in what follows, we define H0:=L2. If (Equation5) holds for r=1,2,,s with some s>r, then (Equation12) also holds for those choices of r. Moreover, we have vGk1L2(Ik,l)vL2(Iˆk,l)for all vH1(Ωˆ).

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 (vGk1)nkL2(Ik,l)vL2(Iˆk,l) for all vH2(Ωˆ).

Proof.

We have (vGk1)nkL2(Ik,l)vGk1L2(Ik,l)nkL(Ik,l), where certainly nkL(Ik,l)=1 because the length of nk is always 1. The estimate vGk1L2(Ik,l)vL2(Iˆk,l) 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 (,)Ah is coercive and bounded, i.e. a scalar product.

Theorem 3.3

Coercivity and boundedness

There is some σ0>0 that only depends on CG, CI, and Ch such that (uh,uh)AhuhQh2and(uh,vh)AhuhQhvhQhhold for all uh,vhVh and all σp2σ0.

Proof.

Note that (uh,vh)Ah=(uh,vh)Qh(uh,vh)Bh(vh,uh)Bh. Using Lemma 3.2, [Citation14, Lemma 4.4], [Citation26, Corollary 3.94], and Lemma 3.1, we obtain (13) vhnkL2(Ik,l)2(vhGk)nkL2(Iˆk,l)2vhGkH2(Ωˆ)vhGkH1(Ωˆ)p2hvhGkH1(Ωˆ)2p2hvhH1(Ωk)2(13) for all vhVh, k=1,,K, and lN(k). As VhL2(Ω), the Poincaré inequality (see, e.g. Ref. [Citation26, Theorem A.25]) yields also vhnkL2(Ik,l)2p2h|vh|H1(Ωk)2.The Cauchy–Schwarz inequality, the triangle inequality, (Equation13), and |N(k)|4 yield (14) |(uh,vh)Bh|(k,l)NuhL2(Ik,l)21/2(k,l)N{vh}nkL2(Ik,l)21/2(k,l)NuhL2(Ik,l)21/2k=1KjN(k)vhnkL2(Ik,l)21/2p2h1/2(k,l)NuhL2(Ik,l)21/2k=1K|vh|H1(Ωk)21/2pσ1/2uhQhvhQh(14) for all uh,vhVh. Let c01 be the hidden constant, i.e. such that (15) |(uh,vh)Bh|c0pσ1/2uhQhvhQh.(15) For σ16c0p2, we obtain (uh,uh)Ah=uhQh22(uh,uh)Bh12uhQh2,i.e. coercivity. Using (Equation14) and the Cauchy–Schwarz inequality, we obtain further (uh,vh)Ah=(uh,vh)Qh(uh,vh)Bh(vh,uh)Bh32uhQhvhQh,i.e. boundedness.

As we have boundedness and coercivity (Theorem 3.3), the Lax Milgram theorem (see, e.g. Ref. [Citation26, 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 (Equation8) has exactly one solution uhVh.

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

Theorem 3.5

Consistency

The solution uH1,(Ω)H2(Ω) of the original problem (Equation1) satisfies (u,vh)Ah=(f,vh)L2(Ω)for all vhVh.

For a proof, see, e.g. Ref. [Citation4, Proposition 2.9]; the proof requires elliptic regularity (cf. Assumption 2.4).

If boundedness of the bilinear form (,)Ah was also satisfied for uH2,(Ω), Ceá's Lemma (see, e.g. Ref. [Citation26, Theorem 2.19.iii]) would allow to bound the discretization error. However, the bilinear form is not bounded in the norm Qh, but only in the stronger norm Qh+, given by (16) uQh+2:=uQh2+h2σ2|u|H2(Ω)2.(16)

Theorem 3.6

If σ is chosen as in Theorem 3.3, (u,vh)AhuQh+vhQhholds for all uH2,(Ω) and all vhVh.

Proof.

