High-accuracy quadrature methods for solving boundary integral equations of steady-state anisotropic heat conduction problems with Dirichlet conditions

2.1 Closed smooth curve Γ

Assume that C_Γ≠1 and Γ be parameterized by x(t)=(x₁(t), x₂(t)):[0, 2π]→Γ, with , where μ̄ and μ are two constants. Let C˜^ℓ[0, 2π] consist of periodic functions with period 2π, which has derivatives up to 2ℓ, i.e. v∈C˜^ℓ[0, 2π]={v(t)|v^(μ)(t)∈C[0, 2π], v^(μ)(t+2π)=v^(μ)(t), μ=0, 1, …, 2ℓ}.

Define the integral operator

with where w(t)=v(x(τ))| x′(τ) |.

The integral operator Kw can be split into a singularity part and a compact perturbation part as follows

where with and with where Z={0,±1,±2, …}. With the definitions of the operators mentioned above, EquationEquation (4) may be rewritten or where f̄=|κ^ij|^−1/2f.

For the logarithmically singular operator A, the associated integral is needed to be evaluated by means of Cauchy Principal Value. High accuracy for numerical computation of finite-range integrals with weakly singular kernels can be achieved by Sidi–Israeli quadrature formula. Now, we describe the Sidi–Israeli quadrature formula for weakly singular integrals. Let

Assume that the functions H₁(t, τ) and H₂(t, τ) are 2ℓ times differentiable on [0, 2π]. Assume also that the functions H(t, τ) are periodic with period T=2π, and that they are 2ℓ times differentiable on . If a(t, τ)=H₁(t, τ)ln |t−τ|+H₂(t, τ), then and where ζ(z) is a Riemann function and .

Let τ_j=jh (j=0, 1, …, n). From the representation of a(t, τ), we know H₁(t, τ)=−1/2π and H₂(t, τ)=1/4π. Thus, we can construct the Fredholm approximation of A,

which has the following error bounds:

For the integral operators B with periodic kernels, we can construct the Nyström approximation by the trapezoidal rule Citation12. The Nyström approximate operator B^h of B can be defined as:

and the error is Thus, we obtain the discrete approximation of EquationEquation (8) where w^h=(w(t₀), w(t₂), …, w(t_n))^T, , , and f̄_h=|κ^ij|^−1/2(f(t₀), …, f(t_n))^T with f(t_i)=f(x(t_i)). Obviously, EquationEquation (12) is a linear equation system with n unknowns. Once w^h is solved from EquationEquation (12), the solution of EquationEquation (3) φ(y) (y∈Ω) can be computed by

From EquationEquation (9), we have

By Citation11–15, we know that A^h is a symmetric circulant matrix and the eigenvalues λ_i of A^h are positive, and 1/4nπ≤λ_i≤c (i=0, …, n), where c is a constant independent of h. Hence, A^h is invertible and |(A^h)⁻¹|=O (n), where |·| denotes the spectral norm.

In order to prove the existence and convergence of numerical approximations, we first introduce two special operators. Let V^h=span{e_j(t), j=0, 1, …, n}⊂C[0, 2π) be a piecewise linear function subspace with nodes , where e_j(t) is the basis function satisfying e_j(t_i)=δ_ji. Define a prolongation operator satisfying , and a restricted operator satisfying .

Lemma 2.1

The operator sequence {I^h(A^h)⁻¹R^hA^h:C³[0, 2π)→C[0, 2π)} is uniformly bounded and convergent to the embedding operator I.

Proof

Let ω∈C³[0, 1] and ω^h be solutions of auxiliary equations Aω=η and A^hω^h=R^hη, respectively, where η∈C⁴[0, 1]. By using Sidi–Israeli quadrature formula, we have

where

Let e=ω^h−ω, we have Because |(A^h)⁻¹|=O (n), we have e=(A^h)⁻¹ϵ with ||e||=O (h²), and By I^hR^h→I as h→0 in , where is a linear bound operator space, the proof of Lemma 2.1 is completed.   ▪

Now we give two concepts, which can be seen in Citation3Citation4. Let denote the closed unit ball in X. Then is collectively compact if and only if the set has compact closure in Y. Define collectively compact convergence :

where denotes the pointwise convergence.

