An adaptive dual reciprocity scheme for the numerical solution of the Poisson equation

Abstract

An adaptive refinement technique is proposed for the dual reciprocity boundary element method. Numerical and computational techniques are developed from error analyses based on sound mathematical grounds. Error indicators are derived from a collocation scheme using uniform norms that compare values of field variables at two successive mesh iterations. The adaptation is achieved by simultaneous h-refinements of the boundary elements and the dual reciprocity (DR) internal domain points. When a new mesh i is constructed and solved during the adaptive process, the field variable approximations obtained for all new collocation nodes and DR points are compared with the values calculated at the same coordinates using the solution of the previous smaller mesh i−1. The comparison is performed using ad hoc indicators derived from error bounds of the piecewise polynomial collocation approximation of Fredholm integral equations. For problems governed by the Poisson equation the internal domain is divided into non-overlapping clusters containing a number of DR points. Error indicators are evaluated for each cluster to determine whether an increase of DR points is needed. The proposed analyses and the case studies show that the numerical error is reduced when the boundary elements are refined simultaneously with the DR internal points.