# Convergence of the High-Accuracy Algorithm for Solving the Dirichlet Problem of the Modified Helmholtz Equation

Authors: Hu Li; Guang Zeng
Journal: Complexity
 (2020)

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## Abstract

In this paper, we derive the convergence for the high-accuracy algorithm in solving the Dirichlet problem of the modified Helmholtz equation. By the boundary element method, we transform the system to be a boundary integral equation. The high-accuracy algorithm using the specific quadrature rule is developed to deal with weakly singular integrals. The convergence of the algorithm is proved based on Anselone’s collective compact theory. Moreover, an asymptotic error expansion shows that the algorithm is of orderOh03. The numerical examples support the theoretical analysis.


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## Body

## 1. Introduction

Modified Helmholtz equation arises in many important fields of science and engineering, for instance, wave propagation and scattering [1], structural vibration [2], implicit marching schemes for the heat equation [3], the Navier–Stokes equations [4], and so on. With the rapid development of computer power, many numerical methods [5–11] can be used to solve the modified Helmholtz equation. Among all methods for numerically solving the boundary value problem, the boundary integral equation approach is one of the most fundamental treatments. By the fundamental solutions, it can transform the boundary problem into an integral equation defined on the boundary. In the literature, several popular numerical methods for solving the boundary integral equations reformulated from the modified Helmholtz equation are used. For instance, Steinbach and Tchoualag [12] used the Galerkin method with one-periodic B-spline as basis functions and a spectral collocation method to solving the modified Helmholtz equation with the mixed boundary value problem; Kropinski and Quaife [13] used the fast multipole-accelerated integral equation methods to solving the modified Helmholtz equation with linear boundary conditions, and so on. Motivation of this work is to present an accuracy algorithm for solving the following modified Helmholtz problem:(1)Δux−α2ux=0,∈Ω,ux=gx,∈Γ,where a is a real and positive constant and called the acoustic wave number. Ω⊂ℜ2 is a bounded domain with the boundary Γ=∪m=1dΓm,d>1 which is a closed curve. gx is a given function.Based on the boundary element method, the functionuy satisfies(2)uy=∫ΓK∗y,xvxdsx,y∈Ω,where x=x1,x2, y=y1,y2, and vx is the density function, and(3)K∗y,x=−12πK0αx−y,is the fundamental solution of the modified Helmholtz equation. K0 is the modified Bessel function [14]:(4)K0z=−lnz+ln2−γ,z⟶0,where γ is Euler’s constant.Sinceuy is a continuous function in Ω∪Γ, we have(5)gy=∫ΓK∗y,xvxdsx,y∈Γ.The aforementioned equation is weakly singular boundary integral equation of first kind, whose solution exists and is unique as long asCT≠1 [15], where CT is the logarithmic capacity. As soon as vx is solved from (5), the function uyy∈Ω can be calculated by (2). Galerkin methods and collocation methods have been used to solve (5) based on the projective theory. However, there exist the following disadvantages: (a) the discrete matrix is full and each element has to calculate the weakly singular integral for collocation methods or the double weakly singular integral for Galerkin methods; (b) the order of accuracy is lower [16].In this paper, we present a high-accuracy algorithm to solve (5). By specific quadrature rule, the high-accuracy algorithm is constructed to discrete the boundary integral equation, and a linear system is obtained. The calculation of the discrete matrix becomes very simple and straightforward without any singular integrals. We prove the convergence of the linear system by estimating eigenvalues of the discrete matrix and Anselone’s collective compact theory. Moreover, an asymptotic expansion of errors is obtained. Finally, numerical results verify the theoretical analysis.This paper is organized as follows: in Section2, the singularity of integral kernels and solutions are discussed; in Section 3, the high-accuracy algorithm is shown; in Section 4, the convergence analysis of the algorithm is given; in Section 5, an asymptotic expansion of the errors is obtained; and in Section 6, numerical examples are shown.


