A priori error analysis of the BEM with graded meshes for the electric field integral equation on polyhedral surfaces

The Galerkin boundary element discretisations of the electric field integral equation (EFIE) on Lipschitz polyhedral surfaces suffer slow convergence rates when the underlying surface meshes are quasi-uniform and shape-regular. This is due to singular behaviour of the solution to this problem in neighbourhoods of vertices and edges of the surface. Aiming to improve convergence rates of the Galerkin boundary element method (BEM) for the EFIE on a Lipschitz polyhedral closed surface  ΓΓ, we employ anisotropic meshes algebraically graded towards the edges of  ΓΓ. We prove that on sufficiently graded meshes the hh-version of the BEM with the lowest-order Raviart–Thomas elements regains (up to a small order of  ε>0ε>0) an optimal convergence rate (i.e., the rate of the hh-BEM on quasi-uniform meshes for smooth solutions).