Hybrid Galerkin boundary elements on degenerate meshes

Abstract

In recent work ["Discrete boundary element methods on general meshes in 3D", Bath Mathematics Preprint No. 97/19, Univ. Bath, Bath, 1997; "Hybrid Galerkin boundary elements: theory and implementation", Preprint No. 98-6, Univ. Kiel, Kiel, 1998] we have presented a new discretisation scheme for boundary integral equations which has the same energy norm stability and convergence properties as the Galerkin method but has a complexity comparable with discrete collocation or Nyström methods. Our results were for non-quasiuniform but nevertheless shape-regular meshes. Here we extend the theory to much more general meshes, including the degenerate meshes commonly used to handle singularities arising from corners and edges in 3D applications. As an application we give numerical results for the classical problem of computing the capacitance of a two-dimensional plate in R3. These show that the method is capable of attaining the same type of complexity reduction for singular problems as was already attained for smooth applications in [I. G. Graham, W. Hackbusch and S. A. Sauter, op. cit., 1998].