## 2. Singularity of Integral Kernels and Solutions

LetΓ=∪m=1dΓm,d>1 be closed polygons with CΓ≠1 and Γmm=1,…,d be a piecewise smooth curve. Define the boundary integral operators on Γm:(6)Kqmvmy=∫Γmkqmy,xvmxdsx,y∈Γq,m,q=1,…,d,where kqmy,x=K∗y,x.Then, (5) can be converted to a matrix operator equation:(7)Kv=G,where K=Kqmq,m=1d, v=v1x,v2x,…,vdxT, and G=g1y,g2y,…,gdyT. Assume that Γm be described by the parameter mapping: xms=xm1s,xm2s:0,1⟶Γm, with xm′s=xm1′s2+xm2′s21/2>0, m=1,…,d. Using the sinp-transformation [17],(8)ψpτ=ϑpτϑp1:0,1⟶0,1,p∈N,where ϑpτ=∫​0τsinπρpdρ. The operators in (6) are converted to the following integral operators on [0,1]:(9)Kqmωmt=∫01kqmt,τωmτdτ,t∈0,1,where kqmt,τ=−1/2πK0αxqt−xmτ, xqt=xq1ψpt,xq2ψpt, ωmτ=vmψpτxm′ψpτψp′, and xqt−xmτ=xq1t−xm1τ2+xq2t−xm2τ21/2.Hence, (7) is equivalent to(10)Kω=G,where ω=ω1,ω2,…,ωdT. By (4), it is clear that the operator K is a weakly singular integral operator, which is decomposed into the sum of the integral operators A and B, where one carries the main singularity characteristic of A and the other is a compact operator with a smooth kernel.Define(11)Ammωmt=∫01ammt,τωmτdτ,t∈0,1,and(12)Bqmωmt=∫01bqmt,τωmτdτ,t∈0,1,where ammt,τ=1/2πln2e−1/2sinπt−τ and(13)bqmt,τ=kmmt,τ−ammt,τ,forq=m,kmmt,τ,forq≠m,where bmmt,τ ⟶ 1/2πlnxm′ψpτψp′/2πe−1/2+lnα/2+γ as t⟶τ.Then, (10) becomes(14)A+Bω=G,where A=diagA11,A22,…,Add, B=K−A=Bqmq,m=1d, and G=g1t,g2t,…,gdtT. Ammm=1,…,d is an isometry operator from Hr0,1 to Hr+10,1 for any real number r. A is also an isometry operator from Hr0,1d to Hr+10,1d. Hence, A is invertible, and (14) is equivalent to(15)E+A−1Bω=A−1G.Sinceψpt∈C∞0,1 increases monotonously on [0,1] with ψp0=0 and ψp1=1, the solutions of (14) are equivalent to those of (7) [17].Now, let us study the solution singularity for (7). Denote the corner points Q1,…,Qd of the closed polygonal boundary Γ and Q1=Qd. Denote χm∈−1,1 by χ1=χd=−1 for the endpoints Q1 and Qd or 1−χmπ=∠Qm−1QmQm+1 for the middle corner Qm. Based on the potential theory [15, 18], near the corner or endpoint Qm, the solution vm has a singularity Os−smβm, where βm=−χm/1+χm≥−1/2, and s=sm at Qm is the arc parameter. Since the singularity of vm results from the singularities of the potential uy, when uy is nonsingular in the exterior region, the singularity of vm becomes Os−smχm/1−χm, and when it is nonsingular in the interior, the singularity of vm becomes Os−sm−χm/1+χm. Although vms has a singularity, ωms has no singularity by (8). The kernel bmm of Bmm has a singularity at corners, but sin2πsbmm has no singularity.