Lemma 2.2

Let the integral operator and A⁻¹ exist, and let the integral operator C^r+1[0, 2π)) with the kernel b(t, τ) of B satisfy A⁻¹b(., τ)=b̄(t, τ), where is a linear bound operator space. Also assume that b̄(t, τ) and (∂/∂ t)b̄(t, τ) are continuous on [0, 2π)². Let τ_j=(j−1)h (j=1, …, n) be nodes. Then the Nyström approximation of B̄,

is collectively compact convergent to B̄, i.e. where denotes the collectively compact convergence.

Proof

From EquationEquation (15), we have

Since b(t, τ) is continuous on [0, 2π)², we have Citation1 and Based on the continuity of (∂/∂ t)b̄(t, τ), EquationEquation (16) holds.   ▪

Corollary 2.3

For an integral operator B with a periodic smooth kernel b(t, τ), defining the Nyström approximation

and assume that Equation Equation(4) is uniquely solvable. we have and E^h+I^h(A^h)⁻¹R^hB^h is invertible, and the inverse operator is uniformly bounded.

Proof

Because the kernel b(t, τ) of the operator B and its derivatives of higher order are continuous Citation5, and we have

By Lemmas 2.1 and 2.2, there exists a constant M such that where is the norm of the linear bounded operator space . Applying the results of Citation1Citation8Citation18, the operator sequence {(A^h)⁻¹B^h: C[0, 2π), C³[0, 2π)} must be collectively compactly convergent to (A^h)⁻¹B. Hence, the proof of Corollary 2.3 is completed.   ▪

Replacing (A^h)⁻¹, A^h, and B^h by (Â^h)⁻¹=I^h(A^h)⁻¹R^h, Â^h=I^h(A^h)R^h and Bˆ^h=I^hB^hR^h, respectively. Consider the operator equation

with f̄ˆ^h=(Â^h)⁻¹f̄^h. Obviously, if ŵ^h=I^hw^h is a solution of EquationEquation (18), then R^hŵ^h must be a solution of Conversely, if w^h is a solution of EquationEquation (19), then ŵ^h must be a solution of EquationEquation (18). The following theorem shows that there exists a unique solution ŵ^h in EquationEquation (18) such that converges to w in EquationEquation (8).

Theorem 2.4

The operator sequence {(Â^h)⁻¹Bˆ^h} is collectively compact convergent to A⁻¹B in C[0, 2π), i.e.

Proof

Let {χ_h}_h∈H be a bounded sequence, where χ_h={x_h}, and h_j→0 as j→∞. Consider the convergence of

Using the results of Lemma 2.2 and Corollary 2.3, and by we have By , we obtain On the basis of collectively compact theory Citation1, we can find an infinite subsequence in {(Â^h)⁻¹Bˆ^hx_h} which converges as h→0. This shows that {(Â^h)⁻¹Bˆ^h} is a collectively compact sequence, and (Â^h)⁻¹Bˆ^h is pointwisely converge to A⁻¹B. The proof of Theorem 2.4 is completed.   ▪

2.2 Closed curved polygons Γ

Let be the boundary of a polygonal domain Ω in ℝ² with C_Γ≠1, Γ_m∈C^2ℓ+1(m=1, …, d, ℓ∈N), and let . Define the boundary integral operators on Γ_m,

Then EquationEquation (4) can be converted into a matrix operator equation

where and F=(f₁(y), …, f_d(y))^T. Assume that Γ_m can be described by the parameter mapping: x_m(s)=(x_m1(s), x_m2(s)):[0, 1]→Γ_m with . In order to degrade the singularities at corners, we apply the Sidi transformation to the parameter mapping, which is defined by

