# A Fourth-Order Block-Grid Method for Solving Laplace's Equation on a Staircase Polygon with Boundary Functions inCk,λ

Authors: A. A. Dosiyev; S. Cival Buranay
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The integral representations of the solution around the vertices of the interior reentered angles (on the “singular” parts) are approximated by the composite midpoint rule when the boundary functions are fromC4,λ,0<λ<1. These approximations are connected with the 9-point approximation of Laplace's equation on each rectangular grid on the “nonsingular” part of the polygon by the fourth-order gluing operator. It is proved that the uniform error is of order O(h4+ε), where ε>0 and h is the mesh step. For the p-order derivatives (p=0,1,…) of the difference between the approximate and the exact solutions, in each “ singular” part O((h4+ε)rj1/αj-p) order is obtained; here rj is the distance from the current point to the vertex in question and αjπ is the value of the interior angle of the jth vertex. Numerical results are given in the last section to support the theoretical results.


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## Body

## 1. Introduction

In the last two decades, among different approaches to solve the elliptic boundary value problems with singularities, a special emphasis has been placed on the construction of combined methods, in which differential properties of the solution in different parts of the domain are used (see [1, 2], and references therein).In [2–7], a new combined difference-analytical method, called the block-grid method (BGM), is proposed for the solution of the Laplace equation on polygons, when the boundary functions on the sides causing the singular vertices are given as algebraic polynomials of the arc length. In the BGM, the given polygon is covered by a finite number of overlapping sectors around the singular vertices (“singular” parts) and rectangles for the part of the polygon which lies at a positive distance from these vertices (“nonsingular” part). The special integral representation in each “singular” part is approximated, and they are connected by the appropriate order gluing operator with the finite difference equations used in the “nonsingular” part of the polygon.In [8, 9], the restriction on the boundary functions to be algebraic polynomials on the sides of the polygon causing the singular vertices in the BGM was removed. It was assumed that the boundary function on each side of the polygon is given from the Hölder classes Ck,λ,0<λ<1, and on the “nonsingular” part the 5-point scheme is used when k=2 [8] and the 9-point scheme is used when k=6 [9]. For the 5-point scheme a simple linear interpolation with 4 points is used. For the 9-point scheme an interpolation with 31 points is used to construct a gluing operator connecting the subsystems. Moreover, to connect the quadrature nodes which are at a distance of less than 4h from boundary of the polygon, a special representation of the harmonic function through the integrals of Poisson type for a half plane is used (see [9]).In this paper the BGM is developed for the Dirichlet problem when the boundary function on each side of the polygon is fromC4,λ, by using the 9-point scheme on the “nonsingular” part with 16-point gluing operator for all quadrature nodes, including those near the boundary. The paper is organized as follows: in Section 2, the boundary value problem and the integral representations of the exact solution in each “singular” part are given. In Section 3, to support the aim of the paper, a Dirichlet problem on the rectangle for the known exact solution from Ck,λ,  k=3,4, is solved using the 9-point scheme and the numerical results are illustrated. In Section 4, the system of block-grid equations and the convergence theorems are given. In Section 5 a highly accurate approximation of the coefficient of the leading singular term of the exact solution (stress intensity factor) is given. In Section 6 the method is illustrated for solving the problem in L-shaped polygon with the corner singularity. The conclusions are summarized in Section 7.