## 3. The High-Accuracy Algorithm

Lemma 1 (see [19]).
Assume thatgx and hx∈C2ma,b. Gx=lnx−tgx+hx is a periodic function with a period of b−a and at least 2 m differentiable in Ł=−∞,+∞/t+kTk=−∞+∞. There is the following quadrature formula:(16)QnG=h∑j=1,xj≠tGxj+hth+lnh2πgth,and the error(17)EnG=∫abGxdx−QnG=2∑j=1m−1ζ′−2j2j!g2jth2j+1+Oh2m,where h is the mesh width and ζ′t is the derivative of the Riemann zeta function.Lethm=1/nmnm∈N,m=1,…,d be mesh widths and tj=τj=j−1/2hmj=1,…,nm be nodes.(1)
Sincekqmt,τ=bqmt,τ are the smooth functions on [0, 1] for q≠m, by the trapezoidal or the midpoint rule [20], we construct the Nyströms approximate operator Bqmh of the integral operator Bqm defined by(18)Bqmhωmt=hm∑j=1nmbqmt,tjωmtj,t∈0,1q,m=1,…,d.We have the following error bounds [20]:(a)
ForΓq∩Γm=∅,(19)Bqmωmt−Bqmhωmt=Ohm2l,l∈N,(b)
ForΓq∩Γm=Q∈Qm,(20)Bqmωmt−Bqmhωmt=Ohmλ,where [17](21)λ=minp+1βm+1,p+1,p,odd,minp+1βm+1,2p+2,p,even.(2)
Sincekmmt,τ=ammt,τ+bmmt,τ have the logarithmic singularities on [0,1], by Lemma 1, we get the following approximations Ammh of the integral operator Amm defined by(22)Ammhωmt=hm2π∑j=1,t≠τjnmln2e−1/2sinπt−τjωmτj+hm2πln2πe−1/2hm2πωmt,t∈ti,and the error bounds(23)Ammωmt−Ammhωmt=2π∑μ=12l−1ζ′−2μ2μ!ωmt2μhm2μ+1+Ohm2l.The approximationsBmmh of the integral operator Bmm are defined by(24)Bmmhωmt=hm∑j=1,t≠τjnmbmmt,τjωmτj+hm2πlnxm′ψpτψp′2πe−1/2+lnα2+γωmt,t∈ti.Settingt=ti i=1,…,nm, we obtain the discrete equations of (14):(25)Ah+Bhωh=Gh,where Ah=diagA11h,…,Addh, Ammh=ammti,τji,j=1nm, Bh=Bqmhq,m=1d, Bqmh=bqmti,τji,j=1nq,nm, Gh=g1t1,…,g1tn1,…,gdt1,…,gdtndT, and ωh=ωmhm=1d.(26)ωmh=ωmt1,…,ωmtnmT,ammti,τj=hmln2e−1/2sinπti−τj2π,asi≠j,hmln2πe−1/2hm/2π2π,asi=j.Obviously, (25) is a system of linear equations with n=∑m=1dnm unknowns. Once ωh is solved by (25), the solution uyy∈Ω can be computed by(27)uhy=−12π∑m=1dhm∑i=1nmK0αy−xtiωmhti.