Sidi transformation in using the trapezoidal rule to evaluate was proposed in Citation26, where h(x) is not necessarily continuous or differentiable at x=0 and x=1, and h(x) may even behave singularity at the endpoints with different types of singularities. Let the variable transformation ψ_γ(t) have the property that

Also assume that for some q≥1, the variable transformation ψ_γ(t)∼α t^q as t→0+ [hence ψ(t)∼ 1−α(1−t)^q as t→1− as well]. In Citation27, Sidi pointed out that ψ(t)∼α t^q as $t\rightarrow 0$ implies that, when q is large, ψ_γ(ih) (h=1/n, i=1, …, n−1) in the original variable of integration s, are clustered in two very small regions, one to the right of t=0 and the other to the left of t=1, many of them being very close to 0 and to 1. The amount of this clustering is determined by the size of q, the larger q, the larger the density of the ψ_γ(ih) near t=0 and t=1. So Sidi transformation plays a role in refining the local mesh near the singular points s=0 and s=1. In addition, Sidi transformation has also been introduced as an example of ‘integral’ sigmoidal transformations in Citation10. It is well known that the truncation error is given by the Euler–Maclaurin expansion which involves a knowledge of the integrand and its derivatives at the end-points 0 and 1. A suitably choice for γ in EquationEquation (23) will make some derivatives in the Euler–Maclaurin expansion to be zero thereby improving the rate of convergence of the quadrature sum to the integral.

Now we give the following decomposition of K_lm, namely,

where with and with where Then EquationEquation (22) becomes where and

The operator A_ll is defined on the circular contour with radius e^−1/2. From Citation32Citation29, we have

where Z*=Z\ {0}, Z is the set of all integers, and σˆ_ll(t) is the Fourier transform of σ_ll(t)∈C^∞ (i.e. the r order derivative of σ_ll is continuous and 2π-periodic for all r≥0). As is easily shown, |A_llσ_ll|_r+1=|σ_ll|_r, for all σ_ll∈H^r, where |σ_ll|_r on C^∞ is the norm of the usual space H^r. Hence, A_ll is an isometry from H^r to H^r+1, and A is also an isometry operator from (H^r[0, 1])^d to (H^r+1[0, 1])^d. Then EquationEquation (26) is equivalent to where E is the identity operator. From Citation26, we know that on the open interval (0, 1) with ψ_γ(0)=0 and ψ_γEquation(1)=1. Hence, the solution of EquationEquation (27) is equivalent to the solution of EquationEquation (22).

Let P₁, …, P_d denote the corner points of the boundary Γ. Define a function χ_m∈[−1, 1]

where θ_m is the interior angle of the middle corner P_m.

Since the boundary Γ is closed polygon, the solution singularity for EquationEquation (26) occurs at the corner points P₁, …, P_d of the boundary Γ Citation5Citation29Citation32. The solution w_m(x)=∂ φ_m(x)/∂ ν⁻−∂ φ_m(x)/∂ ν⁺ has a singularity when s=s_m, where β_m=−|χ_m|/(1+|χ_m|), β_m∈[−½, 0) and s_m denotes the arc parameter at P_m.

Lemma 2.5 Citation17

•  (1) Suppose that the function where β_m∈[−½, 0) and h_m(s)∈C^∞[0, 1 with h_m(0)≠0. Then the function w_m(t) can be expressed by

  
  where c₁ and c₂ are two constants independent of t.

•  (2) Suppose that the function where β_m∈[−½, 0) and h˜_m(s)∈C^∞[0, 1] with h˜_mEquation(1)≠0. Then the function w_m(t) can be expressed by

  
  where c₃ and c₄ are two constants independent of t.

Proof

• (1) By using Taylor's formula, we can obtain

  
  and
  
  By substituting EquationEquations (30) and Equation(31) into the expression of w_m(t), we obtain EquationEquation (28). The proof for Equation(2) of Lemma 2.5 is similar.

