A High-Order Nodal Galerkin Formulation for the M\"uller Equation: Bypassing Divergence Conformity via Kernel Cancellation

TL;DR

This paper introduces a high-order Galerkin method for the M"uller equation that avoids divergence conformity constraints by leveraging kernel cancellation, leading to efficient and accurate electromagnetic scattering simulations.

Contribution

It develops a novel nodal, high-order Galerkin formulation utilizing kernel cancellation to bypass divergence-conforming basis functions in M"uller boundary integral equations.

Findings

Achieves superlinear GMRES convergence with a Block Jacobi preconditioner.

Demonstrates high-order spatial accuracy and optical-theorem compliance.

Validates the method against semi-analytical references.

Abstract

The M\"{u}ller boundary integral equation for penetrable electromagnetic scattering is conventionally discretized using divergence-conforming basis functions, a restriction inherited from the PMCHWT framework. This paper demonstrates that this constraint can be bypassed. The double-gradient operator in the M\"uller formulation acts on the kernel difference $\varphi_a - \varphi_i$, so that the $\mathcal{O}(R^{-3})$ hypersingularity cancels identically, reducing the operators to weakly singular $\mathcal{O}(R^{-1})$ kernels. Exploiting this cancellation, we develop a nodal, high-order Galerkin formulation using $\mathrm{P}_2$ isoparametric shape functions on curved manifolds. The surface vector field is constructed via a metric-weighted orthonormal tangent frame. The singular integrals are evaluated by Sauter--Schwab quadrature, and a Morton-ordered Block Jacobi preconditioner is introduced. By capturing the dominant near-field interactions within geometrically clustered diagonal blocks, it yields robust, superlinear GMRES convergence under extreme material and geometric parameters. Validation against semi-analytical EBCM references confirms high-order spatial accuracy and optical-theorem satisfaction to high precision.