## 4. Convergence of the Algorithm

Form (11) and (26), we have(28)Ammh=circulatehm2πln2e−1/2hmπ,…,hm2πln2e−1/2sinπnm−1hm.Lemma 2 (see [21]).
(1) There exists a positivec1>0 so that the eigenvalues λββ=1,…,nm of Ammh satisfy c1>λβ>1/2πnm; (2) Amm is invertible, and Ammh−1 is uniformly bounded.Based on Lemma2, we immediately get the following corollary:Corollary 1.
A
h is invertible, and Ah−1 is uniformly bounded.From Corollary1, we know (25) is equivalent to(29)Eh+Ah−1Bhωh=Ah−1Gh,where Eh denotes the unit matrix.Now we give the following definitions to discuss the approximate convergence in (29).Define a subspaceC00,1=vt∈C0,1:vt/sin2πt∈C0,1 of the space C0,1 with a norm v∗=max0≤t≤1vt/sin2πt. Let Smh⊂C00,1 be a piecewise linear function subspace with base points tii=1nm and ejt,j=1,…,nm be base function satisfying ejti=δji. Define a prolongation operator Phm:ℜhm⟶Shm satisfying(30)Phmω=∑j=1nmωjejt,∀ω=ω1,…,ωnm∈ℜnm.Define a restricted operatorRhm:C00,1⟶ℜnm satisfying(31)Rhmvm=vmt1,…,vmtnm∈ℜnm,∀vm∈C00,1.ReplacingA^h−1B^h=PhAh−1RhBh, ωh^=Phωh, and G^=PhGh, we construct an operator equation(32)Eh+A^h−1B^hωh^=A^h−1G^h,where Ph=diagP1,…,Pd, Rh=diagR1,…,Rd. Obviously, if ω^h is the solution of (32), then Rhωh must be the solution of (29); conversely, if ωh is the solution of (29), then Phωh must be the solution of (32).In order to prove the convergence of the algorithm, we give the following lemma.Lemma 3 (see [21]).
The operator sequencePmAmmh−1RmAmm:C20,1⟶C0,1 is uniformly bounded and convergence to the embedding operator I.Lemma 4 (see [22]).
Let the integral operatorMm∈LCγ0,1,Cϱ0,1m=1,…,d and Mm−1 exist, and let the integral operator Lm∈LCγ0,1,Cϱ0,1 with the kernel lmt,s of Lm satisfy Mm−1l.,s=l¯mt,s, where LCγ0,1,Cϱ0,1 is a linear bound operator space, and l¯m is the kernel of L¯m. Also assume that l¯mt,s and ∂/∂tl¯mt,s are continuous on 0,12. Let smi=i−1/2hmi=1,…,nm be nodes. Then, the Nyström approximation of L¯mh,(33)L¯mhωm=hm∑i=1nml¯mt,smiωmsmi,∀ωm∈0,1,is collectively compact convergent to L¯m,(34)Mm−1Lmh⟶Mm−1Lm∈LC00,1,C10,1.Based on Lemma4, we can immediately obtain the following corollary.Corollary 2.
LetΓ=∪m=1dΓm with CT≠1, and let q−m≠1 or d−1. Then, under (8), the Nyström approximation Bqmh of Bqm has(35)PqAqq−1RqBqmh⟶c·cAqq−1Bqm,inC0,1⟶C0,1,by the trapezoidal or the midpoint rule, where ⟶c.c denotes the collectively compact convergence.Lemma 5.
LetΓ=∪m=1dΓm with CT≠1, and let q−m=1 or d−1. Then, we have(36)PqAqq−1RqB˜qm⟶c·cAqq−1B˜qm,in∈LC00,1,C10,1,where the kernel of B˜qm is ψp′bqm.Proof.
the kernel ofBqm has logarithmic singularity at the corner Qm+1=Γq∩Γm with an inner angle θm+1 (replacing m+1≔1 when m=d). Without loss of generality, suppose that Qm+1=0,0, then by (4), the kernel of Bqm can be expressed by(37)bqmt,s=12πlna02t+a1s2−2a0ta1scosθm+1+12πlnα2+γ,where a0t=xqψpt and a1s=xmψps. We assume that a00=a10=0, namely, bqm has the logarithmic singularity at points 0,0 and are continuously differentiable in 0,12/0,0. Consider(38)bqmt,s=12πlna02t+a1s2+12πln1−2a0ta1scosθm+1a02t+a1s2+12πlnα2+γ.
We have(39)2a0ta1scosθm+1a02t+a1s2≤cosθm+1<1.
So, the singularity ofbqmt,s occurs in 1/2πlna02t+a1s2. In order to remove the singularity, define(40)ςt,s=sinpπsψp1lna02t+a1s2.
Sinceψp′ have p order zero, we have the error estimate(41)ςt,s=oεplnε⟶0,asε⟶0,which shows that ςt,s is bounded and continuous on C20,12 when ε/2≤t,s≤ε for any ε>0.
Similarly, we can prove∂/∂tςt,s and ∂2/∂t2ςt,s are continuous on C0,1, which can be obtained by(42)∂∂tςt,s≤sinpπsψp12a0txm+1′ψptψp′ta02t+a12s=Oεp,∀t,s∈ε/2,ε,and ∂2/∂t2ςt,s=Oεp−1. As above, ∂n/∂tnςt,sn=0,1,…,p is also continuous on C0,12. Thus, ψp′bqm and ∂/∂tψp′bqm are continuous on C20,12. By Lemma 4, we complete Lemma 5.Theorem 1.
LetΓ=∪m=1dΓm with CT≠1 and Γmm=1,…,d be the smooth curve. The operator sequence PhAh−1RhBh is collectively convergent to A−1B in V=C00,1d,(43)PhAh−1RhBh⟶c·cA−1B.Proof.
LetΘ=υ:υ≤1,υ∈V be a unit ball. H=h1,h2,… is the grid step sequence, where hn=h1n,h2n,…,hdn with n⟶∞, max1≤m≤dhmn⟶0. We arbitrarily take a sequence Zhh∈H in space Θ, where Zh=Zh1,Zh2,…,ZhdT with Zhm/sin2πt∞≤1, m=1,…,d. We firstly consider the first component of PhAh−1RhBhZh:(44)∑m=1dP1A11h−1R1B1mhZhm.
ForΓm=Γ1 or Γm∩Γ1=∅, by Corollary 2, P1A11h−1R1B1mh⟶c.cA11−1B1m in C0,1, and there exists a convergent subsequence in P1A11h−1R1B1mhZhm. For Γm∩Γ1≠∅, then(45)P1A11h−1R1B1mhZhm0,0=P1A11h−1R1B˜1mhZhmsin2πt0,0≤P1A11h−1R1A110,2A11−1B˜1mh2,0Zhmsin2πt∞.
By Lemma5, there exists a convergent subsequence in P1A11h−1R1B1mhZhm. Based on the above two cases, it is proved that there exists an infinite subsequence H1⊂H such that the first component converge. Similarly, it can be concluded that there exists an infinite subsequence Hd⊂Hd−1⊂⋯⊂H1⊂H such that PhAh−1RhBhZh converge. Therefore, PhAh−1RhBh is collectively compact convergent sequence, and PhAh−1RhBh⟶PA−1B, where ⟶P shows the point convergence. We complete the proof.


