A-Posteriori Error Estimates and Adaptive Approaches for some Boundary Element Methods

Abstract

Many problems in physical science and engineering technology can be formulated as boundary value problems of partial differential equations in domain, and many boundary value problems usually can be reduced into integral equations on the boundary through different ways. Based on these boundary reductions and the classical finite element methods, in recent years boundary element methods hays been ikveloped by many authors (Hsiao and Wendland13; Feng10; Brebbia8; Nedelecl15). At the same time, adaptive finite element methods have been developed (Babuska et al4; Oden et al16). These methods utilize the currently available information for steering the computational process. According to a-posteriori error estimate of approximate solution, the mesh or approximation structure is automatically changed so as to improve the quality of numerical solutions. The aim of these estimates is not only to measure the size of the error but also to give an idea of the error distribution over the mesh. Most a-posteriori error estimates for finite element methods use the residual of the approximate solution. Another is obtained from higher order derivatives of the unknown solution through a currently available finite element solution.