Let uH2,(Ω) and vhVh be arbitrarily but fixed. Note that the arguments from (Equation14) also hold if the first parameter of the bilinear form (,)Bh is not in Vh. So, we obtain |(u,vh)Bh|14uQhvhQh.Using Lemma 3.2, [Citation14, Lemma 4.4], Lemma 3.1, and the Poincaré inequality, we obtain (17) vnkL2(Ik,l)2(vGk)nkL2(Iˆk,l)2vGkH2(Ωˆ)vGkH1(Ωˆ)vH2(Ωk)vH1(Ωk)1βvH2(Ωk)2+βvH1(Ωk)21β|v|H2(Ωk)2+β|v|H1(Ωk)2(17) for all vH2(Ωk), all k=1,,K, all lN(k), and all β>1. Using this estimate, and |N(k)|4, we obtain for β:=h2σ |(vh,u)Bh|σh(k,l)NvhL2(Ik,l)21/2hσk=1KlN(k)unkL2(Ik,l)21/2vhQhk=1K|u|H1(Ωk)2+h2σ2k=1K|u|H2(Ωk)21/2vhQhuQh+.Using these estimates, we obtain (u,vh)Ah=(u,vh)Qh(u,vh)Bh(vh,u)BhuQh+vhQh,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 uuhQhinfvhVhuvhQh+holds, where u is the solution of the original problem (Equation1) and uh is the solution of the discrete problem (Equation8).

Proof.

For any vhVh, the triangle inequality yields (18) uuhQhuvhQh+vhuhQh.(18) Theorem 3.5 and Galerkin orthogonality yield (uuh,wh)Ah=0 for all whVh. So, we obtain using Theorems 3.3 and 3.6 that vhuhQh2(vhuh,vhuh)Ah=(vhu,vhuh)AhvhuQh+vhuhQh,which shows vhuhQhuvhQh+. Together with (Equation18), this shows uuhQhuvhQh+. Since this holds for all vhVh, this finishes the proof.

Theorem 3.8

Approximation error estimate

Let q{1,,pmin}. Provided that σ is as in Theorem 3.3 and that ukHq+1(Ωk)uˆkHq+1(Ωˆ) for k=1,,K (cf. Lemma 3.1), then (19) infvhVhuvhQh+σ1/2πqhquHq+1(Ω)(19) holds for all uH1,(Ω)Hq+1(Ω).

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

Assuming ppmin, then we have for the case q=pmin that uuhQhinfvhVhuvhQh+σ01/2q2πq<1hquHq+1(Ω).For the analysis of linear solvers, like multigrid solvers [Citation15], 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 H2-regularity is usually the best we can hope for. For this case, we obtain infvhVhuvhQh+σ01/2p2huH2(Ω).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) infvhVhuvhQh+(lnσ)2σ1/(2pmin1)h|u|H2(Ω)(20) holds for all uH1,(Ω)H2(Ω).

The proof is given at the end of the next section. Assuming again ppmin2, we obtain (21) uuhQhinfvhVhuvhQh+σ01/2(lnp)2h|u|H2(Ω),(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. [Citation14, Sections 3.1 and 3.2]. Let Πp,Z be the HD1(0,1)-orthogonal projection into Sp,Z(0,1), where (u,v)HD1(0,1)=(u,v)L2(0,1)+u(0)v(0).In what follows, we also write Sp,h(0,1) and Πp,h if we refer to a uniform grid of size h. Takacs [Citation14, Lemma 3.1] states that (Πp,Zu)(0)=u(0) and (Πp,Zu)(1)=u(1). Using 0=(uΠp,Zu,x2)HD1(0,1)=2((uΠp,Zu),x)L2(0,1)=2(uΠp,Zu,1)L2(0,1)+u(1)(Πp,Zu)(1), we obtain for p2 and uH1(0,1) that (22) (uΠp,Zu,1)L2(0,1)=0.(22) The next step is to consider the multivariate case, more precisely the parameter domain Ωˆ=(0,1)2. Let Πkx:H2(Ωˆ)H2(Ωˆ) and Πky:H2(Ωˆ)H2(Ωˆ) be given by (Πkxu)(x,y)=(Πpk,Zk,1u(,y))(x)and(Πkyu)(x,y)=(Πpk,Zk,2u(x,))(y)and let Πˆk:H2(Ωˆ)Spk,Zk,1(Ωˆ) be such that (23) Πˆk=ΠkxΠky.(23) For the physical domain, define Π:H1,(Ω)H2(Ω)Vh to be such that (Πv)|Ωk=(Πˆk(vGk))Gk1for all vH2(Ω) and k=1,,K.Observe that we obtain using (Equation22) that (24) (uΠˆku,1)L2(Ωˆ)=0,Πˆkc=c,andΠc=c(24) for all cR.

The projectors Πˆk satisfy robust error estimates and are almost stable in H2.

Lemma 4.1

Let Z be a grid of size h, p{2,3,}, and r{1,2,,p}. Then, (IΠp,Z)uL2(0,1)πr1hr+1|u|Hr+1(0,1) and|(IΠp,Z)u|H1(0,1)πrhr|u|Hr+1(0,1)hold for all uHr+1(0,1).

