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

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.

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 𝑆^𝑘 is constructed for gluing together the grids and the blocks. The uniform estimate of the error of the BGM is of order 𝑂(ℎ^𝑘) (ℎ is the mesh step) when the given boundary function on the boundary of the “nonsingular” part might be from the Hölder classes 𝐶^𝑘,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(ℎ^𝑘/𝑟^{𝑝−1/𝛼𝑗}_𝑗) is obtained in a finite neighborhood of the vertices, where 𝑟_𝑗 is the distance from the current point to the vertex in question, and 𝛼_𝑗𝜋 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 𝑈⁶_ℎ (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 𝜕𝑈⁶_ℎ/𝜕𝑥, 𝜕𝑈⁶_ℎ/𝜕𝑦, 𝜕²𝑈⁶_ℎ/𝜕𝑥², 𝜕²𝑈⁶_ℎ/𝜕𝑦², 𝜕²𝑈⁶_ℎ/𝜕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

Let 𝐺 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 𝛾_𝑗, 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 Δ𝑢=0on𝐺,𝑢=𝜑_𝑗on𝛾_𝑗,𝑗=1,2,…,7, (1)(2) where Δ≡𝜕²/𝜕𝑥²+𝜕²/𝜕𝑦² and 𝜑_𝑗 is the value of the function 𝑣(𝑟,𝜃)=√2𝑟^1/2sin(1/2)𝜃 on 𝛾_𝑗.

In the neighborhood of 𝑂, we construct two fixed block-sectors 𝑇^𝑖=𝑇(𝑟_𝑖)⊂𝐺, 𝑖=1,2, where 0<𝑟₂<𝑟₁<1, 𝑇(𝜌)={(𝑟,𝜃)∶0<𝑟<𝜌,0<𝜃<2𝜋}⊂𝐺.

Let 𝑅₁=1𝑟,𝜃,𝜂2¹_𝑘=0(−1)^𝑘𝑅𝑟𝑟₂^1/2,𝜃2,(−1)^𝑘𝜂2, (3) where 𝑅𝑟,𝜃,𝜂=1−𝑟²2𝜋1−2𝑟cos𝜃−𝜂+𝑟² (4) is the kernel of the Poisson integral for a unit circle.

Lemma 1. The solution 𝑢 of the boundary value problem (1), (2) can be represented on 𝑇²⧵𝑉, in the form 𝑢(𝑟,𝜃)=₀^2𝜋𝑅₁𝑟,𝜃,𝜂𝑢𝑟₂,𝜂𝑑𝜂, (5) 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 𝐺^S which is called the “singular” part of 𝐺 and the set 𝐺^NS=𝐺⧵𝐺^S 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 𝐺_𝑇=𝐺⧵(𝑇⁴) and Π_𝑙⊂𝐺^NS⊂𝐺_𝑇, 𝑙=1(1)5, be fixed open rectangles (see Figure 1). Then the domain 𝐺 can be represented as 𝐺=(∪⁵_𝑙=1Π_𝑙)∪(𝑇³). Let 𝜂_𝑙 be the boundary of the rectangle Π_𝑙 and 𝑡_𝑙=𝜂_𝑙∩𝑡. We define a square grid on Π_𝑙, 𝑙=1(1)5, with step ℎ such that the boundary 𝜂_𝑙 lies entirely on the grid lines. Π^ℎ_𝑙 denotes the set of grid nodes on Π_𝑙, 𝜂^ℎ_𝑙 denotes the set of nodes on 𝜂_𝑙 and Π^ℎ_𝑙=Π^ℎ_𝑙∪𝜂^ℎ_𝑙. We refer to the set of nodes on the closure of 𝜂_𝑙∩𝐺_𝑇 as 𝜂^ℎ_𝑙0, the set of nodes on 𝑡_𝑙 as 𝑡^ℎ_𝑙 and the set of remaining nodes on 𝜂_𝑙 as 𝜂^ℎ_𝑙1. We also introduce the natural number 𝑛≥4, and 𝜃^𝑞=(𝑞−1/2)2𝜋/𝑛, 1≤𝑞≤𝑛. On the arc 𝑉, we choose the points (0.93,𝜃^𝑞),  1≤𝑞≤𝑛, denote the set of these points by 𝑉^𝑛 and let 𝐺_𝑇^ℎ,𝑛=𝑉^𝑛∪(∪⁵_𝑙=1Π^ℎ_𝑙).

