A fast multigrid-based electromagnetic eigensolver for curved metal boundaries on the Yee mesh

TL;DR

This paper introduces a multigrid-based eigensolver for electromagnetic problems with curved boundaries, demonstrating efficient inversions and analyzing the accuracy of the Dey-Mittra algorithm in complex geometries.

Contribution

It develops a specialized grad-div matrix enabling multigrid methods for Maxwell's equations with curved boundaries, improving inversion efficiency and accuracy analysis.

Findings

Efficient curl-curl inversions achieved within a shift-and-invert eigensolver.

Frequencies converge with second-order error, surface fields nearly second-order.

Neglecting boundary-cut faces reduces convergence order and accuracy.

Abstract

For embedded boundary electromagnetics using the Dey-Mittra algorithm, a special grad-div matrix constructed in this work allows use of multigrid methods for efficient inversion of Maxwell's curl-curl matrix. Efficient curl-curl inversions are demonstrated within a shift-and-invert Krylov-subspace eigensolver (open-sourced at https://github.com/bauerca/maxwell) on the spherical cavity and the 9-cell TESLA superconducting accelerator cavity. The accuracy of the Dey-Mittra algorithm is also examined: frequencies converge with second-order error, and surface fields are found to converge with nearly second-order error. In agreement with previous work, neglecting some boundary-cut cell faces (as is required in the time domain for numerical stability) reduces frequency convergence to first-order and surface-field convergence to zeroth-order (i.e. surface fields do not converge). Additionally and importantly, neglecting faces can reduce accuracy by an order of magnitude at low resolutions.