The Block-Hexagonal Grid Method for Laplace’s Equation with Singularities

A fourth order accurate matching operator is constructed on a hexagonal grid, for the interpolation of the mixed boundary value problem of Laplace’s equation, by using the harmonic properties of the solution. With the application of this matching operator for the connection of the subsystems, the Block-Grid method (BGM), which is a difference-analytical method, has been analysed on a hexagonal grid, for the solution of both the Dirichlet and mixed boundary value problems of Laplace’s equation with singularities. First of all, BGM is considered on staircase polygons and it is justified that when the boundary functions outside the finite neighbourhood of the singular points are from the Hölder classes C6;l ; 0 < l < 1; the error of approximation has an accuracy of O 􀀀��� h4 ; where h is the mesh size. The analysis of this method is extended to special polygons whose interior angles are a jp; a j 2 1 3 ; 2 3 ;1;2 ; and for the Dirichlet problem of Laplace’s equation it is proved that, with the application of BGM, it is possible to lower the smoothness requirement on the boundary functions to C4;l ; 0 < l < 1; outside the finite neighbourhood of the singular points, in order to obtain an accuracy of O 􀀀��� h4 . For the demonstration of the theoretical results on staircase polygons, BGM has been applied on an L-shaped domain for two examples, which has a singularity at the vertex with an interior angle of 3p 2 ; where Dirichlet and mixed boundary conditions are assumed respectively. The slit problem, which has the strongest singularity due to the interior angle of 2p at the vertex of the slit, has been considered on a parallelogram with a slit, in order to illustrate the results obtained on polygons with interior angles of a jp; a j 2 1 3 ; 2 3 ;1;2 : The second example on a parallelogram demonstrates the application of BGM on a domain with two singularities as it is assumed that the vertices with interior angles of 2p 3 are singular points. Solutions of the numerical examples are consistent with the theoretical results obtained. Keywords: Hexagonal grids, Laplace’s equation, singularity problem, block-grid method.