Proof.

The identity (Equation22) implies that the projector Πp,Z coincides with the projector Qp1 from Ref. [Citation12, Equations (3.8) and (3.9)]. Thus, the desired result follows from Ref. [Citation12, Theorem 3.1].

Lemma 4.2

Let Z be a quasi-uniform grid of size h, p{2,3,}, and r{1,2,,p}. Then, |(IΠp,Z)u|H2(0,1)p2πr+1hr1|u|Hr+1(0,1)holds for all uHr+1(0,1).

Proof.

The proof is analogous to the proof of Ref. [Citation27, Theorem 4]. Let Qp2 be the H2-orthogonal projector into Sp,Z(0,1) as introduced in Ref. [Citation12]. Using the triangle inequality and a standard inverse estimate [Citation26, Corollary 3.94], we obtain |(IΠp,Z)u|H2(0,1)|(IQp2)u|H2(0,1)+|(Qp2Πp,Z)u|H2(0,1)|(IQp2)u|H2(0,1)+h1p2|(Qp2Πp,Z)u|H1(0,1)|(IQp2)u|H2(0,1)+h1p2|(IQp2)u|H1(0,1)+h1p2|(IΠp,Z)u|H1(0,1).Thus, the desired result follows from Ref. [Citation12, Theorem 3.1].

Lemma 4.3

Let r{1,2,,pmin}. The estimates (IΠˆk)uL2(Ωˆ)πr1hkr+1|u|Hr+1(Ωˆ),|(IΠˆk)u|H1(Ωˆ)πrhkr|u|Hr+1(Ωˆ) and|(IΠˆk)u|H2(Ωˆ)p2πr+1hkr1|u|Hr+1(Ωˆ)hold for all uHr+1(Ωˆ).

Proof.

The proof is based on the univariate estimates given in Lemmas 4.1 and 4.2 (pmin2 and the quasi-uniformity of the grids have been required in (Equation7)) and follows the standard construction that can be found, e.g. in the proof of Ref. [Citation14, Theorem 3.3].

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

Lemma 4.4

Let r{1,2,,pmin} and (k,l)NN. Then, the estimate (IΠˆk)uL2(Iˆk,l)πrhr|u|Hr(Iˆk,l) holds for all uH2(Ωˆ)Hr(Iˆk,l).

Proof.

Without loss of generality, we assume Iˆk,l=(0,1)×{0}. Because of Ref. [Citation14, Theorem 3.4], we know that ((IΠˆk)u)(,0)=(IΠpk,Zk,1)(u(,0)). So, we have (IΠˆk)uL2(Iˆk,l)2=(IΠpk,Zk,1)u(,0)L2(0,1)2.Using Refs. [Citation14, Equation (3.4)], [Citation28, Lemma 8] and that Πpk,Zk,1 minimizes the H1-seminorm, we further obtain (IΠˆk)uL2(Iˆk,l)2h2infvhSp,h(0,1)|u(,0)vh|H1(0,1)2.Sande et al. [Citation12, 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)N, let Υk,l be given by Υk,lv:=vγk,l. For (k,l)N, we define Υk,lv:=vGk1Glγl,k, i.e. we have (Υk,lv)(t)=v(γl,k(t)) or (Υk,lv)(t)=v(γl,k(1t)), cf. Assumption 2.2. For all cases, Υk,l is a bijective function Hs(0,1)Hs(Iˆk,l) and (25) |u|Hs(0,1)|Υk,lu|Hs(Iˆk,l)(25) holds for all s. For vHs(Ωˆ), we define the abbreviated notation Υk,l1v:=Υk,l1(v|Iˆk,l) and observe Υk,l1uH3/2(0,1)for alluH2(Ωˆ).For (k,l)NN, we define extension operators Ξk,l:Hs(Iˆk,l)Hs(Ωˆ) by (Ξk,lw)(x,y):=ϕ(x)w(0,y)if Iˆk,l={0}×(0,1),ϕ(1x)w(1,y)if Iˆk,l={1}×(0,1),ϕ(y)w(x,0)if Iˆk,l=(0,1)×{0},ϕ(1y)w(x,1)if Iˆk,l=(0,1)×{1},where (26) ϕ(x):=max{0,1η1x}andη(0,1).(26) Now, define for each patch Ωk, a mollifying operator Mˆk by (27) Mˆk:=IlN(k)Ξk,lΥk,l(IΠr,η)Υk,l1.(27) The combination of the patch local operators yields a global operator M: (28) (Mu)|Ωk:=(Mˆk(uGk))Gk1.(28) Observe that M preserves constants, i.e. (29) Mc=cfor allcR.(29)

Lemma 4.5

For all (k,l)NN and all uH01(Iˆk,l):={uH1(Iˆk,l):u=0 on Iˆk,l}, we have Ξk,lu=0on ΩˆIˆk,landΞk,lu=u on Iˆk,l.

