# Analysis of the Block-Grid Method for the Solution of Laplace's Equation on Polygons with a Slit

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

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## Abstract

The error estimates obtained for solving Laplace's boundary value problem on polygons by the block-grid method contain constants that are difficult to calculate accurately. Therefore, the experimental analysis of the method could be essential. The real characteristics of the block-grid method for solving Laplace's equation on polygons with a slit are analysed by experimental investigations. The numerical results obtained show that the order of convergence of the approximate solution is the same as in the case of a smooth solution. To illustrate the singular behaviour around the singular point, the shape of the highly accurate approximate solution and the figures of its partial derivatives up to second order are given in the “singular” part of the domain. Finally a highly accurate formula is given to calculate the stress intensity factor, which is an important quantity in fracture mechanics.


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## Body

## 1. Introduction

In the past few decades, in order to improve the accuracy and resolve the convergence difficulties that appear in the neighbourhood of singular points, many different methods have been proposed for the numerical solution of plane elliptic boundary value problems with singularities. Among many approaches, 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]).In [2–6] a new combined difference-analytical method called the block-grid method (BGM) is given for solving the Laplace equation on polygons, when the boundary functions on the sides causing the singular vertices are given as algebraic polynomials of the arclength. This method is a combination of the exponentially convergent block method (see [7, 8]) in the “singular” part, and the finite difference method, which has a simple structure on the “nonsingular” part of the polygon. A kth order gluing operator Sk is constructed for gluing together the grids and the blocks. The uniform estimate of the error of the BGM is of order O(hk) (h is the mesh step) when the given boundary function on the boundary of  the “nonsingular” part might be from the Hölder classes Ck,λ, 0<λ<1 (see [2–4] for k=6, [6] for k=4, and [5] for k=2). For the errors of p-order derivatives (p=1,2,…) the estimation O(hk/rjp-1/αj) is obtained in a finite neighborhood of the vertices, where rj is the distance from the current point to the vertex in question, and αjπ is the value of the interior angle at the considered vertex. Moreover, BGM can give a simple and highly accurate formula for the stress intensity factor which is an important quantity from an engineering standpoint.The experimental investigation of the block-grid method is important and numerical results could be interesting to support the theoretical results in [2–6]. The objective of this paper is to analyze the real characteristics of the BGM for solving the Laplace equation on polygons with a slit. For this purpose a slit problem on a square domain whose exact solution is known is considered. The computational algorithm by the BGM with 5-point and 9-point schemes is given and implemented. The obtained numerical results justify the theoretical results given in [2–5]. Moreover, for the approximate solution Uh6 (by 9-point scheme with S6) and the error function the graphs are given to demonstrate the high accuracy of the block-grid method. The shapes of the partial derivatives ∂Uh6/∂x, ∂Uh6/∂y, ∂2Uh6/∂x2, ∂2Uh6/∂y2, ∂2Uh6/∂x∂yare given to illustrate the singular behavior in the “singular” part of the domain. Furthermore, a simple and highly accurate formula is given to calculate the stress intensity factor.The experimental analyses of the different methods on slit problems were given in many papers (see [9, 10]).


