Preconditioning for time-harmonic Maxwell's equations using the Laguerre transform

TL;DR

This paper introduces a preconditioning technique using the Laguerre transform to efficiently solve large, ill-conditioned linear systems arising from discretized time-harmonic Maxwell's equations, reducing computational resources.

Contribution

The paper proposes a novel iterative preconditioning method based on the Laguerre transform for Maxwell's equations, enabling efficient solutions with less memory usage.

Findings

Effective reduction in RAM requirements compared to direct methods.

Compatibility with multigrid algorithms enhances computational efficiency.

Improved handling of ill-conditioned matrices in electromagnetic simulations.

Abstract

A method of numerically solving the Maxwell equations is considered for modeling harmonic electromagnetic fields. The vector finite element method makes it possible to obtain a physically consistent discretization of the differential equations. However, solving large systems of linear algebraic equations with indefinite ill-conditioned matrices is a challenge. The high order of the matrices limits the capabilities of the Gaussian method to solve such systems, since this requires large RAM and much calculation. To reduce these requirements, an iterative preconditioned algorithm based on integral Laguerre transform in time is used. This approach allows using multigrid algorithms and, as a result, needs less RAM compared to the direct methods of solving systems of linear algebraic equations.