## 2. Dirichlet Problem on a Staircase Polygon

LetG be an open simply connected polygon, γj, j=1,2,…,N, its sides, including the ends, enumerated counterclockwise, γ=γ1∪⋯∪γN the boundary of G, and αjπ, (αj=1/2,1,3/2,2), the interior angle formed by the sides γj-1 and γj,(γ0=γN). Denote by Aj=γj-1∩γj the vertex of the jth angle and by rj,θj a polar system of coordinates with a pole in Aj, where the angle θj is taken counterclockwise from the side γj.We consider the boundary value problem(1)Δu=0onG,u=φj(s)onγj,1≤j≤N,whereΔ≡∂2/∂x2+∂2/∂y2,φj is a given function on γj of the arc length s taken along γ, and φj∈C4,λ(γj),0<λ<1; that is, φj has the fourth-order derivative on γj, which satisfies a Hölder condition with exponent λ.At some verticesAj,(s=sj) for αj=1/2 the conjugation conditions
(2)φj-1(2q)(sj)=(-1)qφj(2q)(sj),q=0,1are fulfilled. For the remaining verticesAj, the values of φj-1 and φj at Aj might be different. Let E be the set of all j,(1≤j≤N) for which αj≠1/2 or αj=1/2, but (2) is not fulfilled. In the neighborhood of Aj,j∈E, we construct two fixed block sectors Tji=Tj(rji)⊂G,i=1,2, where 0<rj2<rj1<min{sj+1-sj,sj-sj-1},Tj(r)={(rj,θj):0<rj<r,0<θj<αjπ}.Let (see [10])
(3)φj0(t)=φj(sj+t)-φj(sj),φj1(t)=φj-1(sj-t)-φj-1(sj),(4)Qj(rj,θj)=φj(sj)+(φj-1(sj)-φj(sj))θjαjπ+1π∑k=01‍∫0σjk‍y~jφjk(tαj)dt(t-(-1)kx~j)2+y~j2,
where
(5)x~j=rj1/αjcos(θjαj),y~j=rj1/αjsin(θjαj),σjk=|sj+1-k-sj-k|1/αj.The functionQj(rj,θj) is harmonic on Tj1 and satisfies the boundary conditions in (1) on γj-1∩T¯j1 and γj∩T¯j1, j∈E, except for the point Aj whenφj-1(sj)≠φj(sj).We formally set the value ofQj(rj,θj) and the solution u of the problem (1) at the vertex Aj equal to (φj-1(sj)+φj(sj))/2,j∈E.Let(6)Rj(r,θ,η)=1αj∑k=01‍(-1)kR((rrj2)1/αj,θαj,(-1)kηαj),nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnj∈E,
where
(7)R(r,θ,η)=1-r22π(1-2rcos(θ-η)+r2)
is the kernel of the Poisson integral for a unit circle.Lemma 1 (see [10]).
The solutionu of the boundary value problem (1) can be represented on T¯j2∖Vj,j∈E, in the form
(8)u(rj,θj)=Qj(rj,θj)+∫0αjπ‍Rj(rj,θj,η)(u(rj2,η)-Qj(rj2,η))dη,
whereVj is the curvilinear part of the boundary of  Tj2, and Qj(rj,θj) is the function defined by (4).


