Natural hp-BEM for the electric field integral equation with singular solutions
Alexei Bespalov and Norbert Heuer
Numer. Methods Partial Differential Eq. 28 (5), 1466-1480, 2012.
We apply the hp-version of the boundary element
method (BEM) for the numerical solution of the electric field
integral equation (EFIE) on a Lipschitz polyhedral surface Γ.
The underlying meshes are supposed to be quasi-uniform triangulations of
Γ, and the approximations are based on either Raviart-Thomas or
Brezzi-Douglas-Marini families of surface elements.
Non-smoothness of Γ leads to singularities in the solution
of the EFIE, severely affecting convergence rates of the BEM.
However, the singular behaviour of the solution can be explicitly
specified using a finite set of power functions (vertex-, edge-, and
vertex-edge singularities). In this paper we use this fact to perform an
a priori error analysis of the hp-BEM on quasi-uniform meshes.
We prove precise error estimates in terms of the polynomial degree p,
the mesh size h, and the singularity exponents.