Analysis of hypersingular residual error estimates in boundary element methods for potential problems

Abstract

A novel iteration scheme, using boundary integral equations, is developed for error estimation in the boundary element method. The iteration scheme consists of using the boundary integral equation for solving the boundary value problem and iterating this solution with the hypersingular boundary integral equation to obtain a new solution. The hypersingular residual r is consistently defined as the difference in the derivative quantities on the boundary, i.e. r = ∂ φ ( 1 ) ∂ n − ∂ φ ( 2 ) ∂ n where φ is the potential and (∂φ/∂ n) (1), i = 1, 2, is the flux obtained by solution ( i). Here, i = 1 refers to the boundary integral equation, and i = 2 refers to the hypersingular boundary integral equation. The hypersingular residual is interpreted in the sense of the iteration scheme defined above and it is shown to provide an error estimate for the boundary value problem. Error-hypersingular residual relations are developed for Dirichlet and Neumann problems, which are shown to be limiting cases of the more general relation for the mixed boundary value problem. These relations lead to global bounds on the error. Four numerical examples, involving Galerkin boundary elements, are given, and one of them involves a physical singularity on the boundary and preliminary adaptive calculations. These examples illustrate important features of the hypersingular residual error estimate proposed in this paper.