Local defect correction for boundary integral equation methods

Abstract

The aim in this paper is to develop a new local defect correction approach to gridding for problems with localised regions of high activity in the boundary element method. The technique of local defect correction has been studied for other methods as finite difference methods and finite volume methods. The initial attempts to developing such a technique by the authors for the boundary element method was based on block decomposition and manipulation of the coefficient matrix and right hand side of the system of equations in three dimension. It ignored the inherent global nature of the boundary integral equation, that is, each node of the grid contributes to all the others in the grid through integration. In this paper we present a better approach that takes this into account. We use a new integral approach to defect correction and develop the technique for the boundary element method. The technique offers an iterative way for obtaining the solution on an equivalent composite grid. It uses two grids: a global uniform coarse grid covering the whole boundary and a local fine grid covering the local active boundary. The solution of the local problem on the local fine grid is used to estimate the defect on the fine grid. The effect of the defect onto the right hand side is then considered. We demonstrate the technique’s strength using an example and show that it offers a cheaper alternative to either solving on a global uniform grid or directly on a composite grid. Keywords: integral equations, boundary elements, local error, local defect correction, local activity, gridding.