## 3. 9-Point Solution on Rectangles

LetΠ={(x,y):0<x<a,0<y<b} be a rectangle, with a/b being rational, γj,j=1,2,3,4 the sides, including the ends, enumerated counterclockwise, starting from the left side (γ0≡γ4,γ5≡γ1), and γ=∪j=14γj the boundary of Π.We consider the boundary value problem(9)Δu=0onΠ,u=φjonγj,j=1,2,3,4,whereφj is the given function on γj.Definition 2.
One says that the solutionu of the problem (9) belongs to C~4,λ(Π¯) if
(10)φj∈C4,λ(γj),0<λ<1,j=1,2,3,4,
and at the vertices Aj=γj-1∩γj the conjugation conditions
(11)φj(2q)=(-1)qφj-1(2q),q=0,1
are satisfied.Remark 3.
From Theorem 3.1 in [11] it follows that the class of functions C~4,λ(Π¯) is wider than C4,λ(Π¯).Leth>0, with a/h≥2, b/h≥2 integers. We assign to Πh a square net on Π, with step h, obtained with the lines ,y=0,h,2h,…. Let γjh be a set of nodes on the interior of γj and let
(12)γ.jh=γj∩γj+1,γh=∪j=14(γjh∪γ.jh),Π¯h=Πh∪γh.We consider the system of finite difference equations(13)uh=BuhonΠh,uh=φjonγjh,j=1,2,3,4,
where
(14)Bu(x,y)≡(u(x+h,y)+u(x,y+h)+u(x-h,y)+u(x,y-h))5+(20-1((u(x+h,y+h)+u(x-h,y+h)nnnnnnnnn+u(x-h,y-h)+u(x+h,y-h)))nnnnnnn×20-1).On the basis of the maximum principle the unique solvability of the system of finite difference equations (13) follows (see [12, Chapter 4]).Everywhere below we will denote constants which are independent ofh and of the cofactors on their right by c,c0,c1,…, generally using the same notation for different constants for simplicity.Theorem 4.
Letu be the solution of problem (9). If u∈C~4,λ(Π¯), then
(15)maxΠ¯h|uh-u|≤ch4,
whereuh is the solution of the system (13).Proof.
For the proof of this theorem see [13].LetΠ′={(x,y):-0.25<x<0.25,0<y<1} and let γ′ be the boundary of Π′. We consider the Dirichlet problem
(16)Δu=0onΠ′,u=vonγ′,wherev=rk+λcos(k+λ)θ,r=x2+y2,0<λ<1, is the exact solution of this problem, which is from Ck,λ(Π¯′).We solve the problem (16) by approximating 9-point scheme when k=3,4 for the different values of λ.In Figure1, the order of numerical convergence
(17)ℜΠhϱ=maxΠh|u2-ϱ-u|maxΠh|u2-(ϱ+1)-u|
of the 9-point solution uh, for different h=2-ϱ and ϱ=4,5,6,7, is demonstrated. These results show that the order of numerical convergence, when the exact solution u∈Ck,λ(Π¯), depends on k and λ and is O(h4) when k=4, which supports estimation (15). Moreover, this dependence also requires the use of fourth-order gluing operator for all quadrature nodes in the construction of the system of block-grid equations, when the given boundary functions are from the Hölder classes C4,λ.Figure 1
Dependence of the approximate solutions for the boundary functions fromCk,λ.


