A Saddle Point Numerical Method for Helmholtz Equations

TL;DR

This paper introduces a saddle point variational method for Helmholtz equations that produces symmetric positive definite systems and simplifies computation by avoiding the need to solve for the solution's gradient, enabling efficient iterative solutions.

Contribution

The paper develops a new saddle point variational approach that simplifies Helmholtz problem solving by eliminating the gradient computation and leveraging the Conjugate Gradient method.

Findings

Produces symmetric positive definite systems

Eliminates the need to solve for the gradient

Enables efficient iterative solutions with Conjugate Gradient

Abstract

In a previous work, the author and D.C. Dobson proposed a numerical method for solving the complex Helmholtz equation based on the minimization variational principles developed by Milton, Seppecher, and Bouchitte. This method results in a system of equations with a symmetric positive definite coefficient matrix, but at the same time requires solving simultaneously for the solution and its gradient. Herein is presented a method based on the saddle point variational principles of Milton, Seppecher, and Bouchitte, which produces symmetric positive definite systems of equations, but eliminates the necessity of solving for the gradient of the solution. The result is a method for a wide class of Helmholtz problems based completely on the Conjugate Gradient algorithm.