Boundary element methods for the magnetic field integral equation on polyhedra

TL;DR

This paper rigorously analyzes boundary element methods for the magnetic field integral equation on polyhedra, establishing solvability, stability, and convergence properties with practical numerical validation.

Contribution

It introduces a Petrov-Galerkin discretization with Raviart-Thomas and Buffa-Christiansen boundary elements, proving discrete stability and error estimates for the first time.

Findings

Unique solvability of the continuous problem

Stable and well-conditioned discretization scheme

Asymptotically quasi-optimal error estimates

Abstract

This paper provides a rigorous analysis of boundary element methods for the magnetic field integral equation on Lipschitz polyhedra. The magnetic field integral equation is widely used in practical applications to model electromagnetic scattering by a perfectly conducting body. The governing operator is shown to be coercive by means of the electric field integral operator with a purely imaginary wave number. Consequently, the continuous variational problem is uniquely solvable, given that the wave number does not belong to the spectrum of the interior Maxwell's problem. A Petrov-Galerkin discretization scheme is then introduced, employing Raviart-Thomas boundary elements for the solution space and Buffa-Christiansen boundary elements for the test space. Under a mild assumption depending only on the geometrical domain, the corresponding discrete inf-sup condition is proven, implying the unique solvability of the discrete problem. An asymptotically quasi-optimal error estimate for numerical solutions is established, and the convergence rate of the numerical scheme is examined. In addition, the resulting matrix system is shown to be well-conditioned regardless of the mesh refinement. Some numerical results are presented to support the theoretical analysis.