## 2. The Slit Problem and the Integral Representation of the Solution

LetG be an open domain in the plane xOy, that is obtained from the unit square G={(x,y):|x|<1,|y|<1} by making a cut OA along the positive semiaxis Ox from the center (see Figure 1). Let γj, j=1(1)7, be its sides, including the ends, enumerated counterclockwise, γ=γ1∪⋯∪γ7, (γ0=γ7), be the boundary of G, 2π is the interior angle formed by the sides γ1 and γ0. Denote by O=γ0∩γ1 the vertex of this angle and let r,θ be a polar system of coordinates with a pole in O, where the angle θ is taken counterclockwise from the side γ1.Figure 1
Covering the square domain with a slit by overlapping rectangles and sector.We consider the boundary value problem(1)Δu=0onG,(2)u=φjonγj,j=1,2,…,7,
where  Δ≡∂2/∂x2+∂2/∂y2 and φj is the value of the function v(r,θ)=2r1/2sin(1/2)θ on γj.In the neighborhood ofO, we construct two fixed block-sectors Ti=T(ri)⊂G, i=1,2, where 0<r2<r1<1, T(ρ)={(r,θ):0<r<ρ,0<θ<2π}⊂G.Let(3)R1(r,θ,η)=12∑k=01(-1)kR((rr2)1/2,θ2,(-1)kη2),
where
(4)R(r,θ,η)=1-r22π(1-2rcos(θ-η)+r2)
is the kernel of the Poisson integral for a unit circle.Lemma 1.
The solutionu of the boundary value problem (1), (2) can be represented on T¯2∖V, in the form
(5)u(r,θ)=∫02πR1(r,θ,η)u(r2,η)dη,
where V is the curvilinear part of the boundary of T2.Proof.
The proof follows from Theorem 3.1 in [8] by taking into account that φ0=φ1=0.