## 4. System of Block-Grid Equations

In addition to the sectorsTj1 and Tj2 (see Section 2) in the neighborhood of each vertex Aj,j∈E of the polygon G, we construct two more sectors Tj3 and Tj4, where 0<rj4<rj3<rj2, rj3=(rj2+rj4)/2 and Tk3∩Tl3=∅, k≠l,k,l∈E, and let GT=G∖(∪j∈ETj4).We cover the given solution domain (a staircase polygon) by the finite number of sectorsTj1,j∈E, and rectangles Πk⊂GT,  k=1,2,…,M, as is shown in Figure 2, for the case of L-shaped polygon, where j=1,M=4 (see also [2]). It is assumed that for the sides a1k and a2k of Πk the quotient a1k/a2k is rational and G=(∪k=1MΠk)∪(∪j∈ETj3). Let ηk be the boundary of the rectangle Πk, let Vj be the curvilinear part of the boundary of the sector Tj2, and let tkj=ηk∩T¯j3. We choose a natural number n and define the quantities n(j)=max{4,[αjn]},βj=αjπ/n(j), and θjm=(m-1/2)βj,j∈E,1≤m≤n(j). On the arc Vj we take the points (rj2,θjm),1≤m≤n(j), and denote the set of these points by Vjn. We introduce the parameter h∈(0,ϰ0/4], where ϰ0 is a gluing depth of the rectangles Πk,k=1,2,…,M, and define a square grid on Πk,k=1,2,…,M, with maximal possible step hk≤min{h,min{a1k,a2k}/6} such that the boundary ηk lies entirely on the grid lines. Let Πkh be the set of grid nodes on Πk, let ηkh be the set of nodes on ηk, and let Π¯kh=Πkh∪ηkh. We denote the set of nodes on the closure of ηk∩GT by ηk0h, the set of nodes on tkj by tkjh, and the set of remaining nodes on ηk by ηk1h.Figure 2
Description of BGM for the L-shaped domain.Let(18)ωh,n=(∪k=1Mηk0h)∪(∪j∈EVjn),G¯Th,n=ωh,n∪(∪k=1MΠ¯kh).Letφ={φj}j=1N, where φj∈C4,λ(γj),0<λ<1, is the given function in (1). We use the matching operator S4 at the points of the set ωh,n constructed in [14]. The value of S4(uh,φ) at the point P∈ωh,n is expressed linearly in terms of the values of uh at the points Pk of the grid constructed on Πk(P),(P∈Πk(P)) some part of whose boundary located in G is the maximum distance away from P, and in terms of the boundary values of φ(m),m=0,1,2,3 at a fixed number of points. Moreover S4(uh,0) has the representation
(19)S4(uh,0)=∑0≤l≤15‍ξluh,l,
where uh,k=uh(Pk),
(20)ξl≥0,∑0≤l≤15‍ξl=1,(21)u-S4(u,φ)=O(h4).LetωIh,n⊂ωh,n be the set of such points P∈ωh,n, for which all points Pl in expression (19) are in ∪k=1MΠ¯kh. If some points Pl in (19) emerge through the side γm, then the set of such points P is denoted by ωDh,n. According to the construction of S4 in [14], the expression S4(uh,φ) at each point P∈ωh,n=ωIh,n∪ωDh,n can be expressed as follows:
(22)S4(uh,φ)={S4uh,P∈ωIh,n,S4(uh-∑k=03‍akRezk)+(∑k=03‍akRezk)P,P∈ωDh,n,
where
(23)ak=1k!dkφm(s)dsk|s=sP,k=0,1,2,3andsP corresponds to such a point Q∈γm for which the line PQ is perpendicular to γm.Let(24)Qj=Qj(rj,θj),Qj2q=Qj(rj2,θjq).
The quantities in (24) are given by (4) and (5), which contain integrals that have not been computed exactly in the general case. Assume that approximate values Qjε and Qj2qε of the quantities in (24) are known with accuracy ε>0; that is,
