A fourth order blockhexagonal grid approximation for the solution of Laplace’s equation with singularities
 Adiguzel A Dosiyev^{1}Email author and
 Emine Celiker^{1}
https://doi.org/10.1186/s1366201504079
© Dosiyev and Celiker; licensee Springer. 2015
Received: 1 December 2014
Accepted: 9 February 2015
Published: 24 February 2015
Abstract
The hexagonal grid version of the blockgrid method, which is a differenceanalytical method, has been applied for the solution of Laplace’s equation with Dirichlet boundary conditions, in a special type of polygon with corner singularities. It has been justified that in this polygon, when the boundary functions away from the singular corners are from the Hölder classes \(C^{4,\lambda}\), \(0<\lambda<1\), the uniform error is of order \(O(h^{4})\), h is the step size, when the hexagonal grid is applied in the ‘nonsingular’ part of the domain. Moreover, in each of the finite neighborhoods of the singular corners (‘singular’ parts), the approximate solution is defined as a quadrature approximation of the integral representation of the harmonic function, and the errors of any order derivatives are estimated. Numerical results are presented in order to demonstrate the theoretical results obtained.
Keywords
hexagonal grid Laplace’s equation singularity problem blockgrid method1 Introduction
It is well known that angular singularities arise in many applied problems when the solution of Laplace’s equation is considered, and that finitedifference and finiteelement methods may become less accurate when singularities are not taken into account. In the last two decades, for the solution of singularity problems, various combined and highly accurate methods have been proposed (see [1–8], and references therein).
Among these methods the blockgrid method (BGM), presented in [6–8], on polygons with interior angles \(\alpha_{j}\pi\), \(j=1,2,\ldots,N\), where \(\alpha_{j}\in \{ \frac{1}{2},1,\frac{3}{2},2 \} \) (staircase polygons), requires the finite neighborhood of the singular vertices to be covered by sectors (blocks), and the remaining part of the domain by overlapping rectangles (‘nonsingular’ part). The finitedifference method with square grids is used for the approximate solution in the ‘nonsingular’ part, and in the blocks the integral representations of the harmonic function are approximated by the exponentially convergent midpoint quadrature rule (see [9]). Finally these subsystems are connected together by constructing an appropriate order matching operator. BGM is a highly accurate method not only for the approximation of the solution, but also for the approximation of its derivatives around singular points.
In this paper, the fourth order BGM is extended and justified for the Dirichlet problem of Laplace’s equation on polygons with interior angles \(\alpha_{j}\pi\), where \(\alpha_{j}\in \{ \frac{1}{3},\frac{2}{3} ,1,2 \} \) (nonstaircase), by gluing with the matching operator the 7point approximation on a hexagonal grid in the ‘nonsingular’ part and the approximation of the integral representations around the singular points (on ‘singular’ parts).
An advantage of using the hexagonal grid version of BGM in this domain is that a highly accurate approximation on the irregular grids is not required as in [8]. Thus the realization of the total system of algebraic equations becomes simpler. This may not be the case for this type of domain when square or rectangular grids are applied.
Furthermore it is justified that, when the boundary functions on the sides except the adjacent sides of the singular vertices are given in \(C^{4,\lambda}\), \(0<\lambda<1\), the proposed hexagonal grid version of BGM has an accuracy of \(O ( h^{4} ) \), h is the mesh step. The same order of accuracy is obtained for the 9point scheme on a square grid (see [10, 11]).
Finally in the last section of the paper, numerical experiments are demonstrated to support the theoretical results obtained.
