Boundary element methods for the magnetic field integral equation on polyhedra

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.

Figures (5)

• Figure 1: A Buffa-Christiansen basis element expressed as a linear combination of Raviart-Thomas basis elements on the barycentric refinement $\widetilde{\Gamma}_h$ of the triangulation $\Gamma_h$, with coefficient of each edge multiplied by the number indicated at its origin. The edges of the refinement $\widetilde{\Gamma}_h$ are oriented away from the central edge.
• Figure 1: Triangulation of polyhedral boundaries used in numerical experiments. Left: A cuboid of dimension $1 \mathrm{m} \times 1 \mathrm{m} \times 0.25 \mathrm{m}$ with a concentric cuboid hole of dimension $0.5 \mathrm{m} \times 0.5 \mathrm{m} \times 0.25 \mathrm{m}$. Right: A pyramid of height $0.5 \mathrm{m}$ and 24-pointed star base (bottom-right corner), whose vertices lie on two concentric circles of radius $1 \mathrm{m}$ and $0.3 \mathrm{m}$.
• Figure 2: Approximated constants $\alpha$ and $\beta$ for the verification of Assumption \ref{['ast:inf_sup_const']}. Left: A cuboid of dimension $\mathrm{1\mathrm{m} \times 1\mathrm{m} \times 0.25\mathrm{m}}$ with a concentric cuboid hole of dimension $\mathrm{0.5\mathrm{m} \times 0.5\mathrm{m} \times 0.25\mathrm{m}}$. Right: A pyramid of height $0.5 \mathrm{m}$ and a 24-pointed star base, whose vertices lie on two circles of radius $1\mathrm{m}$ and $0.3\mathrm{m}$. For each domain, there are some $\kappa^\prime$ at which Assumption \ref{['ast:inf_sup_const']} is satisfied.
• Figure 3: Relative error of numerical solutions to the MFIE with respect to meshwidth $h$. Left: A cuboid of dimension $\mathrm{1\mathrm{m} \times 1\mathrm{m} \times 0.25\mathrm{m}}$ with a concentric cuboid hole of dimension $\mathrm{0.5\mathrm{m} \times 0.5\mathrm{m} \times 0.25\mathrm{m}}$. The numerical solutions converge to the reference solution with an optimal rate. Right: A pyramid of height $0.5 \mathrm{m}$ and a 24-pointed star base, whose vertices lie on two circles of radius $1\mathrm{m}$ and $0.3\mathrm{m}$. The numerical solutions exhibit a sub-optimal convergence rate due to the singularities of solution at the bottom vertex.
• Figure 4: Condition number and corresponding number of GMRES iterations required to solve the linear systems produced by the Galerkin discretization of the EFIE and the MFIE. Left: A cuboid of dimension $\mathrm{1m \times 1m \times 0.25m}$ with a concentric hole of dimension $\mathrm{0.5m \times 0.5m \times 0.25m}$. Right: A pyramid of height $0.5 \mathrm{m}$ and 24-pointed star base, whose vertices lie on two circles of radius $1\mathrm{m}$ and $0.3\mathrm{m}$. Whereas the condition number and the corresponding GMRES iteration counts of the EFIE grow when the meshwidth $h$ decreases, those of the MFIE stay almost constant.

Theorems & Definitions (26)

• Theorem 2.1: self-duality of the space $\mathop{\mathrm{\mathbf H}}\nolimits^{-1/2}_{\times}(\textup{div}_{\Gamma}, \Gamma)$
• Theorem 2.2: integration by parts formula
• Definition 3.1
• Lemma 3.2
• Lemma 3.3: $\mathop{\mathrm{\mathbf H}}\nolimits_\times^{-1/2}(\textup{div}_\Gamma, \Gamma)$-ellipticity of $S_{i\kappa^\prime}$
• Lemma 3.4: Contraction property
• Proof 1
• Corollary 3.5
• Lemma 3.6: $S_{i\kappa^\prime}$-coercivity
• Proof 2