(25)|Qjε-Qj|≤c1ε,|Qj2qε-Qj2q|≤c2ε,wherej∈E, 1≤q≤n(j), and c1, c2 are constants independent of ε.Consider the system of linear algebraic equations(26)uhε=BuhεonΠkh,uhε=φmonηk1h∩γm,uhε(rj,θj)=Qjε+βj∑q=1n(j)‍(uhε(rj2,θjq)-Qj2qε)nnnnnnnnnnnn×Rj(rj,θj,θjq)on(rj,θj)∈tkjh,uhε=S4uhεonωh,n,where1≤k≤M,1≤m≤N, and  j∈E.LetTj*=Tj(rj*) be the sector, where rj*=(rj2+rj3)/2,j∈E, and let uhε(rj2,θjq),  1≤q≤n(j),  j∈E, be the solution values of the system (26) on Vjh (at the quadrature nodes). The function
(27)Uhε(rj,θj)=Qj(rj,θj)+βj∑q=1n(j)‍Rj(rj,θj,θjq)(uhε(rj2,θjq)-Qj2qε),
defined on Tj*, is called an approximate solution of the problem (1) on the closed block T¯j3,  j∈E.Definition 5.
The system (26) and (27) is called the system of block-grid equations.Theorem 6.
There is a natural numbern0, such that for all n≥n0 and for any ε>0 the system (26) has a unique solution.Proof.
From the estimation(2.29) in [15] follows the existence of the positive constants n0 and σ, such that for all n≥n0(28)max(rj,θj)∈T¯j3βj∑q=1n(j)‍Rj(rj,θj,θjq)≤σ<1.
The proof is obtained on the basis of principle of maximum by taking into account (14), (19), (20), and (28).Theorem 7.
There exists a natural numbern0, such that for all
(29)n≥max{n0,[ln1+ϰh-1]+1},
where ϰ>0 is a fixed number, and for any ε>0 the following inequalities are valid:
(30)maxG¯Th,n|uhε-u|≤c(h4+ε),(31)|∂p∂xp-q∂yq(Uhε(rj,θj)-u(rj,θj))|≤cp(h4+ε)onT¯j3,
for integer 1/αj when p≥1/αj,
(32)|∂p∂xp-q∂yq(Uhε(rj,θj)-u(rj,θj))|≤cp(h4+ε)rp-1/αjonT¯j3,
for any 1/αj, if 0≤p<1/αj,
(33)|∂p∂xp-q∂yq(Uhε(rj,θj)-u(rj,θj))|≤cp(h4+ε)rp-1/αjonT¯j3∖Aj,
for noninteger 1/αj, when p>1/αj. Everywhere 0≤q≤p,u is the exact solution of the problem (1) and Uhε(rj,θj) is defined by formula (27).Proof.
Let(34)ξhε=uhε-u,
where uhε is a solution of system (26) and u is the trace on  G¯Th,n of the solution of (1). On the basis of (1), (26), and (34) the error ξhε satisfies the system of difference equations
(35)ξhε=Bξhε+rh1onΠkh,ξhε=0onηk1h,ξhε(rj,θj)=βj∑q=1n(j)‍ξhε(rj2,θjq)Rj(rj,θj,θjq)nnnnnnnn+rjh2,(rj,θj)∈tkjh,ξhε=S4ξhε+rh3onωh,n,
where 1≤k≤M,  j∈E,
(36)rh1=Bu-uon∪k=1MΠkh,(37)rjh2=βj∑q=1n(j)‍(u(rj2,θjq)-Qj2qε)Rj(rj,θj,θjq)-(u-Qjε)on∪k=1M(∪j∈Etkjh),(38)rh3={S4u-uonωIh,n,S4(u-∑k=03‍akRezk)-(u-∑k=03‍akRezk)P,P∈ωDh,n.
On the basis of estimations (15), (21), (25), and Lemma 1 by analogy to the proof of Theorem 4.3 in [9] the proof of inequality (30) follows.
The functionUhε(rj,θj)given by formula (27), defined on the closed sector T¯j*,j∈E, where rj*=(rj2+rj3)/2, and the integral representation (8) of the exact solution of the problem (1) is given on T¯j2∖Vj,  j∈E, and then the difference function ζhε(rj,θj)=Uhε(rj,θj)-u(rj,θj) is defined on T¯j*,  j∈E and
(39)ζhε(rj*,0)=ζhε(rj*,αjπ)=0,j∈E.
On the basis of Lemma 6.11 from [16], (25), and (28), for n≥max{n0,[ln1+ϰh-1]+1}, ϰ>0 is a fixed number, and we obtain
(40)|ζhε(rj,θj)|≤c(h4+ε)onT¯j*,j∈E.
Furthermore, the function ζhε(rj,θj) continuous on T¯j* is a solution of the following Dirichlet problem:
(41)Δζhε=0onTj*,ζhε=0onγm∩T¯j*,m=j-1,j,ζhε(rj*,θj)=Uhε(rj*,θj)-u(rj*,θj),0≤θj≤αjπ.
Since Tj3⊂T¯j*, on the basis of (39) and (40), from Lemma 6.12 in [16], inequalities (31)–(33) of Theorem 7 follow.


