A fourth order block-hexagonal grid approximation for the solution of Laplace's equation with singularities

Abstract

The hexagonal grid version of the block-grid method, which is a difference-analytical 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 Holder classes C-4,C-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.