## 5. The Error of the Algorithm

In this section, we derive the asymptotic expansion of the errors. We first provide the main result.Theorem 2.
LetΓ=∪m=1dΓm with CΓ≠1. There exists a vector function Φ=ϕ1,ϕ2,…,ϕdT independent of h=h1,h2,…,hd such that the following multiparameter asymptotic expansion holds at nodes(46)ω^h−ω=diagh13,…,hd3Φ+oh03,h0=max1≤m≤dhm,where ω^h∈Sd, ω∈C30,1.Proof.
By (19), (20), and (23), we can obtain(47)PhAhRh+PhRhBhω^h−ω=PhRhf−PhAhRh+PhRhBhω=PhRhA+Bω−PhAhRh+RhBhω=PhRhA+B−AhRh+RhBhω=diagh13,…,hddPhRhη+oh03,where η=η1,…,ηd, ηm=ζ′−2ωm″. Since PhRhω^h=ω^h, we have(48)E+PhAh−1RhBhω^h−ω=diagh13,…,hddPhAh−1Rhη+oh03.
Constructing the auxiliary equation(49)E+A−1Bω¯=A−1η,and its approximate equation(50)E+PhAh−1RhBhω¯h=PhAh−1Rhη,and substituting (50) in (49), we obtain(51)E+PhAh−1RhBhωh−ω−diagh13,…,hddω¯h=oh03.
SinceI+PhAh−1RhBh is uniformly bounded from Theorem 1, we get(52)ωh−ω−diagh13,…,hddω¯h=oh03.
Replacingω¯h with ω¯, we can complete the proof.