## 5. Stress Intensity Factor

Let, in the conditionφj∈C4,λ(γj), the exponent λ be such that
(42){αj(4+λ)}≠0,{2αj(4+λ)}≠0,where{·} is the symbol of fractional part. These conditions for the given αj can be fulfilled by decreasing λ.On the basis of Section 2 of [11], a solution of the problem (1) can be represented in T¯j*,j∈E, as follows:
(43)u(xj,yj)=u~(xj,yj)+∑k=04‍μk(j)Im{zklnz}+∑k=1nαj‍τk(j)rjk/αjsinkθjαj,
where nαj=[αj(4+λ)],  [·] is the integer part, z=xj+iyj,  μk(j) and τk(j) are some numbers, and u~(xj,yj)∈C4,λ(Tj2) is the harmonic on Tj2. By taking θj=αjπ/2, from the formula (43), it follows that the coefficient τ1(j) which is called the stress intensity factor can be represented as
(44)τ1(j)=limrj→01rj1/αj(∑k=04‍μk(j)Im{zklnz}u(xj,yj)-u~(xj,yj)nnnnnnnnnnnn-∑k=04‍μk(j)Im{zklnz}).From formula (44) it follows that τ1(j) can be approximated by
(45)τ1,n(j)ε=limrj→01rj1/αj((φj(sj)+(φj-1(sj)-φj(sj))θjαjπ)Uhε(rj,θj)nnnnnnnnnnn-(φj(sj)+(φj-1(sj)-φj(sj))θjαjπ)).Using formula (3), (4), and (27) from (45) for the stress intensity factor (see [17]), we obtain the next formula:
(46)τ1,n(j)ε=1π∫0σj0‍φj0(tαj)dtt2+1π∫0σj1‍φj1(tαj)dtt2+2n(j)rj21/αj∑q=1n(j)‍(uhε(rj2,θjq)-Qj2qε)sin1αjθjq.
This formula is obtained for the second-order BGM in [8].