2 Boundary value problem on a special type of polygon
Let D be an open simply connected polygon, \(\gamma_{j}\), \(j=1,2,\ldots,N\), be its sides, including the ends, enumerated counterclockwise (\(\gamma _{0}\equiv\gamma_{N}\), \(\gamma_{1}\equiv\gamma_{N+1}\)), and let \(\alpha _{j}\pi\), \(\alpha_{j}\in \{ \frac{1}{3},\frac{2}{3},1,2 \} \), be the interior angles formed by the sides \(\gamma_{j1}\) and \(\gamma_{j}\). Furthermore, let \(\dot{\gamma}_{j}=\gamma_{j1}\cap\gamma_{j}\) be the jth vertex of D, \(\gamma=\bigcup_{j=1}^{N}\gamma_{j}\) be the boundary of D; s is the arclength measured along the boundary of D in the positive direction, and \(s_{j}\) is the value of s at \(\dot{\gamma }_{j}\). We denote by \(r_{j}\), \(\theta_{j}\) the polar system of coordinates with poles in \(\dot{\gamma}_{j}\) and the angle \(\theta _{j}\) is taken counterclockwise from the side \(\gamma_{j}\).
Let \(E= \{ j:\alpha_{j}\neq1/3, 1\leq j\leq N \} \). We construct two fixed block sectors in the neighborhood of \(\dot{\gamma}_{j}\), \(j\in E\), denoted by \(T_{j}^{i}=T_{j}(r_{ji})\subset D\), \(i=1,2\), where \(0< r_{j2}<r_{j1}<\min\{ s_{j+1}s_{j},s_{j}s_{j1} \} \), \(T_{j}(r)= \{ ( r_{j},\theta _{j} ) :0< r_{j}<r, 0<\theta_{j}<\alpha_{j}\pi \} \). On the closed sector \(\overline{T}_{j}^{1}\), \(j\in E\), we consider the carrier function \(Q_{j}(r_{j},\theta_{j})\) in the form given in [12], which satisfies the boundary conditions (2) on \(\gamma_{j1}\cap\overline{T}_{j}^{1}\) and \(\gamma_{j}\cap\overline{ T}_{j}^{1}\), \(j\in E\).
The following lemma acts as a basis for the approximation of the solution around the vertices \(\dot{\gamma}_{j}\) with angle \(\alpha_{j}\pi\), \(\alpha_{j}\neq1/3\).
Lemma 2.1
([12])
 (1)
We blockade the singular corners \(\dot{\gamma}_{j}\), \(j\in E\), by the double sectors \(T_{j}^{i}(r_{ji})\), \(i=2,3\), with \(T_{k}^{2}\cap T_{l}^{3}=\emptyset\), \(k\neq l\), \(k,l\in E\), and cover the polygon D by overlapping parallelograms denoted by \(D_{l}^{\prime}\), \(l=1,2,\ldots ,M\), and sectors \(T_{j}^{3}\), \(j\in E\), such that the distance from \(\overline{ D}_{l}^{\prime}\) to \(\dot{\gamma}_{j}\) is not less that \(r_{j4}\) for all \(l=1,2,\ldots,M\).
 (2)
On the parallelograms \(\overline{D}_{l}^{\prime}\), \(l=1,2,\ldots,M\), we use the 7point scheme for the hexagonal grid with step size \(h_{l}\leq h\), h a parameter, for the approximation of Laplace’s equation, and the singular part \(\overline{T}_{j}^{3}\), \(j\in E\), is approximated by using the harmonic function defined in Lemma 2.1.
 (3)
The fourth order matching operator in a hexagonal grid is applied for connecting the subsystems.
 (4)
Schwarz’s alternating procedure is used for solving the finitedifference system formed for Laplace’s equation on the parallelograms covering \(D_{T}\).
Let \(D_{l}^{\prime}\subset D_{T}\), \(l=1,2,\ldots,M\), be open fixed parallelograms and \(D\subset ( \bigcup_{l=1}^{M}D_{l}^{\prime} ) \cup ( \bigcup_{j\in E}T_{j}^{3} ) \subset D\). We denote by \(\eta _{l}\) the boundary of \(D_{l}^{\prime}\), \(l=1,2,\ldots M\), by \(V_{j}\) the curvilinear part of the boundary of the sector \(T_{j}^{2}\), and let \(t_{j}= ( \bigcup_{l=1}^{M}\eta_{l} ) \cap\overline{T}_{j}^{3}\). For the arrangement of the parallelograms \(D_{l}^{\prime}\), \(l=1,2,\ldots ,M\), it is required that any point P lying on \(\eta_{l}\cap D_{T}\), \(1\leq l\leq M\), or lying on \(V_{j}\cap D\), \(j\in E\), falls inside at least on of the parallelograms \(D_{l(P)}^{\prime}\), \(1\leq l(P)\leq M\), depending on P, where the distance from P to \(D_{T}\cap\eta_{l(P)}\) is not less than some constant \(\kappa_{0}\) independent of P.