Proof.

Assume without loss of generality that Iˆk,l={0}×(0,1). For this case, we have (Ξk,lu)(x,y)=ϕ(x)u(0,y).As uH01(Iˆk,l), we obtain u(0,0)=u(0,1)=0. This shows the first statement for the two boundary segments adjacent to Iˆk,l, i.e. [0,1]×{0} and [0,1]×{1}. Since η<1 yields ϕ(1)=0, we also have the first statement for the boundary segment {1}×(0,1). This finishes the proof for the first statement. The proof for the second statement follows directly from ϕ(0)=1.

Lemma 4.6

Υk,l1Mˆk=Πr,ηΥk,l1 holds for all (k,l)NN.

Proof.

Equation (Equation27) implies Υk,l1Mˆk=Υk,l1jN(k)Υk,l1Ξk,jΥk,j(IΠr,η)Υk,j1. Observe that the projector Πr,η is interpolatory on the boundary [Citation14, Lemma 3.1]. So, (IΠr,η) maps into H01(0,1) and Υk,j(IΠr,η) maps into H01(Iˆk,j). Therefore, Lemma 4.5 yields Υk,l1Mˆk=Υk,l1Υk,l1Υk,l(IΠr,η)Υk,l1, which immediately implies the desired result.

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

Lemma 4.7

The estimate Ψ(u):=infvH1(Iˆk,l)uvL2(Iˆk,l)2+θ2|v|H1(Iˆk,l)2θ|u|H1(Ωˆ)2holds for all uH1(Ωˆ) and (k,l)NN and all θ>0.

Proof.

A trace theorem [Citation14, Lemma 4.4] yields (30) Ψ(u)infvH2(Ωˆ)uvL2(Ωˆ)|uv|H1(Ωˆ)+θ2|v|H1(Ωˆ)|v|H2(Ωˆ).(30) Case 1. Assume θ<1. In this case, we choose v to be the H1-orthogonal projection of u into S3,θ11(Ωˆ). Since the spline degree of that space is fixed, we obtain using a standard inverse inequality [Citation26, Corollary 3.94] and a standard approximation error estimate (like from Ref. [Citation13]) that Ψ(u)(θ11+θ2θ1)|v|H1(Ωˆ)2θ|v|H1(Ωˆ)2.Case 2. Assume θ1. In this case, we choose v:=(u,1)L2(Ω) and obtain from (Equation30) directly Ψ(u)uvH1(Ωˆ)|u|H1(Ωˆ).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 |Mˆkuˆk|Hr(Iˆk,l)2(23r2η1)2r3r2|uˆk|H2(Ωˆ)2 holds for all uˆkHr(Ωˆ) and all (k,l)NN.

Proof.

We have using (Equation25) and Lemma 4.6 |Mˆkuˆk|Hr(Iˆk,l)2|Υk,l1Mˆkuˆk|Hr(0,1)2=|Πr,ηΥk,l1uˆk|Hr(0,1)2.Now, a standard inverse estimate [Citation26, Corollary 3.94] yields |Mˆkuˆk|Hr(Iˆk,l)2ψ2(rs)|Πr,ηΥk,l1uˆk|Hs(0,1)2for s{1,2},where ψ:=23r2η1. Lemma 4.2 and the H1-stability of Πr,η yield |Πr,ηw|H2(0,1)2r4|w|H2(0,1)2and |Πr,ηw|H1(0,1)2|w|H1(0,1)2,so we obtain |Mˆkuˆk|Hr(Iˆk,l)2ψ2r2infvH2(0,1)|Υk,l1uˆkv|H1(0,1)2+ψ2r4|v|H2(0,1)2.Using (Equation25), we obtain |Mˆkuˆk|Hr(Iˆk,l)2ψ2r2infvH2(Iˆk,l)|uˆkv|H1(Iˆk,l)2+ψ2r4|v|H2(Iˆk,l)2.By applying Lemma 4.7 to the derivative of uˆk, we obtain the desired result.