## 6. Numerical Examples

In this section, in order to verify the errors in the previous sections, we carry out some numerical examples for the modified Helmholtz equation with Dirichlet boundary value problems by the high-accuracy algorithm.Example 1.
We consider the modified Helmholtz equation withα2=8 and α2=18 on a plate domain Ω with the boundary Γ=∪m=12Γm and Γ1=x1,x2=t,0:0≤t≤1 and Γ2=x1,x2=0.5cosπt+0.5,0.5sinπt:0≤t≤1. We consider the Dirichlet boundary conditions corresponding to the exact solution e−2α/2x1+x2.
We calculate the errors ofux at the points (0.5, 0.3) and (0.5, 0.35) by ψ3t. The errors e1,2=uhx−ux are given in Table 1. Average errors of 100 points are given in Table 2. We let ratio denote ein1,2=2m/ein1,2=2m+1,i=1,2. From the numerical results, we can see that log2ratio≈3, which shows that the accuracy order of uhx is Oh03 for the high-accuracy algorithm.Table 1
The simulation results for Example 1.
0.5
,
0.3
0.5
,
0.35
α
2
=
8
α
2
=
18
α
2
=
8
α
2
=
18
n
1
,
n
2
e
1
Ratio
e
2
Ratio
e
1
Ratio
e
2
Ratio
(8, 8)
1.3001e − 3
—
1.3000e − 3
—
1.5002e − 3
—
1.5000e − 3
—
(16, 16)
1.5737e − 4
8.2162
1.6175e − 4
8.0786
1.7826e − 4
8.5450
1.6922e − 4
8.8212
(32, 32)
1.9508e − 5
8.0668
2.0317e − 5
7.9613
2.2899e − 5
7.7844
2.1061e − 5
8.0350
(64, 64)
2.4256e − 6
8.0428
2.5402e − 6
7.9983
2.8596e − 6
8.0078
2.6396e − 6
7.9789
(128, 128)
2.9535e − 7
8.2125
3.1267e − 7
8.1243
3.5122e − 7
8.1420
3.2650e − 7
8.0844
(256, 256)
3.2949e − 8
8.9638
3.6475e − 8
8.5720
4.0697e − 8
8.6301
3.8832e − 8
8.4081Table 2
The simulation results for Example 1.
α
2
=
8
α
2
=
18
n
1
,
n
2
Average errors
Ratio
Average errors
Ratio
(8, 8)
1.4512e − 4
—
1.4195e − 4
—
(16, 16)
1.7060e − 5
8.5065
1.6596e − 5
8.5534
(32, 32)
2.1358e − 6
7.9874
2.0716e − 6
8.0113
(64, 64)
2.6607e − 7
8.0271
2.5935e − 7
7.9876
(128, 128)
3.2564e − 8
8.1709
3.2011e − 8
8.1019
(256, 256)
3.7158e − 9
8.7636
3.7756e − 9
8.4786Example 2.
We consider the modified Helmholtz equation withα2=8 and α2=18 on a plate domain Ω with the boundary Γ=∪m=12Γm and Γ1=x1,x2=0.5cosπt+π+0.5,0.5sinπt+π+0.5:0≤t≤1 and Γ2=x1,x2=0.5cosπt+0.5,sinπt+0.5:0≤t≤1. We consider the Dirichlet boundary conditions corresponding to the exact solution e−2α/2x1+x2.
We calculate the errors ofux at the points (0.6, 0.6) and (0.6, 0.5) by ψ3t. The errors e1,2=uhx−ux are given in Table 3. Average errors of 100 points are given in Table 4. We let ratio denote ein1,2=2m/ein1,2=2m+1,i=1,2. From the numerical results, we can see that log2ratio≈3, which shows that the accuracy order of uhx is Oh03 for the high-accuracy algorithm.Table 3
The simulation results for Example 2.
0.6
,
0.6
0.6
,
0.5
α
2
=
8
α
2
=
18
α
2
=
8
α
2
=
18
n
1
,
n
2
e
1
Ratio
e
2
Ratio
e
1
Ratio
e
2
Ratio
(8, 8)
4.2759e − 4
—
5.0447e − 4
—
5.7375e − 4
—
7.8778e − 4
—
(16, 16)
4.2257e − 5
10.1188
5.9981e − 5
8.4105
6.7300e − 5
8.5252
9.6859e − 5
8.1332
(32, 32)
5.2327e − 6
8.0756
7.5075e − 6
7.9895
8.3529e − 6
8.0571
1.2148e − 5
7.9733
(64, 64)
6.3995e − 7
8.1768
9.3323e − 7
8.0446
1.0259e − 6
8.1423
1.5119e − 6
8.0349
(128, 128)
7.3252e − 8
8.7362
1.1356e − 7
8.2182
1.1946e − 7
8.5879
1.8458e − 7
8.1913
(256, 256)
5.7950e − 9
12.6406
1.2628e − 8
8.9928
1.0553e − 8
11.3198
2.0826e − 8
8.8628Table 4
The simulation results for Example 2.
α
2
=
8
α
2
=
18
n
1
,
n
2
Average errors
Ratio
Average errors
Ratio
(8, 8)
4.9001e − 4
—
6.3035e − 4
—
(16, 16)
5.3408e − 5
9.1748
7.6396e − 5
8.2511
(32, 32)
6.6211e − 6
8.0664
9.5714e − 6
7.9817
(64, 64)
8.1158e − 7
8.1583
1.1905e − 6
8.0396
(128, 128)
9.3748e − 8
8.6571
1.4511e − 7
8.2045
(256, 256)
7.8774e − 9
11.9008
1.6256e − 8
8.9266


## 7. Conclusions

In the paper, we introduce a high-accuracy algorithm using the specific quadrature rule to solve the modified Helmholtz equation with Dirichlet boundary value problems. A few concluding remarks can be made: (1) evaluation on entries of discrete matrices is very simple and straightforward without any singular integrals by the algorithm. Hence, the algorithm is appropriate to solve singularity problems; (2) the numerical results show that the algorithm retains the optimal convergence orderOh03. These are remarkable advantages of the present algorithm, and other existing numerical methods, such as Galerkin methods and collocation methods, did not possess; and (3) this algorithm will be used to solve the three-dimensional axisymmetric boundary integral equations in the future.



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*Source: 6484890-2020-01-20.xml*