  ▪

By Lemma 2.5 and β_m≥−½, we can obtain (γ+1)(β_m+1)−1≥0 for γ≥1. Hence, we have the following remark.

Remark 1 Although v_m(s) has singularities at endpoints s=0 and s=1, w_m(t) has no singularity by Sidi transformation at t=0 and t=1.

Bellow we study singularity for the kernels b_lm(t, τ) (l, m=1, …, d). Obviously, if Γ_l∩Γ_m=∅, b_lm(t, τ) are continuous in [0, 1]², and if Γ_l∩Γ_m≠∅, b_lm(t, τ) have singularities at the points (t, τ)=(0, 1) and (t, τ)=(1, 0). For convenience of analysis, we only discuss the case in which (t, τ)=(1, 0). Defining the following function

Lemma 2.6

Let b˜_lm(t, τ) be defined by Equation Equation(32), then b˜_lm(t, τ) and (∂^kb˜_lm(t, τ)/∂ τ^k) (k=1, 2) are smooth on [0, 1]².

Proof

By using the continuity of b˜_ll(t, τ) and the boundness of sin ^γ(π t), we can immediately complete the proof for the case l=m. Let Γ_l−1∩Γ_l=P_l=(0, 0) and θ_l∈(0, 2π) is the corresponding interior angle. Then we have

which shows the kernel b_{l−1, l}(t, τ) has a logarithmic singularity at (t, τ)=(1, 0). Suppose that a_l−1(t)=|x_l−1(t)| and a₁(τ)=|y_l(τ)|. Then we can obtain Because This shows that I₂(t, τ) and its second derivative are bounded. Noting that we have Let (t, τ)∈[ϵ/2, ϵ]×[1−ϵ, 1−ϵ/2] for all ϵ>0, we have so I₁(t, τ) is also bounded. In addition, we have and Similarly, from EquationEquation (33) we also have This shows (∂^jb˜_lm(t, τ)/∂ τ^j) (j=0, 1, 2) are also continuous in (C²[0, 1])². The proof of Lemma 2.6 is completed.   ▪

Remark 2 The kernel b_lm of B_lm(t, τ) has singularities at corners (t, τ)=(0, 1) and (t, τ)=(1, 0), but sin^γ(π t)b_lm(t, τ) has no singularities.

Let h_m=(1/n_m) (n_m∈N, m=1, …, d) and t_mj=τ_mj=(j−1/2)h_m (j=1, …, n_m) be the step sizes and boundary nodes, respectively. The trapezoidal rule approximation for the boundary integral operator B_lm is

which has the error bounds Citation12Citation26 where In addition, by using Sidi–Israeli quadrature formula, the approximate operator of the weakly singular operators A_ll can be constructed by which has the errors asymptotic expansion Citation28 where c_μ=−(2/π)(ζ′(−2μ)/(2μ)!)[w_l(t)]^(2μ) and ζ′(t) is the derivative of the Riemann zeta function.

Letting t=t_li (l=1, …, d, i=1, …, n_l), we have the discrete equations of Equation(26)

where and

EquationEquation (38) has n () unknowns. Once W^h is solved from EquationEquation (38), the solution φ(y) (y∈Ω) can be calculated by

From EquationEquations (37) and Equation(39), we can obtain the circulate matrix

For the properties of , we have the following lemma.

Lemma 2.7 Citation15Citation17

\

• Equation(1) Let λ_lk (k=1, …, n_l; l=1, …, d) be the eigenvalues of . Then there holds

  
  where c₁ and c₂ are two positive constants.

•  (2)  and A^h are invertible, and the following holds

  
  where h_min=min _1≤l≤dh_l and ‖·‖ denotes the spectral norm.

