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.