A Numerical Minimization Scheme for the Complex Helmholtz Equation

TL;DR

This paper introduces a finite element minimization approach for the complex Helmholtz equation, resulting in a symmetric positive-definite matrix suitable for efficient iterative solutions, with proven error bounds and numerical validation.

Contribution

It develops a novel variational-based finite element method for the complex Helmholtz equation that ensures a symmetric positive-definite system, enabling efficient numerical solutions.

Findings

The method produces a symmetric positive-definite matrix.

Error bounds are established for the numerical scheme.

Numerical experiments validate the effectiveness of the approach.

Abstract

We use the work of Milton, Seppecher, and Bouchitt\'{e} on variational principles for waves in lossy media to formulate a finite element method for solving the complex Helmholtz equation that is based entirely on minimization. In particular, this method results in a finite element matrix that is symmetric positive-definite and therefore simple iterative descent methods and preconditioning can be used to solve the resulting system of equations. We also derive an error bound for the method and illustrate the method with numerical experiments.