Let C₀[0, 1]={g(t)∈C[0, 1]:g(t)/sin ^γ(π t)∈C[0, 1]} denote the subspace of the space C[0, 1] with the associated norm ||g||*=max _0≤t≤1|g(t)/sin ^γ(π t)|, and also let span{N_mj(t), j=1, …, n_m}⊂C₀[0, 1] be a piecewise linear function subspace with the basis nodes , where N_mj(t) are the basis functions satisfying N_mj(t_mi)=δ_ji. Define a prolongation operator satisfying for all , and a restricted operator satisfying for all g_m∈C₀[0, 1].

Lemma 2.8

Suppose that satisfy C_Γ≠1. Then there exists a constant M₁ independent of h_l such that

where ‖·‖_{0, 2} is the norm of the linear bounded operator space .

Proof

From Citation17, the operator sequence is uniformly bounded and convergent to the embedding operator I. Hence, the proof of Lemma 2.8 is completed.   ▪

Lemma 2.9 Citation14

Let the integral operator and exist, and let the integral operator with the kernel l_m(t, τ) of L_m satisfy where is a linear bound operator space, and l̄_m is the kernel of L̄_m. Also assume that l̄_m(t, τ) and (∂/∂ t)l̄_m(t, τ) are continuous on [0, 1]². Let τ_m=(m−½)h_l (j=1, …, n_m) be nodes. Then the Nyström approximation of L̄_j,

is collectively compact convergent to L̄_m, i.e.

Corollary 2.10

\

• Equation(1) For Γ_m=Γ_l or Γ_m∩Γ_l=∅, let be the Nystr öm's approximation of B_lm, and also let b_lm be the kernel of the integral operator B_lm. Then under the transformation Equation(23), there holds

  
  where M₂ is a constant independent of h_m.

•  (2) For Γ_m∩Γ_l∈{P_m}, let be the approximation of B˜_lm, and also let b˜_lm=sin^γ(πτ) be the kernel of integral . Then under the transformation Equation(23), there exists a constant M₃ independent of h_m such that

  

Proof

• (1) Since |l−m|≠1 or d−1, by the definition of B_lm we know the kernel b_lm(t, τ) of the operator B_lm and its derivatives of higher order are continuous. By Lemma 2.9, we have

  
  which implies that the sequence is uniformly bounded. The proof is completed.

• (2) From Lemma 2.6, we know that b˜_lm(t, τ) are continuous on (C²[0, 1])². Hence, the proof for Equation(2) of Corollary 2.10 is similar to the proof for Equation(1).

  ▪

Lemma 2.11 Citation17

Under the same conditions of Corollary 2.10, by the trapezoidal or the midpoint rule, we have

and where denotes the collectively compact convergence.

Proof

Using the results of Lemma 2.8 and Corollary 2.10, we have

and

Thus, we complete the proof of Lemma 2.11.   ▪

Using and to replace and (m, l=1, …, d), respectively. Define an operator (Â^h)⁻¹Bˆ^h: (, where (Â^h)⁻¹Bˆ^h=I^h (A^h)⁻¹R^hB^h, and . Then the associated equation of EquationEquation (38) is

From Citation17, if Ŵ^h is a solution of EquationEquation (48), R^hŴ^h is also a solution of EquationEquation (38). Reversely, if W^h is a solution of EquationEquation (38), I^hW^h is also a solution of EquationEquation (48).

Theorem 2.12 Citation13

Assume that there exists a unique solution in Equation Equation(4), and Γ_m∈C^2ℓ+1 (m=1, …, d, ℓ∈N). Then the operator sequence {(Â^h)⁻¹Bˆ^h} is collectively compact convergent to A⁻¹B in the space V=(C₀[0.1])^d. That is, the following holds