## 3. The Block-Grid Method for the Slit Problem

The realization of the BGM for the solution of the problem (1), (2) is as follows. Let T2=T(0.93) and t be a polygonal line abcde which lies on T2 with a positive distance from the vertex O and from the curvilinear boundary V={(r,θ):r=0.93,0<θ<2π} of T2. The set of points T2 from O up to t is denoted by GS which is called the “singular” part of G and the set GNS=G∖GS is the “nonsingular” part of G. In addition to the sector T2 in the neighborhood of the vertex O of the polygon G we construct two more sectors T3=T(0.85) and T4=T(5). Let GT=G∖(T4) and Πl⊂GNS⊂GT, l=1(1)5, be fixed open rectangles (see Figure 1). Then the domain G can be represented as G=(∪l=15Πl)∪(T3). Let ηl be the boundary of the rectangle Πl and tl=ηl∩t. We define a square grid on Πl, l=1(1)5, with step h such that the boundary ηl lies entirely on the grid lines. Πlh denotes the set of grid nodes on Πl, ηlh denotes the set of nodes on ηl and Π¯lh=Πlh∪ηlh. We refer to the set of nodes on the closure of ηl∩GT as ηl0h, the set of nodes on tl as tlh and the set of remaining nodes on ηl as ηl1h. We also introduce the natural number n≥4, and  θq=(q-1/2)2π/n, 1≤q≤n. On the arc V, we choose the points (0.93,θq),  1≤q≤n, denote the set of these points by Vn and let G¯Th,n=Vn∪(∪l=15Π¯lh).Letφ={φj}j=17, where φj is the given function in (2). We introduce a gluing operator Sk, k=2,6 ([5] for k=2 and [2–4] for k=6) at the points of the set Vn. We denote by uhk(Uhk) the approximate solution of the problem (1), (2) obtained by the 5-point scheme with S2 for k=2, and by the 9-point scheme with S6 for k=6, on the “singular” (“nonsingular”) part of G. The operator S2 is defined at each point P∈Vn in the following way: we consider the set of all rectangles {Πl} in the intersections of which the point P lies, and we choose one of these rectangles Πl(P) part of whose boundary situated in GT is furthest away from P. The value S2(uh2,φ) at the point P∈Vn is computed according to the values of the function at the four vertices Pκ, κ=1(1)4, of the closure of the cell, containing the point P of the grid constructed on Π¯l(P) by multilinear interpolation.The value ofS6(uh6,φ) at the point P∈Vn is expressed linearly in terms of the values of uh6 at the points Pκ, κ=1(1)31, of the grid constructed on Πl(P)∋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(τ), τ=0,1,…,5 at a fixed number of points. Moreover, Sk(uhk,0) has the representation
(6)Sk(uhk,0)={∑1≤κ≤31ξκuh,κk,fork=6,∑1≤κ≤4λκuh,κk,fork=2,
where uh,κk=uhk(Pκ),
(7)ξκ≥0,∑1≤κ≤31‍ξκ=1,λκ≥0,∑1≤κ≤4‍λκ=1,
and for the exact solution u of the problem (1), (2), we have
(8)u-S6(u,φ)=O(h6),u-S2(u,φ)=O(h2).Remark 2.
LetVIn⊂Vn be the set of such points P∈Vn, for which all points Pκ in the expression (6) are in ∪l=15Π¯lh. If some of the points Pκ in (6) emerge through the side γj when u=φj, 1≤j≤7, we denote the set of such points P by VDn. Then, according to the construction of S6 in [4] the expression S6(uh6,φ) at each point P∈Vn=VIn∪VDn can be expressed as follows:
(9)S6(uh6,φ)={S6uh6,P∈VIn,S6(uh6-∑τ=05aτRezτ)+(∑τ=05aτRezτ)P,P∈VDn,
where
(10)aτ=1τ!dτφj(s)dsτ|s=sP,τ=0,1,…,5.sP corresponds to such point Q∈γj for which the line PQ is perpendicular to γj.Consider for eachk=2,6 the following system of linear algebraic equations:
(11)uhk=BkuhkonΠlh,(12)uhk=2r1/2sin12θonηl1h∩γj,(13)uhk(r,θ)=2πn∑q=1n‍uhk(0.93,θq)2πn∑q=1n‍×R1(r,θ,θq)on(r,θ)∈tlh,(14)uhk=SkuhkonVn,1≤l≤5,1≤j≤7,
where
(15)B6u(x,y)≡(u(x+h,y)+u(x,y+h)+u(x-h,y)+u(x,y-h))/5+(u(x+h,y+h)+u(x-h,y+h)+u(x-h,y-h)+u(x+h,y-h))/20,B2u(x,y)≡(u(x+h,y)+u(x,y+h)+u(x-h,y)+u(x,y-h))/4.Theorem 3.
There is a natural numbern0 such that for all n≥n0, and for each k=2,6, the system (11)–(14) has a unique solution.Proof.
The proof follows whenk=2 from [5], and when k=6 from [3, 4].We consider the sectorT*=T(0.89), and let uhk(0.93,θq), 1≤q≤n, be the values of the solution of the system (11)–(14) on Vn (at the quadrature nodes). The function
(16)Uhk(r,θ)=2πn∑q=1n‍R1(r,θ,θq)uhk(0.93,θq),
defined on T* is called an approximate solution of the problem (1), (2) on the closed block T¯3.Everywhere below we will denote constants which are independent ofh and of the cofactors on their right by c,c0,c1 for simplicity.Theorem 4.
There exists a natural numbern0 such that for
(17)n≥max{n0,[ln1+ϰh-1]+1},
where ϰ>0 is a fixed number, the following inequalities are valid:
(18)maxG¯Th,n|uhk-u|≤chk,(19)|(Uhk(r,θ)-u(r,θ))|≤c0r1/2hkonT¯3,(20)|∂p∂xp-q∂yq(Uhk(r,θ)-u(r,θ))|≤c1hk/rp-(1/2)onT¯3∖O,
for all p=1,2,…. Everywhere 0≤q≤p, u is a solution of the problem (1), (2).Proof.
The proof is carried out analogically to the proof of Theorems 1 and 2 in [3].


