This paper provides a rigorous analysis on 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 Galerkin discretization scheme is then introduced, employing Raviart-Thomas basis functions for the solution space and Buffa-Christiansen functions for the test space. A discrete inf-sup condition is proven, implying the unique solvability of the discrete variational problem. An asymptotically quasi-optimal error estimate for the 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.

翻译：本文针对Lipschitz多面体上的磁场积分方程，对边界元方法进行了严格的分析。磁场积分方程广泛应用于实际中模拟完美导体的电磁散射问题。通过引入纯虚波数的电场积分算子，证明了控制算子的强制性。因此，当波数不属于内部Maxwell问题的谱时，连续变分问题具有唯一可解性。随后引入Galerkin离散化方案，采用Raviart-Thomas基函数作为解空间，Buffa-Christiansen函数作为测试空间。证明了离散inf-sup条件，从而确保了离散变分问题的唯一可解性。建立了数值解的渐近拟最优误差估计，并检验了数值方案的收敛速率。此外，证明无论网格如何加密，所得矩阵系统都是良态的。文中给出了若干数值结果以支持理论分析。