A posteriori error estimate and h-adaptive algorithm on surfaces for Symm's integral equation

Abstract

A residual-based a posteriori error estimate for boundary integral equations on surfaces is derived in this paper. A localisation argument involves a Lipschitz partition of unity such as nodal basis functions known from finite element methods. The abstract estimate does not use any property of the discrete solution, but simplifies for the Galerkin discretisation of Symm's integral equation if piecewise constants belong to the test space. The estimate suggests an isotropic adaptive algorithm for automatic mesh-refinement. An alternative motivation from a two-level error estimate is possible but then requires a saturation assumption. The efficiency of an anisotropic version is discussed and supported by numerical experiments.