Let 𝜑={𝜑_𝑗}⁷_𝑗=1, where 𝜑_𝑗 is the given function in (2). We introduce a gluing operator 𝑆^𝑘, 𝑘=2,6 ([5] for 𝑘=2 and [2–4] for 𝑘=6) at the points of the set 𝑉^𝑛. We denote by 𝑢^𝑘_ℎ(𝑈^𝑘_ℎ) 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∈𝑉^𝑛 in the following way: we consider the set of all rectangles {Π_𝑙} in the intersections of which the point 𝑃 lies, and we choose one of these rectangles Π_𝑙(𝑃) part of whose boundary situated in 𝐺^𝑇 is furthest away from 𝑃. The value 𝑆²(𝑢²_ℎ,𝜑) at the point 𝑃∈𝑉^𝑛 is computed according to the values of the function at the four vertices 𝑃_𝜅, 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 of 𝑆⁶(𝑢⁶_ℎ,𝜑) at the point 𝑃∈𝑉^𝑛 is expressed linearly in terms of the values of 𝑢⁶_ℎ at the points 𝑃_𝜅, 𝜅=1(1)31, of the grid constructed on Π_𝑙(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, 𝑆^𝑘(𝑢^𝑘_ℎ,0) has the representation 𝑆^𝑘𝑢^𝑘_ℎ=∑,0^{1≤𝜅≤31}𝜉_𝜅𝑢^𝑘_ℎ,𝜅∑,for𝑘=6,_1≤𝜅≤4𝜆_𝜅𝑢^𝑘_ℎ,𝜅,for𝑘=2, (6) where 𝑢^𝑘_ℎ,𝜅=𝑢^𝑘_ℎ(𝑃_𝜅), 𝜉_𝜅≥0,_{1≤𝜅≤31}𝜉_𝜅=1,𝜆_𝜅≥0,_1≤𝜅≤4𝜆_𝜅=1, (7) and for the exact solution 𝑢 of the problem (1), (2), we have 𝑢−𝑆⁶𝑢,𝜑=𝑂ℎ⁶,𝑢−𝑆²𝑢,𝜑=𝑂ℎ². (8)

Remark 2. Let 𝑉^𝑛_𝐼⊂𝑉^𝑛 be the set of such points 𝑃∈𝑉^𝑛, for which all points 𝑃_𝜅 in the expression (6) are in ∪⁵_𝑙=1Π^ℎ_𝑙. If some of the points 𝑃_𝜅 in (6) emerge through the side 𝛾_𝑗 when 𝑢=𝜑_𝑗, 1≤𝑗≤7, we denote the set of such points 𝑃 by 𝑉^𝑛_𝐷. Then, according to the construction of 𝑆⁶ in [4] the expression 𝑆⁶(𝑢⁶_ℎ,𝜑) at each point 𝑃∈𝑉^𝑛=𝑉^𝑛_𝐼∪𝑉^𝑛_𝐷 can be expressed as follows: 𝑆⁶𝑢⁶_ℎ=𝑆,𝜑₆𝑢⁶_ℎ,𝑃∈𝑉^𝑛_𝐼,𝑆⁶𝑢⁶_ℎ−∑⁵_𝜏=0𝑎_𝜏Re𝑧^𝜏+∑⁵_𝜏=0𝑎_𝜏Re𝑧^𝜏_𝑃,𝑃∈𝑉^𝑛_𝐷, (9) where 𝑎_𝜏=1𝑑𝜏!^𝜏𝜑_𝑗(𝑠)𝑑𝑠^𝜏_{𝑠=𝑠𝑃},𝜏=0,1,…,5. (10)𝑠_𝑃 corresponds to such point 𝑄∈𝛾_𝑗 for which the line 𝑃𝑄 is perpendicular to 𝛾_𝑗.