Lemma 4.9

(IM)uL2(Ik,l)=0 holds for all uH1,(Ω)H2(Ω) and (k,l)N.

Proof.

Let uH1,(Ω)H2(Ω) be arbitrary but fixed.

We obtain using the definition of Υk,l and Υl,k and Lemma 3.1 that (31) wL2(Ik,l)=wkwlL2(Ik,l)wˆkwˆlGl1GkL2(Iˆk,l)=Υk,l1(wˆkwˆlGl1Gk)L2(0,1)=Υk,l1wˆkΥl,k1wˆlL2(0,1)(31) holds, where wˆk:=wkGk and wˆl:=wlGl. Since uH1(Ω), a standard trace theorem yields (32) Υk,l1uˆkΥl,k1uˆl=0.(32) Thus, (Equation31) implies uL2(Ik,l)=0. By plugging Mu into (Equation31), we obtain using Lemma 4.6 MuL2(Ik,l)Πr,η(Υk,l1uˆkΥl,k1uˆl)L2(0,1).Using (Equation32) and Πr,η0=0, we obtain MuL2(Ik,l)=0 and consequently also (IM)uL2(Ik,l)=0.

Lemma 4.10

The estimate Πˆk(IMˆk)uH1,(Ωˆ)2(1+η2h2)h2|u|H2(Ωˆ)2 holds for all uH2(Ωˆ) and k=1,,K.

Proof.

Using the definition of Mˆk and of the H1,-norm, we obtain (33) Πˆk(IMˆk)uH1,(Ωˆ)lN(k)ΠˆkΞk,lΥk,l(IΠr,η)Υk,l1uH1,(Ωˆ)lN(k)(Ψx,l+Ψy,l+Ψ,l),(33) where Ψ,l:=ΠˆkΞk,lΥk,l(IΠr,η)Υk,l1uL2(Ωˆ) for {x,y} and Ψ,l:=(ΠˆkΞk,lΥk,l(IΠr,η)Υk,l1u,1)L2(Ωˆ). We estimate the terms Ψx,l, Ψy,l, and Ψ,l separately. Let without loss of generality Iˆk,l={0}×(0,1).

Step 1. Using (Equation23) and the H1-stability of the H1,D-orthogonal projection, and w:=Υk,l(IΠr,η)Υk,l1u, we obtain Ψx,l2=xΠkxΠkyΞk,lwL2(Ωˆ)2xΠkyΞk,lwL2(Ωˆ)2=0101ϕ(x)(Πpk,Zk,2w(0,))(y)2dxdy=|ϕ|H1(0,1)2Πpk,Zk,2(w(0,))L2(0,1)2Ψˆx,l2:=η1Πpk,Zk,2(w(0,))L2(0,1)2,where we use |ϕ|H1(0,1)2η1. The triangle inequality yields Ψx,l2Ψˆx,l2η1w(0,)L2(0,1)2+η1(IΠpk,Zk,2)(w(0,))L2(0,1)2.Lemma 4.1 yields Ψx,l2Ψˆx,l2η1w(0,)L2(0,1)2+η1h2|w(0,)|H1(0,1)2.The definition of w and (Equation25) yield Ψx,l2Ψˆx,l2η1(IΠr,η)Υk,l1uL2(0,1)2+η1h2|(IΠr,η)Υk,l1u|H1(0,1)2.Lemma 4.1 yields Ψx,l2Ψˆx,l2(η+η1h2)infvH2(0,1)|Υk,l1uv|H1(0,1)2+η2|v|H2(0,1)2. Equation (Equation25) yields further Ψx,l2Ψˆx,l2(η+η1h2)infvH2(Iˆk,l)|uv|H1(Iˆk,l)2+η2|v|H2(Iˆk,l)2.Now, Lemma 4.7 applied to the derivative of u yields (34) Ψx,l2Ψˆx,l2(η+η1h2)η|u|H2(Ωˆ)2=(1+η2h2)h2|u|H2(Ωˆ)2.(34) Step 2. Using (Equation23) and the H1-stability of the H1,D-orthogonal projection and w:=Υk,l(IΠr,η)Υk,l1u, we obtain Ψy,l2=yΠkxΠkyΞk,lwL2(Ωˆ)2yΠkxΞk,lwL2(Ωˆ)2=0101(Πpk,Zk,1ϕ)(x)yw(0,y)2dxdyΠpk,Zk,1ϕL2(0,1)2|w|H1(Iˆk,l)2.Using Πpk,Zk,1ϕL2(0,1)2ϕL2(0,1)2+(IΠpk,Zk,1)ϕL2(0,1)2ϕL2(0,1)2+h2|ϕ|H1(0,1)2η+h2η1=(η2h2+1)h2η1 and the definition of w, we obtain Ψy,l2(η2h2+1)h2η1|Υk,l(IΠr,η)Υk,l1u|H1(Iˆk,l)2.Using (Equation25), we obtain further Ψy,l2(η2h2+1)h2η1|(IΠr,η)Υk,l1u|H1(0,1)2.Using the H1-stability of Πr,η and the approximation error estimate [Citation14, Theorem 3.1], we obtain Ψy,l2(η2h2+1)h2infvH2(0,1)η1|Υk,l1uv|H1(0,1)2+η|v|H2(0,1)2.Using (Equation25) and Lemma 4.7 applied to the derivative of Υk,l1u, we obtain (35) Ψy,l2(η2h2+1)h2infvH2(Iˆk,l)η1|uv|H1(Iˆk,l)2+η|v|H2(Iˆk,l)2(η2h2+1)h2|u|H2(Ωˆ)2.(35) Step 3. Using w:=Υk,l(IΠr,η)Υk,l1u and (Equation22), we obtain Ψ,l2=(ΠkxΠkyΞk,lw,1)L2(Ωˆ)2=0101(Πpk,Zk,1ϕ)(x)(Πpk,Zk,2w(0,))(y)dxdy=01(Πpk,Zk,1ϕ)(x)dx01(Πpk,Zk,2w(0,))(y)dy=01ϕ(x)dx01(Πpk,Zk,2w(0,))(y)dyη2Πpk,Zk,2w(0,)L2(0,1)2=η22Ψˆx,l2.Using (Equation34), and using η1, we further obtain (36) Ψ,l2(1+η2h2)h2|u|H2(Ωˆ)2.(36) Concluding step. The combination of (Equation33)–(Equation36) yields the desired result.