## 4. Computational Algorithm

LetΠ={(x,y):a1<x<a2,b1<y<b2}, where a2-a1=2ph0, b2-b1=2qh0, h0>0 is a fixed number, and p and q are integers. We introduce a square grid with the lines x=a1+ih, y=b1+jh, h=h02-m, m≥0 is an integer, i=0,1,…,2p+m, j=0,1,…,2q+m. Let Πh={(x,y):x=xi=a1+ih,0<i<2p+m, y=yj=b1+jh,0<j<2q+m} and Γh=Γ1h∪Γ2h∪Γ3h∪Γ4h be a set of nodes on Γ (the boundary of Π) where
(21)Γ1h={(x,y):x=a1+ih,1≤i≤2p+m,y=b1},(22)Γ2h={(x,y):x=a2,y=b1+jh,1≤j≤2q+m},(23)Γ3h={(x,y):x=a1+ih,1≤i≤2p+m,y=b2},(24)Γ4h={(x,y):x=a1,y=b1+jh,1≤j≤2q+m}.We consider for eachk=2,6 the finite difference problem
(25)uhk=BkuhkonΠh,uhk=φjhonΓjh,j=1,2,3,4,
where φjh is a given function on Γjh that vanishes at the end points.The solution of the problem (25) can be found using the superposition principle uhk=u1hk+u2hk+u3hk+u4hk as the sum of solution of four problems of the type
(26)ujhk=BkuhonΠh,ujhk={φjhonΓjh,0onΓh∖Γjh,j=1,2,3,4.The solution of the problem (26), when j=1 has the representation
(27)u1hk(x,y)=∑n=12p+m-1dnsinh(βnk(1-y/(b2-b1)))sinhβnksinnπxa2-a1,dn=21-p-m∑r=12p+m-1‍φ1h(a1+rh)sinnπ(a1+rh)a2-a1,
where
(28)βn2=2(b2-b1)hsinh-1(sinnπh2(a2-a1)),
for the 5-point approximation [11],
(29)βn6=2(b2-b1)h×sinh-1(sinnπh/2(a2-a1)1-2sin2(nπh/2(a2-a1))/3),
for the 9-point approximation [12].The Discrete Fast Fourier Transform is used for the realization of the finite sums in (27). The solution of the problem (26), for j=2,3,4 can be represented analogously.Now we describe the algorithm of implementing the BGM for the slit problem.Step 1.
Suppose that we have zero approximationuhk(0) to the exact solution uhk of (11)–(14).Step 2.
Findinguhk(1) by the formula (13) on tlh we solve the system (11), (12) on each grid Π¯lh by using the representation of finite difference solution described before Step 1.Step 3.
Using (6) we calculate the values uhk(1)(0.93,θq) at the quadrature nodes for each θq=(q-1/2)2π/n, 1≤q≤n by the formula (14).Step 4.
Repeating Steps2 and 3 we have the sequence uhk(1),uhk(2),…, of Schwarz's iterations defined as follows:
(30)uhk(m)(r,θ)=2πn∑q=1n‍R1(r,θ,θq)2πn∑q=1n‍×(r,θ)uhk(m-1)(0.93,θq)ontlh,uhk(m)=Skuhk(m-1)onVn,uhk(m)=Bkuhk(m)onΠlh,uhk(m)=φonηl1h,k=2,6,1≤l≤5,m=1,2,….As a stopping criteria of the Schwarz's iterations (30), we use the inequality maxηl0,l=1,2,…,5|uhk(m)-uhk(m-1)|≤ϵ for the prescribed accuracy of ϵ>0.Step 5.
Letuhk(M)(0.93,θq), θq=(q-1/2)2π/n, 1≤q≤n,  in (14) be the values at the quadrature nodes on Vn for the final iteration m=M. Using these values we can calculate the value of the solution at any point in the singular part by the explicit formula
(31)Uhk(M)(r,θ)=2πn∑q=1n‍R1(r,θ,θq)uhk(M)(0.93,θq).