Consider for each 𝑘=2,6 the following system of linear algebraic equations: 𝑢^𝑘_ℎ=𝐵_𝑘𝑢^𝑘_ℎonΠ^ℎ_𝑙,𝑢^𝑘_ℎ=√2𝑟^1/21sin2𝜃on𝜂^ℎ_𝑙1∩𝛾_𝑗,𝑢^𝑘_ℎ(𝑟,𝜃)=2𝜋𝑛^𝑛_𝑞=1𝑢^𝑘_ℎ0.93,𝜃^𝑞×𝑅₁𝑟,𝜃,𝜃^𝑞on(𝑟,𝜃)∈𝑡^ℎ_𝑙,𝑢^𝑘_ℎ=𝑆^𝑘𝑢^𝑘_ℎon𝑉^𝑛,1≤𝑙≤5,1≤𝑗≤7, (11)(12)(13)(14) where 𝐵₆𝑢≡𝑢+𝑢𝐵𝑥,𝑦𝑥+ℎ,𝑦+𝑢𝑥,𝑦+ℎ+𝑢𝑥−ℎ,𝑦+𝑢𝑥,𝑦−ℎ/5𝑥+ℎ,𝑦+ℎ+𝑢𝑥−ℎ,𝑦+ℎ+𝑢𝑥−ℎ,𝑦−ℎ+𝑢𝑥+ℎ,𝑦−ℎ/20,₂𝑢𝑥,𝑦≡𝑢𝑥+ℎ,𝑦+𝑢𝑥,𝑦+ℎ+𝑢𝑥−ℎ,𝑦+𝑢𝑥,𝑦−ℎ/4. (15)

Theorem 3. There is a natural number 𝑛₀ such that for all 𝑛≥𝑛₀, and for each 𝑘=2,6, the system (11)–(14) has a unique solution.

Proof. The proof follows when 𝑘=2 from [5], and when 𝑘=6 from [3, 4].

We consider the sector 𝑇^∗=𝑇(0.89), and let 𝑢^𝑘_ℎ(0.93,𝜃^𝑞), 1≤𝑞≤𝑛, be the values of the solution of the system (11)–(14) on 𝑉^𝑛 (at the quadrature nodes). The function 𝑈^𝑘_ℎ(𝑟,𝜃)=2𝜋𝑛^𝑛_𝑞=1𝑅₁𝑟,𝜃,𝜃^𝑞𝑢^𝑘_ℎ0.93,𝜃^𝑞, (16) defined on 𝑇^∗ is called an approximate solution of the problem (1), (2) on the closed block 𝑇³.

Everywhere below we will denote constants which are independent of ℎ and of the cofactors on their right by 𝑐,𝑐₀,𝑐₁ for simplicity.

Theorem 4. There exists a natural number 𝑛₀ such that for 𝑛≥max𝑛₀,ln^1+𝜘ℎ⁻¹+1, (17) where 𝜘>0 is a fixed number, the following inequalities are valid: max_{𝐺𝑇ℎ,𝑛}𝑢^𝑘_ℎ−𝑢≤𝑐ℎ^𝑘,𝑈^𝑘_ℎ(𝑟,𝜃)−𝑢(𝑟,𝜃)≤𝑐₀𝑟^1/2ℎ^𝑘𝑜𝑛𝑇³,𝜕^𝑝𝜕𝑥^𝑝−𝑞𝜕𝑦^𝑞𝑈^𝑘_ℎ(𝑟,𝜃)−𝑢(𝑟,𝜃)≤𝑐₁ℎ^𝑘/𝑟^𝑝−(1/2)𝑜𝑛𝑇³⧵𝑂, (18)(19)(20) 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

Let Π={(𝑥,𝑦)∶𝑎₁<𝑥<𝑎₂,𝑏₁<𝑦<𝑏₂}, where 𝑎₂−𝑎₁=2^𝑝ℎ₀, 𝑏₂−𝑏₁=2^𝑞ℎ₀, 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,…,2^𝑝+𝑚, 𝑗=0,1,…,2^𝑞+𝑚. Let Π_ℎ={(𝑥,𝑦)∶𝑥=𝑥_𝑖=𝑎₁+𝑖ℎ,0<𝑖<2^𝑝+𝑚, 𝑦=𝑦_𝑗=𝑏₁+𝑗ℎ,0<𝑗<2^𝑞+𝑚} and Γ_ℎ=Γ_1ℎ∪Γ_2ℎ∪Γ_3ℎ∪Γ_4ℎ be a set of nodes on Γ (the boundary of Π) where Γ_1ℎ=𝑥,𝑦∶𝑥=𝑎₁+𝑖ℎ,1≤𝑖≤2^𝑝+𝑚,𝑦=𝑏₁,Γ_2ℎ=𝑥,𝑦∶𝑥=𝑎₂,𝑦=𝑏₁+𝑗ℎ,1≤𝑗≤2^𝑞+𝑚,Γ_3ℎ=𝑥,𝑦∶𝑥=𝑎₁+𝑖ℎ,1≤𝑖≤2^𝑝+𝑚,𝑦=𝑏₂,Γ_4ℎ=𝑥,𝑦∶𝑥=𝑎₁,𝑦=𝑏₁+𝑗ℎ,1≤𝑗≤2^𝑞+𝑚. (21)(22)(23)(24)

We consider for each 𝑘=2,6 the finite difference problem 𝑢^𝑘_ℎ=𝐵_𝑘𝑢^𝑘_ℎonΠ_ℎ,𝑢^𝑘_ℎ=𝜑_𝑗ℎonΓ_𝑗ℎ,𝑗=1,2,3,4, (25) where 𝜑_𝑗ℎ is a given function on Γ_𝑗ℎ that vanishes at the end points.