4.3. The approximation error estimate

The following three lemmas give approximation error estimates (Equation20) for the choice uh:=ΠMu separately for the individual parts of Qh+.

Lemma 4.11

|(IΠM)u|H1(Ω)2(1+η2h2)h2|u|H2(Ω)2 holds for all uH1,(Ω)H2(Ω).

Proof.

First note that the Poincaré inequality yields (37) H1(Ωˆ)2H1,(Ωˆ)2:=||H1(Ωˆ)2+(,1)L2(Ωˆ)2.(37) Let uH1,(Ω)H2(Ω) be arbitrary but fixed and let uˆk:=uGk. Using Lemma 3.1, the triangle inequality, (Equation37), and (Equation24), we obtain |(IΠM)u|H1(Ω)2k=1K|(IΠˆk)uˆk|H1(Ωˆ)2+k=1KΠˆk(IMˆk)uˆkH1,(Ωˆ)2.We further obtain using Lemmas 4.3 and 4.10, |(IΠM)u|H1(Ω)2h2k=1K|uˆk|H2(Ωˆ)2+k=1K(1+η2h2)h2uˆkH2(Ωˆ)2.Lemma 3.1, (Equation24), (Equation29), and the Poincaré inequality finish the proof.

Lemma 4.12

|(IΠM)u|H2(Ω)2(1+η2h2)p4|u|H2(Ω)2 holds for all uH1,(Ω)H2(Ω).

Proof.

Using Lemma 3.1, the triangle inequality, (Equation37), (Equation24), and a standard inverse inequality ([Citation26, Corollary 3.94]), we obtain |(IΠM)u|H2(Ω)2k=1K(IΠˆkMˆk)uˆkH2(Ωˆ)2k=1K(IΠˆk)uˆkH2(Ωˆ)2+k=1KΠˆk(IMˆk)uˆkH2(Ωˆ)2k=1K|(IΠˆk)uˆk|H2(Ωˆ)2+p4h2k=1KΠˆk(IMˆk)uˆkH1,(Ωˆ)2.By again applying Lemmas 4.3 and 4.10, we obtain |(IΠM)u|H2(Ω)2p2k=1K|uˆk|H2(Ωˆ)2+p4(1+η2h2)k=1KuˆkH2(Ωˆ)2p4(1+η2h2)k=1KuˆkH2(Ωˆ)2.Lemma 3.1, (Equation24), (Equation29) and the Poincaré inequality finish the proof.