Proof

Let Θ={z:||z||≤1, z∈(C[0, 1])^d} be the unit ball in the space V, and

are multi-parameter sequences. Also let as n_m→∞. Choosing the sequence {Z_h, h∈H}⊂Θ and satisfying Bellow we will verify that there exists one convergent subsequence in . So we consider the first component of (Â^h)⁻¹Bˆ^hZ_h If Γ₁∩Γ_m=∅, from EquationEquation (46) we obtain If Γ₁∩Γ_m≠∅, using EquationEquation (47) and by we know that must be collectively convergent, we can find an infinite subsequence in which converges as h→0. Hence, there exists an infinite subsequence H₁⊂H such that (2.47) converges, As above, there exists an infinite subsequence H_d⊂H_d−1⊂···⊂H₁ such that {(Â^h)⁻¹Bˆ^h, h∈H_d} is a convergent sequence in the space V=(C₀[0, 1])^d. This shows that {(Â^h)⁻¹Bˆ^h} is a collectively compact sequence, and (Â^h)⁻¹Bˆ^h is pointwisely convergent to A⁻¹B. The proof of Theorem 2.12 is completed.   ▪

3.1 Errors analysis for MQMs

In the case of closed smooth boundary Γ, we have the single parameter asymptotic expansions of the solution errors.

Theorem 3.1

If there exists a unique solution in Equation Equation(4), f̄, f̄^h are computed by Equations Equation(8) and Equation(12), respectively, x_i∈C⁶[0, 2π) (i=1, 2) and f(s)∈C⁵[0, 2π), then there exists a function ϱ∈C⁵[0, 2π) independent of h such that

Furthermore, we have

Proof

By the midpoint trapezoidal rule, the asymptotic expansion holds Citation13Citation14

with ϕ₁=−ξ′(−2)f̄^′′(t)/π. Using EquationEquations (8), Equation(12), and Equation(18), we can obtain where ϕ₂=−ξ′(−2)w^′′(t)/π, and ϕ=ϕ₁+ϕ₂. From Theorem 2.4, we have

Define the auxiliary equation

and its approximate equation Substituting EquationEquation (58) into EquationEquation (56) yields Since (E^h+(Â^h)⁻¹B^h)⁻¹ is uniformly bounded, by Theorem 2.4 we have Replacing ϱ^h in EquationEquation (60) with ϱ, we obtain EquationEquation (54). In addition, EquationEquation (60) can be written as Replacing h in EquationEquation (61) with h/2, we get Hence, we can obtain EquationEquation (55) by EquationEquations (61) and Equation(62). The proof of Theorem 3.1 is completed.   ▪

Similar to Theorem 3.1, we have the following theorem in the case of closed curved polygon Γ.

Theorem 3.2 Citation16

Let the boundary satisfy C_Γ≠1 and suppose . Then there exists functions ϱ_m∈C₀[0, 1] (m=1, …, d) independent of h_m (m=1, …, d) such that the following multi-parameter asymptotic expansions hold at nodes, provided ω_m(γ+1)>3 (m=1, …, d) for γ≥1,

where h_m=1/n_m, n_m∈N, m=1, …, d.

Furthermore, we have

3.2 Stability analysis for MQMs

For simplicity, we give the stability of MQMs for smooth boundary and polygon together.

Theorem 3.3 Citation15Citation17

Suppose that satisfy C_Γ≠1. When d=1, the boundary Γ (=Γ₁) is a smooth and closed curve, and when d>1, the boundary is a closed polygon, and Γ_m (m=1, …, d) be smooth curves. Suppose that A^h̄ and B^h̄ are the discrete matrices defined by Equations Equation(9) and Equation(11), or Equation(38), respectively, where h̄=h when d=1, and h̄=h_min =min _1≤m≤dh_m when d>1. Here, h and h_m (=1/n_m, m=1, …, d) is the mesh step size of the smooth curve and the curved edge Γ_m, respectively. Then there exists two positive constants ĉ₁ and ĉ₂ independent of h̄ such that where λ_i (i=1, …, n_m) are the eigenvalues of discrete matrix K^h̄=|κ^ij|^1/2(A^h̄+B^h̄). In addition, the following hold

where Cond is the traditional 2-norm condition number.

The proof of Theorem 3.3 is similar to the proof of Theorem 3.4 in Citation17.