The solution of the problem (25) can be found using the superposition principle 𝑢^𝑘_ℎ=𝑢^𝑘_1ℎ+𝑢^𝑘_2ℎ+𝑢^𝑘_3ℎ+𝑢^𝑘_4ℎ as the sum of solution of four problems of the type 𝑢^𝑘_𝑗ℎ=𝐵_𝑘𝑢_ℎonΠ_ℎ,𝑢^𝑘_𝑗ℎ=𝜑_𝑗ℎonΓ_𝑗ℎ,0onΓ_ℎ⧵Γ_𝑗ℎ,𝑗=1,2,3,4. (26)

The solution of the problem (26), when 𝑗=1 has the representation 𝑢^𝑘_1ℎ=𝑥,𝑦_{2𝑝+𝑚−1𝑛=1}𝑑_𝑛sinh𝛽^𝑘_𝑛1−𝑦/𝑏₂−𝑏₁sinh𝛽^𝑘_𝑛sin𝑛𝜋𝑥𝑎₂−𝑎₁,𝑑_𝑛=2^{21−𝑝−𝑚𝑝+𝑚−1}_𝑟=1𝜑_1ℎ𝑎₁𝑎+𝑟ℎsin𝑛𝜋₁+𝑟ℎ𝑎₂−𝑎₁, (27) where 𝛽²_𝑛=2𝑏₂−𝑏₁ℎsinh⁻¹sin𝑛𝜋ℎ2𝑎₂−𝑎₁, (28) for the 5-point approximation [11], 𝛽⁶_𝑛=2𝑏₂−𝑏₁ℎ×sinh⁻¹sin𝑛𝜋ℎ/2𝑎₂−𝑎₁1−2sin²𝑛𝜋ℎ/2𝑎₂−𝑎₁,/3 (29) 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 approximation 𝑢_ℎ^𝑘(0) to the exact solution 𝑢^𝑘_ℎ of (11)–(14).

Step 2. Finding 𝑢_ℎ^𝑘(1) by the formula (13) on 𝑡^ℎ_𝑙 we solve the system (11), (12) on each grid Π^ℎ_𝑙 by using the representation of finite difference solution described before Step 1.

Step 3. Using (6) we calculate the values 𝑢_ℎ^𝑘(1)(0.93,𝜃^𝑞) at the quadrature nodes for each 𝜃^𝑞=(𝑞−1/2)2𝜋/𝑛, 1≤𝑞≤𝑛 by the formula (14).

Step 4. Repeating Steps 2 and 3 we have the sequence 𝑢_ℎ^𝑘(1),𝑢_ℎ^𝑘(2),…, of Schwarz's iterations defined as follows: 𝑢_ℎ^𝑘(𝑚)(𝑟,𝜃)=2𝜋𝑛^𝑛_𝑞=1𝑅₁𝑟,𝜃,𝜃^𝑞×(𝑟,𝜃)𝑢_ℎ^{𝑘(𝑚−1)}0.93,𝜃^𝑞on𝑡^ℎ_𝑙,𝑢_ℎ^𝑘(𝑚)=𝑆^𝑘𝑢_ℎ^{𝑘(𝑚−1)}on𝑉^𝑛,𝑢_ℎ^𝑘(𝑚)=𝐵_𝑘𝑢_ℎ^𝑘(𝑚)onΠ^ℎ_𝑙,𝑢_ℎ^𝑘(𝑚)=𝜑on𝜂^ℎ_𝑙1,𝑘=2,6,1≤𝑙≤5,𝑚=1,2,…. (30)

As a stopping criteria of the Schwarz's iterations (30), we use the inequality max_{𝜂𝑙0,𝑙=1,2,…,5}|𝑢_ℎ^𝑘(𝑚)−𝑢_ℎ^{𝑘(𝑚−1)}|≤𝜖 for the prescribed accuracy of 𝜖>0.