Lemma 4.13

(k,l)I(IΠM)uL2(Ik,l)2π2rh2r(23r2η1)2r3r2|u|H2(Ω)2 holds for all uH1,(Ω)H2(Ω).

Proof.

Let uH1,(Ω)H2(Ω) be arbitrary but fixed and let uˆk:=uGk. Observe that the triangle inequality, Lemmas 4.9 and 3.1 yield (k,l)N(IΠM)uL2(Ik,l)2(k,l)N(IΠ)MuL2(Ik,l)2(k,l)NN((IΠ)Mu)|ΩkL2(Ik,l)2(k,l)NN(IΠˆk)MˆkuˆkL2(Iˆk,l)2.Lemma 4.4 yields (k,l)I(IΠM)uL2(Ik,l)2π2rh2r(k,l)NN|Mˆkuˆk|Hr(Iˆk,l)2.Since Assumption 2.2 yields |Mˆkuˆk|Hr(Iˆk,l)2|Mˆluˆl|Hr(Iˆl,k)2, we obtain |Mˆkuˆk|Hr(Iˆk,l)2+|Mˆluˆl|Hr(Iˆl,k)2|Mˆkuˆk|Hr(Iˆk,l)2+|Mˆluˆl|Hr(Iˆl,k)2and therefore also (k,l)I(IΠM)uL2(Ik,l)2π2rh2r(k,l)NN|Mˆkuˆk|Hr(Iˆk,l)2.Now, Lemma 4.8 yields (k,l)I(IΠM)uL2(Ik,l)2π2rh2r(23r2η1)2r3r2k=1K|uˆk|H2(Ωˆ)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 uH2,(Ω)H1(Ω) be arbitrary but fixed and define uh:=ΠMu.

First, we show that (38) uuhQh+21+η02h2+σ(3r2hη01)2r3r2Ψ:=h2|u|H2(Ω)2(38) holds for any r{2,,pmin} and all η0>0.

Case 1. Assume η01. In this case, we define η:=η011 and observe 12η0ηη0.Equation (Equation16), Lemmas 4.11–4.13, and σp2 yield uuhQh+21+η2h2+σh3π2rh2r(23r2η1)2r3r2h2|u|H2(Ω)2and since 43π13 further (Equation38).

Case 2. Assume η0>1. Define W:={uH1(Ω):uGkS1,1(Ωˆ) for all k=1,,K},i.e. the set of all globally continuous functions which are locally just linear. Observe that WVh. Using u and w being continuous, we obtain uwQh+2=|uw|H1(Ω)2+h2σ2|uw|H2(Ω)2.For the choice wH1(Ω)withw|Ωk:=wk=wˆkGk1,where wˆk(x,y):=i=01j=01ϕˆi(x)ϕj(y)uˆk(i,j)whereϕˆ0(t):=1tandϕˆ1(t):=t,we further obtain using standard approximation error estimates and Lemma 3.1 infvhVhuvhQh+2uw+cQh+2=uwQh+2(1+h2σ2)|u|H2(Ω)2,where c:=(w,1)L2(Ω)/(1,1)L2(Ω). Using σ2p41/16, h1, and η0>1, we have infvhVhuvhQh+2η02+h216|u|H2(Ω)2(1+η02h2)h2|u|H2(Ω)2,which shows (Equation38) also for the second case.

Finally, we show that Ψ is such that the desired bound (Equation20) follows. We again consider two cases.

Case 1. Assume ln(σ1/2)pmin. In this case, we choose r:=max{2,ln(σ1/2)}andη0:=3er2h,where e is Euler's number (lne=1), and obtain Ψr4+σe2rr4lnσ4lnσ4σ2/(2pmin1),which finishes the proof for Case 1.

Case 2. Assume ln(σ1/2)pmin. In this case, we choose r:=pminandη0:=3σ1/(2r1)r2hand obtain immediately Ψ=1+32σ2/(2r1)r4+σ2/(2r1)r2σ2/(2pmin1)(lnσ)4,which finishes the proof for Case 2.

Proof of Theorem 3.8.

Proof of Theorem 3.8.