## 5. Numerical Results

The computational algorithm in Section4 is applied and the implementation of the block-grid method is carried out using double precision. Let εh=Uhk(M)-u, εh(1)=r1/2((∂Uhk(M)/∂x)-(∂u/∂x)), εh(2)=r3/2((∂2Uhk(M)/∂x2)-(∂2u/∂x2)) be the errors in the “singular” part and εh=uhk(M)-u be the error in the “nonsingular” part of the domain G.In Table1 the errors are given by the BGM when 5-point scheme with S2 interpolation is used, and the iterations are terminated by using ϵ=5×10-8. Table 2 represents the errors by the BGM when 9-point scheme is used with S6 and the stopping criteria for the Schwarz's iterations is taken as ϵ=5×10-14.Table 1
The errors by BGM when 5-point scheme is used withS2 interpolation.
(
h
-
1
,
n
)
∥
ε
h
∥
(
G
NS
)
∥
ε
h
∥
(
G
S
)
∥
ε
h
(
1
)
∥
(
G
S
)
∥
ε
h
(
2
)
∥
(
G
S
)
iter
(16, 85)
6.138
×
10
-
5
2.3127
×
10
-
5
6.432
×
10
-
5
4.464
×
10
-
4
12
(16, 100)
5.264
×
10
-
5
1.393×10-5
5.126
×
10
-
5
3.244
×
10
-
4
12
(16, 120)
5.599
×
10
-
5
1.150
×
10
-
5
5.385
×
10
-
5
3.430
×
10
-
4
12
(32, 85)
1.317
×
10
-
5
4.676
×
10
-
6
1.129
×
10
-
5
2.272
×
10
-
5
13
(32, 100)
1.488
×
10
-
5
1.889
×
10
-
6
2.056
×
10
-
5
1.523
×
10
-
4
13
(32, 120)
1.491
×
10
-
5
1.956
×
10
-
6
2.053
×
10
-
5
1.740
×
10
-
4
13
(32, 130)
1.508
×
10
-
5
5.172
×
10
-
6
1.728
×
10
-
5
4.659
×
10
-
5
13
(32, 140)
1.571
×
10
-
5
3.319
×
10
-
6
2.407
×
10
-
5
2.155
×
10
-
4
13
(64, 130)
3.720
×
10
-
6
7.391
×
10
-
7
2.306
×
10
-
6
1.941
×
10
-
5
14
(64, 140)
3.583
×
10
-
6
7.071
×
10
-
7
3.852
×
10
-
6
3.364
×
10
-
5
14Table 2
The errors by BGM when 9-point scheme is used withS6 interpolation.
(
h
-
1
,
n
)
∥
ε
h
∥
(
G
NS
)
∥
ε
h
∥
(
G
S
)
∥
ε
h
(
1
)
∥
(
G
S
)
∥
ε
h
(
2
)
∥
(
G
S
)
iter.
(16, 100)
2.741
×
10
-
9
4.895
×
10
-
10
6.778
×
10
-
10
2.293
×
10
-
9
23
(16, 145)
2.789
×
10
-
9
3.786
×
10
-
10
5.108
×
10
-
10
1.859
×
10
-
9
23
(32, 100)
4.706
×
10
-
11
7.158
×
10
-
12
2.593
×
10
-
10
4.691
×
10
-
9
24
(32, 125)
4.805
×
10
-
11
2.694
×
10
-
12
6.623
×
10
-
12
4.358
×
10
-
11
24
(32, 130)
4.838
×
10
-
11
1.831
×
10
-
12
7.257
×
10
-
12
5.186
×
10
-
11
24
(32, 145)
4.745
×
10
-
11
3.903
×
10
-
12
8.808
×
10
-
12
5.379
×
10
-
11
24
(64, 125)
7.856
×
10
-
13
2.220
×
10
-
14
1.296
×
10
-
12
3.364
×
10
-
11
25
(64, 130)
7.545
×
10
-
13
4.097
×
10
-
14
5.376
×
10
-
13
4.157
×
10
-
11
25
(64, 145)
7.503
×
10
-
13
4.396
×
10
-
14
1.614
×
10
-
13
1.312
×
10
-
12
25The order of convergence in the “nonsingular” part, and the order of convergence in the “singular” part ofG are
(32)ℜGNSk,ϱ=maxGNS|u2-ϱk(M)-u|maxGNS|u2-(ϱ+1)k(M)-u|,ℜGSk,ϱ=maxGS|U2-ϱk(M)-u|maxGS|U2-(ϱ+1)k(M)-u|,
respectively, where ϱ is a positive integer, M is the final iteration number (Section 4), k=2,6. Taking h=2-ϱ, ϱ=4,5,6, Tables 3 and 4 represent the order of convergence of the BGM in the “nonsingular” part and the “singular” part of the domain G for k=2 and k=6, respectively.Table 3
The order of convergenceRGNS2,ϱ, and RGS2,ϱ when h=2-ϱ.
(
2
-
ϱ
,
n
)
ℜ
G
NS
2
,
ϱ
ℜ
G
S
2
,
ϱ
(2−4, 85)
4.659
4.9459
(2−5, 85)
(2−4, 100)
3.5376
7.3742
(2−5, 100)
