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 eld integral equation (EFIE) on Lipschitz polyhedral surfaces su er 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 Γ, we employ anisotropic meshes algebraically graded towards the edges of Γ. We prove that on su ciently graded meshes the h-version of the BEM with the lowest-order RaviartThomas elements regains (up to a small order of ε > 0) an optimal convergence rate (i.e., the rate of the h-BEM on quasi-uniform meshes for smooth solutions).