## 6. Numerical Results

LetG be L-shaped and defined as follows:
(47)G={(x,y):-1<x<1,-1<y<1}∖Ω,whereΩ={(x,y):0≤x≤1,-1≤y≤0} and γ is the boundary of G.We consider the following problem:(48)Δu=0inG,u=v(r,θ)onγ,
where
(49)v(r,θ)=r2/3sin(23θ)+0.0051r16/3cos(163θ)is the exact solution of this problem.We choose a “singular” part ofG as
(50)GS={(x,y):-0.5<x<0.5,-0.5<y<0.5}∖Ω1,whereΩ1={(x,y):0≤x≤0.5,-0.5≤y≤0}. Then GNS=G∖GS is a “nonsingular” part of G.The given domainG is covered by four overlapping rectangles Πk, k=1,…,4, and by the block sector T13 (see Figure 2). For the boundary of GS on G is the polygonal line t1=abcde. The radius r12 of sector T12 is taken as 0.93. According to (49), the function Q(r,θ) in (4) is
(51)Q(r,θ)=0.0051π∫01‍y~t8dt(t-x~)2+y~2+0.0051π∫01‍y~t8dt(t-x~)2+y~2,wherex~=r2/3cos(2θ/3) and  y~=r2/3sin(2θ/3). Since we have only one singular point, we omit subindices in (51). We calculate the values Qε(r12,θq) and Qε(r,θ) on the grids t1h, with an accuracy of ε using the quadrature formulae proposed in [10].On the basis of (46) and (51), for the stress intensity factor, we have
(52)τ1,nε=0.01027π+2n(0.93)2/3∑q=1n‍(uhε(0.93,θjq)-Qj2qε)sin23θjq.Taking the zero approximationuhε(0)=0, the results of realization of the Schwarz iteration (see [2]) for the solution of the problem (48) are given in Tables 1, 2, 3, and 4. Tables 1 and 2 represent the order of convergence. Table 6 shows a highly accurate approximation of the stress intensity factor by the proposed fourth order BGM
(53)ℜGNSϱ=maxGNS|u2-ϱε-u|maxGNS|u2-(ϱ+1)ε-u|
in the “nonsingular” and the order of convergence
(54)ℜGSϱ=maxGS|U2-ϱε-u|maxGS|U2-(ϱ+1)ε-u|Table 1
The order of convergence in the “nonsingular” part whenh=2-ϱ and ε=5×10-13.
(
2
-
ϱ
,
n
)
∥
ζ
h
ε
∥
G
N
S
ℜ
G
N
S
ϱ
(
2
-
4
,
60
)
1.609
×
1
0
-
8
15.577
(
2
-
5
,
170
)
1.033
×
1
0
-
9
(
2
-
5
,
130
)
1.191
×
1
0
-
9
16.690
(
2
-
6
,
150
)
7.136
×
1
0
-
11
(
2
-
5
,
140
)
1.136
×
1
0
-
9
16.259
(
2
-
6
,
170
)
6.991
×
1
0
-
11
(
2
-
6
,
100
)
2.169
×
1
0
-
10
17.096
(
2
-
7
,
130
)
1.269
×
1
0
-
11Table 2
The order of convergence in the “singular” part whenh=2-ϱ and ε=5×10-13.
(
2
-
ϱ
,
n
)
∥
ζ
h
ε
∥
G
S
ℜ
G
S
ϱ
(
2
-
4
,
100
)
1.931
×
1
0
-
8
16.078
(
2
-
5
,
150
)
1.182
×
1
0
-
9
(
2
-
5
,
130
)
1.294
×
1
0
-
9
16.789
(
2
-
6
,
150
)
7.708
×
1
0
-
11
(
2
-
5
,
140
)
1.312
×
1
0
-
9
17.967
(
2
-
6
,
170
)
7.304
×
1
0
-
11
(
2
-
6
,
100
)
2.389
×
1
0
-
10
18.164
(
2
-
7
,
130
)
1.315
×
1
0
-
11Table 3
The minimum errors of the solution over the pairs(h-1,n) in maximum norm when ε=5×10-13.
(
h
-
1
,
n
)
∥
ζ
h
ε
∥
G
N
S
∥
ζ
h
ε
∥
G
S
Iteration
(
16,70
)
1.139
×
1
0
-
8
1.572
×
1
0
-
8
22
(
32,170
)
1.033
×
1
0
-
9
1.184
×
1
0
-
9
23
(
64,170
)
6.990
×
1
0
-
11
7.304
×
1
0
-
11
24
(
128,200
)
8.628
×
1
0
-
12
8.833
×
1
0
-
12
25Table 4
InGS∩r≥0.2, the minimum errors of the derivatives over the pairs (h-1,n) in maximum norm when ε=5×10-13.
(
h
-
1
,
n
)
Ma
x
G
S
∩
{
r
≥
0.2
}
r
1
/
3
∥
∂
U
h
ε
∂
x
-
∂
u
∂
x
∥
Max
G
S
∩
{
r
≥
0.2
}
r
1
/
3
∥
∂
U
h
ε
∂
y
-
∂
u
∂
y
∥
(
16,70
)
3.895
×
1
0
-
7
3.895
×
1
0
-
7
(
32,170
)
4.627
×
1
0
-
8
4.627
×
1
0
-
8
(
64,170
)
1.124
×
1
0
-
9
3.125
×
1
0
-
9
(
128,200
)
2.214
×
1
0
-
10
2.233
×
1
0
-
10in the “singular” parts ofG, respectively, for ε=5×10-13, where ϱ is a positive integer. In Table 3, the minimal values over the pairs (h-1,n) of the errors in maximum norm, of the approximate solution when ε=5×10-13, are presented. The similar values of errors for the first-order derivatives are presented in Table 4, when ∂Q/∂x and ∂Q/∂y are approximated by fourth-order central difference formula on GS for r≥0.2. For r<0.2, the order of errors decreases down to 10-6, which are presented in Table 5. This happens because the integrands in (51) are not sufficiently smooth for fourth-order differentiation formula. The order of accuracy of the derivatives for r<0.2 can be increased if we use similar quadrature rules, which we used for the integrals in (51) for the derivatives of integrands also.Table 5
InGS, the minimum errors of the derivatives over the pairs (h-1,n) in maximum norm when ε=5×10-13.
(
h
-
1
,
n
)
Max
G
S
r
1
/
3
∥
∂
U
h
ε
∂
x
-
∂
u
∂
x
∥
Max
G
S
r
1
/
3
∥
∂
U
h
ε
∂
y
-
∂
u
∂
y
∥
(
16,70
)
9.663
×
1
0
-
6
9.663
×
1
0
-
6
(
32,170
)
9.653
×
1
0
-
6
9.653
×
1
0
-
6
(
64,170
)
9.649
×
1
0
-
6
9.649
×
1
0
-
6
(
128,200
)
9.648
×
1
0
-
6
9.648
×
1
0
-
6Table 6
The stress intensity  factorτ1,nϵ  for  n=70,170,200 when  ε=5×10-13.
h
-
1
τ
1,70
ε
τ
1,170
ε
τ
1,200
ε
16
1.000000014856688
1.000000017180415
1.000000017197438
32
1.000000005800844
1.000000001230267
1.000000001236709
64
1.000000004138169
1.000000000073107
1.000000000079938
128
1.000000004053153
1.000000000003531
1.000000000003267Figures3 and 4 show the dependence on ε for different mesh steps h. Figure 5 demonstrates the convergence of the BGM with respect to the number of quadrature nodes for different mesh steps h. The approximate solution and the exact solution in the “singular” part are given in Figure 6, to illustrate the accuracy of the BGM. The error of the block-grid solution, when the function Q(r,θ) in (51) is calculated with an accuracy of ε=5×10-13, is presented in Figure 7. Figures 8 and 9 show the singular behaviour of the first-order partial derivatives in the “singular” part. The ratios ℜGSϱ and ℜGNSϱ, when ϱ=5 with respect to different n values for h-1=64 and for a fixed value of n of h-1=32, are illustrated in Figures 10 and 11, respectively. These ratios show that the order of the convergence in both the “singular” and the “nonsingular” parts is asymptotically equal to 16 when n is kept fixed for h-1=32, and it is selected as large as possible (n>100) for h-1=64.Dependence onε for h-1=16,32.
(a)
(b)Dependence onε for h-1=64,128.
(a)
(b)Maximum error depending on the number of quadrature nodesn.
(a)
(b)
(c)
(d)The approximate solutionUhε and the exact solution u in the “singular” part for ε=5×10-13.
(a)
(b)Figure 7
The error function in “singular” part whenε=5×10-13.Figure 8
∂
U
h
ε
/
∂
x in the “singular” part.Figure 9
∂
U
h
ε
/
∂
y in the “singular” part.Figure 10
ℜ
G
S
ϱ when ϱ=5 by fixing n for h-1=32 for different n values of h-1=64.Figure 11
ℜ
G
N
S
ϱ when ϱ=5 by fixing n for h-1=32 for different n values of h-1=64.