(2−4, 120)
3.7551
5.8793
(2−5, 120)
(2−5, 130)
4.0756
6.9976
(2−6, 130)
(2−5, 140)
4.3845
4.6938
(2−6, 140)Table 4
The order of convergenceRGNS6,ϱ, and RGS6,ϱ when h=2-ϱ.
(
2
-
ϱ
,
n
)
ℜ
G
NS
6
,
ϱ
ℜ
G
S
6
,
ϱ
(
2
-
4
,
100
)
58.253
68.386
(
2
-
5
,
100
)
(
2
-
4
,
145
)
58.788
97.005
(
2
-
5
,
145
)
(
2
-
5
,
125
)
61.164
65.711
(
2
-
6
,
125
)
(
2
-
5
,
130
)
64.140
74.926
(
2
-
6
,
130
)
(
2
-
5
,
145
)
63.241
88.92
(
2
-
6
,
145
)The obtained numerical results in Tables3 and 4 show that the order of convergence of the approximate solution is O(h2) for the 5-point scheme with S2 interpolation (k=2) and it is O(h6) for the 9-point scheme with S6 interpolation (k=6) in the “nonsingular” part. In both tables, the order of convergence in the “singular” part is higher than the order of convergence in the “nonsingular” part of the domain, which justifies the estimation (19) in Theorem 4. The errors with respect to the number of quadrature nodes n in the “singular” part and in the “nonsingular” part by BGM for k=2, and k=6 are given in Figures 2 and 3, respectively. These figures demonstrate that the error in the “singular” part is less than the error in the “nonsingular” part for sufficiently large n as it follows from the estimation (19) in Theorem 4. The graphical results in Figures 4–9 are obtained by the BGM when 9-point scheme is used with S6 interpolation for h-1=64, n=140. In Figure 4, the highly accurate approximate solution Uh6(M) and the exact solution u is illustrated. Figure 5 represents the decrease of the error function |Uh6(M)-u| in the “singular” part of the domain as r approaches to zero, which agrees with the estimation (19) in Theorem 4. Moreover, on the “singular” part, up to second order derivatives of the solution at grid points are approximated effectively by a simple differentiation of the function (31). The shapes of the first partial derivatives ∂Uh6(M)/∂x, ∂Uh6(M)/∂y are demonstrated in Figure 6 and the shapes (∂2Uh6(M)/∂x2)(∂2Uh6(M)/∂y2), (∂2Uh6(M)/∂x∂y) are given in Figures 7, 8, and 9, respectively, to show the singular behaviour of the solution around the singular point.Figure 2
The errors with respect to number of quadrature nodesn, in  the “singular” part and in the “nonsingular” part by the BGM when 5-point scheme is used with S2 for h-1=32,64.Figure 3
The errors with respect to number of quadrature nodesn, in the “singular” part and in the “nonsingular” part by the BGM when 9-point scheme is used with S6 for h-1=32,64.The highly accurate approximate solutionUh6(M) and the exact solution u in the “singular” part for h-1=64, n=140.
(a)
(b)Figure 5
The error|Uh6(M)-u| in the “singular” part for h-1=64, n=140.The first partial derivatives∂Uh6(M)/∂x and ∂Uh6(M)/∂y in the “singular” part for h-1=64, n=140.
(a)
(b)Figure 7
The second partial derivative∂2Uh6(M)/∂x2 in the “singular” part for h-1=64, n=140.Figure 8
The second partial derivative∂2Uh6(M)/∂y2 in the “singular” part for h-1=64, n=140.Figure 9
The mixed partial derivative∂2Uh6(M)/∂x∂y in the “singular” part for h-1=64, n=140.