Let \(h\in( 0,\kappa_{0}/4 ] \) be a parameter, and define a hexagonal grid on \(D_{l}^{\prime}\), \(1\leq l\leq M\), with maximal positive step \(h_{l}\leq h\), such that the boundary \(\eta_{l}\) lies entirely on the grid lines. Let \(D_{lh}^{\prime}\) be the set of grid nodes on \(D_{l}^{\prime}\), \(\eta_{l}^{h}\) be the set of nodes on \(\eta_{l}\), and let \(\overline{D}_{lh}^{\prime}=D_{lh}^{\prime}\cup\eta_{l}^{h}\). Furthermore, \(\eta_{l0}^{h}\) denotes the set of nodes on \(\eta_{l}\cap D_{T}\), \(\eta_{l1}^{h}=\eta_{l}^{h}\backslash\eta_{l0}^{h}\), and \(t_{j}^{h}\) denotes the set of nodes on \(t_{j}\).
The solution of this system is the approximation of the solution of problem (1), (2) on \(\overline{D}_{\ast}^{h,n}\).
Theorem 2.2
There is a natural number \(n_{0}\) such that for all \(n\geq n_{0}\) the system of equations (8)(11) has a unique solution.
Proof
Taking into account the corresponding homogeneous system of system (8)(10), the proof follows by analogy to Lemma 2 in [6]. □
3 Error analysis of the 7point approximation on the special parallelogram
Since (17) has nonnegative coefficients and their sum is equal to 1, the solution of system (15), (16) exists and is unique (see [13]).
Everywhere below we will denote constants which are independent of h and of the cofactors on their right by \(c,c_{0},c_{1},\ldots\) , generally using the same notation for different constants for simplicity.
Lemma 3.1
Proof
The closed parallelogram \(\overline{D}^{\prime}\) lies inside the polygon D defined in Section 2, and the vertices \(\dot{\gamma} _{m}^{\prime}\) with an interior angle of \(\alpha_{m}\pi\neq\pi/3\) are located either inside of D or on the interior of a side \(\gamma_{m}\) of D, \(1\leq m\leq N\). Since the boundary functions (14), by the definition of the boundary functions \(\varphi_{j}\) in problem (1), (2) satisfy conditions (3), (4), from the results in [14], (20) follows. □
Let \(D_{h,k}^{\prime}\) be the set of nodes whose distance from the point \(P\in D_{h}^{\prime}\) to \(\gamma_{h}^{\prime}\) is \(\frac{\sqrt{3}}{2} kh\), \(1\leq k\leq a^{\ast}\), where \(a^{\ast}= [ \frac{2d_{t}}{\sqrt{3} h} ] \), \([ c ] \) denotes the integer part of c, and \(d_{t}\) is the minimum of the halflengths of the sides of the parallelogram.
Lemma 3.2
Proof
The proof follows analogously to the proof of the comparison theorem given in [13]. □
Lemma 3.3
Proof
Let \(D_{1h}^{\prime}\) contain the set of nodes whose distance from the boundary \(\gamma^{\prime}\) is \(\frac{\sqrt{3}h}{2}\), and hence for \((x,y)\in D_{1h}^{\prime}\), \((x+sH,y+sK)\in\overline{D}^{\prime}\) for \(0\leq s\leq1\), \(H=\pm\frac{h}{2}, \pm h\), \(K=0, \pm\frac{\sqrt{3}h}{2} \), \(H^{2}+K^{2}>0\), and \(D_{2h}^{\prime}=D_{h}^{\prime}\backslash D_{1h}^{\prime}\).