Step 5. Let 𝑢_ℎ^𝑘(𝑀)(0.93,𝜃^𝑞), 𝜃^𝑞=(𝑞−1/2)2𝜋/𝑛, 1≤𝑞≤𝑛, in (14) be the values at the quadrature nodes on 𝑉^𝑛 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 𝑈_ℎ^𝑘(𝑀)(𝑟,𝜃)=2𝜋𝑛^𝑛_𝑞=1𝑅₁𝑟,𝜃,𝜃^𝑞𝑢_ℎ^𝑘(𝑀)0.93,𝜃^𝑞. (31)

5. Numerical Results

The computational algorithm in Section 4 is applied and the implementation of the block-grid method is carried out using double precision. Let 𝜀_ℎ=𝑈_ℎ^𝑘(𝑀)−𝑢, 𝜀_ℎ⁽¹⁾=𝑟^1/2((𝜕𝑈_ℎ^𝑘(𝑀)/𝜕𝑥)−(𝜕𝑢/𝜕𝑥)), 𝜀_ℎ⁽²⁾=𝑟^3/2((𝜕²𝑈_ℎ^𝑘(𝑀)/𝜕𝑥²)−(𝜕²𝑢/𝜕𝑥²)) be the errors in the “singular” part and 𝜀_ℎ=𝑢_ℎ^𝑘(𝑀)−𝑢 be the error in the “nonsingular” part of the domain 𝐺.

In Table 1 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 with 𝑆² interpolation.

Table 2 

The errors by BGM when 9-point scheme is used with 𝑆⁶ interpolation.

The order of convergence in the “nonsingular” part, and the order of convergence in the “singular” part of 𝐺 are ℜ_𝐺^𝑘,𝜚_NS=max_𝐺^NS𝑢₂^𝑘(𝑀)_−𝜚−𝑢max_𝐺^NS𝑢₂^𝑘(𝑀)_−(𝜚+1),ℜ−𝑢_{𝐺𝑘,𝜚S}=max_𝐺^S𝑈₂^𝑘(𝑀)_−𝜚−𝑢max_𝐺^S𝑈₂^𝑘(𝑀)_−(𝜚+1),−𝑢 (32) 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 convergence 𝑅_𝐺^2,𝜚_NS, and 𝑅_𝐺^2,𝜚_S when ℎ=2^−𝜚.

Table 4 

The order of convergence 𝑅_𝐺^6,𝜚_NS, and 𝑅_𝐺^6,𝜚_S when ℎ=2^−𝜚.

The obtained numerical results in Tables 3 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 𝑈_ℎ^6(𝑀) and the exact solution 𝑢 is illustrated. Figure 5 represents the decrease of the error function |𝑈_ℎ^6(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 𝜕𝑈_ℎ^6(𝑀)/𝜕𝑥, 𝜕𝑈_ℎ^6(𝑀)/𝜕𝑦 are demonstrated in Figure 6 and the shapes (𝜕²𝑈_ℎ^6(𝑀)/𝜕𝑥²)(𝜕²𝑈_ℎ^6(𝑀)/𝜕𝑦²), (𝜕²𝑈_ℎ^6(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 nodes 𝑛, 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 nodes 𝑛, 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.

(a)

(b)

Figure 4 

The highly accurate approximate solution 𝑈ℎ6(𝑀) and the exact solution 𝑢 in the “singular” part for ℎ−1=64, 𝑛=140.

Figure 5 

The error |𝑈ℎ6(𝑀)−𝑢| in the “singular” part for ℎ−1=64, 𝑛=140.

(a)

(b)

Figure 6 

The first partial derivatives 𝜕𝑈ℎ6(𝑀)/𝜕𝑥 and 𝜕𝑈ℎ6(𝑀)/𝜕𝑦 in the “singular” part for ℎ−1=64, 𝑛=140.

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 factor 𝜎. 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 𝜎_𝑘 for 𝑘=2,6: 𝜎_𝑘=lim_𝑟→0𝑈_ℎ^𝑘(𝑀)(𝑟,𝜋)𝑟^1/2=2𝑛(0.93)^𝑛_𝑞=1𝑢_ℎ^𝑘(𝑀)0.93,𝜃^𝑞𝜃sin^𝑞2, (33) where 𝜃^𝑞=(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 fixed 𝑛=140.

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 interpolation 𝑆², 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.