### 5.1. Stress Intensity Factor

In engineering problems a very important constant is the so-called stress intensity factorσ. This constant gives a measure of “the amount of torsion the beam can endure before fracture occurs” [10, 13]. On the basis of (31) we give a simple and highly accurate formula for the stress intensity factor σ denoting by σk for k=2,6:
(33)σk=limr→0Uhk(M)(r,π)r1/2=2n(0.93)∑q=1n‍uhk(M)(0.93,θq)sinθq2,
where θq=(q-1/2)2π/n and M is the final iteration number. The exact value of the stress intensity factor σ is 2. For fixed number of quadrature nodes n=140, the second column in Table 5 represents the error of the stress intensity factor when 5-point scheme is used with S2 and the last column represents this error when 9-point scheme is used with S6.Table 5
The error of the stress intensity factor for fixedn=140.
h
-
1
|
σ
2
-
σ
|
|
σ
6
-
σ
|
16
3.4791
×
10
-
6
3.7655
×
10
-
10
32
3.0364
×
10
-
6
1.9451
×
10
-
12
64
1.4899
×
10
-
8
5.9952
×
10
-
15


## 5.1. Stress Intensity Factor

In engineering problems a very important constant is the so-called stress intensity factorσ. This constant gives a measure of “the amount of torsion the beam can endure before fracture occurs” [10, 13]. On the basis of (31) we give a simple and highly accurate formula for the stress intensity factor σ denoting by σk for k=2,6:
(33)σk=limr→0Uhk(M)(r,π)r1/2=2n(0.93)∑q=1n‍uhk(M)(0.93,θq)sinθq2,
where θq=(q-1/2)2π/n and M is the final iteration number. The exact value of the stress intensity factor σ is 2. For fixed number of quadrature nodes n=140, the second column in Table 5 represents the error of the stress intensity factor when 5-point scheme is used with S2 and the last column represents this error when 9-point scheme is used with S6.Table 5
The error of the stress intensity factor for fixedn=140.
h
-
1
|
σ
2
-
σ
|
|
σ
6
-
σ
|
16
3.4791
×
10
-
6
3.7655
×
10
-
10
32
3.0364
×
10
-
6
1.9451
×
10
-
12
64
1.4899
×
10
-
8
5.9952
×
10
-
15


## 6. Conclusion

For the solution of the Laplace equation on polygons with a slit, the real characteristics of the block-grid method is investigated. The given polygon is decomposed into five overlapping rectangles and one sector. In the sector, we approximate the special integral representation of the solution, which takes into account the behaviour of the exact solution near the end point of the slit. On the rectangles, to approximate Laplace's equation on square grids either 5-point scheme is used which is simpler by means of sparsity, or 9-point scheme is used, which gives a highly accurate approximation. In correspondence with the finite difference scheme used, a gluing together of the subsystems is carried out effectively by a sufficiently simple linear interpolationS2, or a highly accurate interpolation S6. By choosing the step size h=2-4,2-5,2-6, the obtained numerical results show that the order of convergence of the approximate solution is O(h2) for the 5-point scheme with S2 and it is O(h6) for the 9-point scheme with S6 in the “nonsingular” part. The results also show that the order of convergence in the “singular” part is higher than the order of convergence in the “nonsingular” part of the domain. This conclusion justifies the theoretical results obtained in [2–5]. Moreover, the shapes up to the second-order derivatives of the highly accurate solution obtained by the BGM are shown to display the singular behaviour at the end point of the slit. Finally the stress intensity factor is approximated by the given highly accurate formula.



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*Source: 948564-2013-06-04.xml*