4 Error analysis of the hexagonal blockgrid equations
Lemma 4.1
Proof
The function \(S^{4}(u,\varphi)\) is defined as (3.14) in [8]. Keeping in mind the positioning of the points in \(\omega^{h,n}\), conditions (3), (4), and estimation (4.64) in [14], it follows that the fourth order partial derivatives of the exact solution of problem (1), (2) are bounded on \(D_{T}\). Then estimation (41) follows from the construction of the operator \(S^{4}\). □
Lemma 4.2
Proof
The proof follows by analogy to the proof of Lemma 6.2 in [7]. □
Theorem 4.3
Proof
Hence (42) follows. □
For the approximation of (12), we consider the following theorem.
Theorem 4.4

For \(\alpha_{j}=1\), \(p\geq1\),$$ \biggl\vert \frac{\partial^{p}}{\partial x^{pq}\, \partial y^{q}} \bigl( U_{h}(r_{j},\theta _{j})u(r_{j},\theta_{j}) \bigr) \biggr\vert \leq c_{p}h^{4}\quad\textit{on }\overline{T}_{j}^{3}. $$

For \(\alpha_{j}=\frac{2}{3},1,2\), \(0\leq p\leq\frac{1}{\alpha_{j}}\),$$ \biggl\vert \frac{\partial^{p}}{\partial x^{pq}\, \partial y^{q}} \bigl( U_{h}(r_{j},\theta _{j})u(r_{j},\theta_{j}) \bigr) \biggr\vert \leq c_{p}h^{4}r_{j}^{1/\alpha_{j}p}\quad\textit{on }\overline{T}_{j}^{3}. $$

For \(\alpha_{j}=\frac{2}{3},2\), \(p>\frac{1}{\alpha_{j}}\),$$ \biggl\vert \frac{\partial^{p}}{\partial x^{pq}\, \partial y^{q}} \bigl( U_{h}(r_{j},\theta _{j})u(r_{j},\theta_{j}) \bigr) \biggr\vert \leq c_{p}h^{4}/r_{j}^{p1/\alpha_{j}}\quad \textit{on }\overline{T}_{j}^{3}\backslash\dot{\gamma }_{j}, $$
5 Numerical results
Two examples have been solved in order to test the effectiveness of the proposed method. In Example 5.1, it is assumed that there is a slit in the domain D, thus causing a strong singularity at the origin. The vertex \(\dot{\gamma}_{1}\) containing the singularity has an interior angle of \(\alpha_{1}\pi=2\pi\). The exact solution of this problem is assumed to be known. In Example 5.2, we consider a problem with two singularities. The vertices which contain the singularities have interior angles of \(\alpha_{j}\pi=\frac{2}{3}\pi\), \(j=2,4\). In this example, the exact solution is not known.
After separating the ‘singular’ part in Example 5.1, the remaining part of the domain is covered by 5 overlapping parallelograms, whereas in Example 5.2, the ‘nonsingular’ part of the domain is covered by only two parallelograms. For the solution of the blockgrid equations, Schwarz’s alternating method is used. In each Schwarz iteration the system of equations on the parallelograms are solved by the block GaussSeidel method. The carrier function is constructed for each example, taking into consideration the boundary conditions given on the adjacent sides of the vertices in the ‘singular’ parts. Furthermore, the derivatives are approximated in the ‘singular’ parts for both of the examples.