## 7. Conclusions

In the block-grid method (BGM) for solving Laplace's equation, the restriction on the boundary functions to be algebraic polynomials on the sides of the polygon causing the singular vertices is removed. This condition is replaced by the functions from the Hölder classesC4,λ,  0<λ<1. In the integral representations around singular vertices (on the “singular” part), which are combined with the 9-point finite difference equations on the “nonsingular” part of the polygon, the boundary conditions are taken into account with the help of integrals of Poisson type for a half-plane. To connect the subsystems, a homogeneous fourth-order gluing operator is used. It is proved that the final uniform error is of order O(h4+ε), where ε is the error of the approximation of the mentioned integrals and h is the mesh step. For the p-order derivatives (p=0,1,…) of the difference between the approximate and the exact solutions, in each “singular” part O((h4+ε)rj1/αj-p) order is obtained. The method is illustrated in solving the problem in L-shaped polygon with the corner singularity. Dependence of the approximate solution and its errors on ε,h and the number of quadrature nodes n are demonstrated. Furthermore, by the constructed approximate solution on the “singular” part of the polygon, a highly accurate formula for the stress intensity factor is given.From the error estimation formula (33) of Theorem 7 it follows that the error of the approximate solution on the block sectors decreases as rj1/αj(h4+ε), which gives an additional accuracy of the BGM near the singular points.The method and results of this paper are valid for multiply connected polygons.



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*Source: 864865-2013-06-20.xml*