Let uH2,(Ω)H1(Ω) be arbitrary but fixed and define uh:=Πu. The definition of the Qh+-norm and the Qh-norm yield uvhQh+2=(IΠ)uH12+σh(k,l)N(IΠ)uL2(Ik,l)2+h2σ2|(IΠ)u|H22.Using the triangle inequality and Lemma 3.1, we have further uvhQh+2k=1KwˆkH1(Ωˆ)2+σhwˆkL2(Ωˆ)2+h2σ2|wˆk|H2(Ωˆ)2,where wˆk:=(IΠˆk)uˆk. Using a trace theorem [Citation14, Lemma 4.4] and ABγA2+γ1B2, we obtain wˆkL2(Ωˆ)2wˆkL2(Ωˆ)wˆkH1(Ωˆ)h1wˆkL2(Ωˆ)2+hwˆkH1(Ωˆ)2.Using this, h1, and 2p2σ, we obtain uvhQh+2k=1Kσh2wˆkL2(Ωˆ)2+σ|wˆk|H1(Ωˆ)2+h2p2|wˆk|H2(Ωˆ)2.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 Ω:={(x,y):x,y>0,1<x2+y2<4}. This domain is uniformly split into 4×4 patches in the obvious way. We solve the Poisson equation Δu=200π2sin(10πx)sin(10πy) in Ωandu=u on Ω,where u(x,y):=sin(10πx)sin(10πy) is the exact solution. On each patch, we introduce a coarse discretization space Vh0 for =0, which only consists of global polynomials of degree p. We then refine all grids =1,2, 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  and Figures  and , we depict the discretization errors e,p in the H1-norm relative to that of the solution and the rates r,p, given by e,p:=|uh,pu|H1|u|H1andr,p:=|uh1,pu|H1|uh,pu|H1for the case that the smoothness is Cp1 on the patches with spline degree p and Cp on the patches with spline degree p + 1 (maximum smoothness). The numerical experiments show that the error decreases like h2 or even faster. Whether or not the error bound depends on p2 or lnp 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. [Citation15].

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

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.

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

Table 1. Discretization errors in case of maximum smoothness.

In Table  and Figure , we choose C0 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.

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

Table 2. Discretization errors in case of minimum smoothness.

In Figure , one can observe the dependence of the discretization error on the number of patches. Here, we consider a decomposition of the initial domain into 2ι×2ι patches, where ι=1,2,,6. We again consider a heterogeneous discretization (different spline degrees, different grid sizes), where we apply =7ι 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 4ι patches and =7ι.

Figure 4. Errors for splines of maximum smoothness with 4ι patches and ℓ=7−ι.

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 Ω1:=(0,1)3, Ω2:=(1,2)×(0,1)×(0,1), Ω3:=(0,1)×(1,2)×(0,1), and Ω4:=(0,1)×(0,1)×(1,2). Each of the patches is parameterized with the appropriately shifted identity function. We solve (39) (αu)=27π2αsin(3πx)sin(3πy)sin(3πz) in Ωandu=u on Ω,(39) where u(x,y,z)=sin(3πx)sin(3πy)sin(3πz) is the exact solution. We choose α=100 on the patch Ω1:=(0,1)3 and α=1 on the remaining patches. The SIPG formulation reads as follows. Find uVh,:={vVh:v|Ω=u} such that (40) (u,v)H1(u,v)Bh(v,u)Bh+(u,v)Ch=(f,v)L2(Ω)(40) holds for all vVh,0:={vVh:v|Ω=0}, where (41) (u,v)Hr:=k=1Kαk(ru,rv)L2(Ωk)for r=1,2,,(u,v)Bh:=(k,l)N(u,{αu}nk)L2(Ik,l),and(u,v)Ch:=(k,l)N(αk+αl)δ2h(u,v)L2(Ik,l).(41) The discretization spaces on the patches are obtained by =0,1,2, uniform refinement steps, where we use splines of maximum smoothness with degree p + 1 on the patch Ω1 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 . 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.

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 H2-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 Ik,l are two-dimensional faces between patches, so the set 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 (Equation23) to Πˆk:=ΠkxΠkyΠkz yields a projector that is well defined only in H3. Based on this observation, Theorem 3.8 (approximation error estimate) can be extended to three dimensions, provided that q2. For the extension of Theorem 3.8 (low-order approximation error estimate), one would need to construct an H2-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 H1. If the differential equation is parameter-dependent, like (Equation39), a parameter-dependent SIPG formulation can be introduced, cf. (Equation40)–(Equation41). For the problem (Equation40)–(Equation41), 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 (Equation11), (Equation16), and (Equation41). 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 max(k,l)Nαk/αl and the Hr-norms are parameter-dependent norms as defined in (Equation41).

Disclosure statement

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).

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