Results obtained for the slit problem
\(\boldsymbol{(h^{1},n)}\)  \(\boldsymbol{\Vert uu_{h}\Vert _{\overline{D}_{NS}}}\)  \(\boldsymbol{\Vert uu_{h}\Vert _{\overline{D}_{S}}}\)  \(\boldsymbol{R_{D_{NS}}^{m}}\)  \(\boldsymbol{R_{D_{S}}^{m}}\) 

(16,70)  5.924280 × 10^{−5}  5.191270 × 10^{−7}  
(32,70)  3.910378 × 10^{−6}  4.794595 × 10^{−8}  15.1501  10.8273 
(64,110)  2.478126 × 10^{−7}  2.558563 × 10^{−9}  15.7796  18.7394 
(128,130)  1.56560 × 10^{−8}  1.27915 × 10^{−10}  15.8286  20.0021 
Example 5.1
Consider the open parallelogram \(D= \{ ( x,y ) :\frac{\sqrt{3}}{2}< y<\frac{\sqrt{3}}{2},1 \frac{y}{\sqrt{3}}<x<1\frac{y}{\sqrt{3}} \} \). We assume that there is a slit along the straight line \(y=0\), \(0\leq x\leq1\). Let \(\gamma_{j}\), \(j=1,2,\ldots,7\), be the sides of D, including the ends, enumerated counterclockwise starting from the upper side of the slit (\(\gamma _{0}\equiv\gamma_{7} \)), \(\gamma=\bigcup_{j=1}^{7}\gamma_{j}\), and \(\dot{\gamma}_{j}=\gamma_{j}\cap\gamma_{j1}\) be the vertices of D. Let \(( r,\theta ) \equiv ( r_{1},\theta_{1} ) \) be a polar system of coordinates with pole in \(\dot{\gamma}_{1}\), where the angle θ is taken counterclockwise from the side \(\gamma_{1}\).
Results obtained for first derivative of the slit problem
\(\boldsymbol{(h^{1},n)}\)  (16,70)  (32,70)  (64,110)  (128,130) 

\(\Vert \epsilon_{h}^{ ( 1 ) }\Vert _{\overline{D}_{S}}\)  7.89831 × 10^{−7}  9.78871 × 10^{−8}  4.29502 × 10^{−9}  2.94108 × 10^{−10} 
Results obtained for second derivative of the slit problem
\(\boldsymbol{(h^{1},n)}\)  (16,70)  (32,70)  (64,110)  (128,130) 

\(\Vert \epsilon_{h}^{ ( 2 ) }\Vert _{\overline{D}_{S}}\)  3.7119 × 10^{−6}  9.736 × 10^{−7}  2.03211 × 10^{−8}  9.30597 × 10^{−10} 
Example 5.2
Order of convergence of Example 5.2
\(\boldsymbol{2^{m}}\)  \(\boldsymbol{2^{5}}\)  \(\boldsymbol{2^{6}}\) 

\(\widetilde{R}_{P_{NS}^{1}}^{m}\)  16.257  15.9884 
\(\widetilde{R}_{P_{NS}^{2}}^{m}\)  16.2387  16.0086 
\(\widetilde{R}_{P_{S}^{1}}^{m}\)  19.3268  12.7771 
\(\widetilde{R}_{P_{S}^{2}}^{m}\)  18.2604  14.0755 
Order of convergence of derivatives in ‘singular’ parts of Example 5.2
\(\boldsymbol{2^{m}}\)  \(\boldsymbol{2^{5}}\)  \(\boldsymbol{2^{6}}\) 

\(\widetilde{R}_{P_{S}^{1}}^{m}\)  13.8404  19.6426 
\(\widetilde{R}_{P_{S}^{2}}^{m}\)  13.7489  19.6505 
6 Conclusion
A fourth order square and hexagonal grid version of the blockgrid method, for the solution of the boundary value problem of Laplace’s equation on staircase polygons, with interior angles \(\alpha_{j}\in \{ \frac{1}{2},1,\frac {3}{2},2 \}\), is extended for the polygons with interior angles \(\alpha_{j}\pi\), \(\alpha_{j}\in \{ \frac{1}{3},\frac{2}{3},1,2 \}\), by constructing and justifying the blockhexagonal grid method. Moreover, the smoothness requirement on the boundary functions away from the singular vertices (outside of the ‘singular’ parts) is lowered down from the Hölder classes \(C^{6,\lambda}\), \(0<\lambda<1\), as in [8], to \(C^{4,\lambda}\), \(0<\lambda<1\), which was proved for the 9point scheme on square grids (see [10, 11]).
The proposed version of the BGM can be applied for the mixed boundary value problem of Laplace’s equation on the above mentioned polygons. Furthermore, by this method any order derivatives of the solution can be highly approximated on the ‘singular’ parts, which are difficult to obtain in other numerical methods.
This method can also be used for the solution of the biharmonic equation by representing the problem with two problems for the Laplace and Poisson equations.
Declarations
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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