High order eigenvalues for the Helmholtz equation in complicated non-tensor domains through Richardson Extrapolation of second order finite differences

TL;DR

This paper introduces a method combining second order finite differences with Richardson extrapolation to accurately compute the lowest eigenvalues of the Helmholtz equation in complex non-tensor domains, surpassing existing results.

Contribution

It demonstrates a novel application of Richardson and Padé-Richardson extrapolation to finite difference eigenvalues for complex domains, achieving unprecedented precision.

Findings

Achieved highly accurate eigenvalues surpassing previous literature.

Validated the asymptotic nature of the finite difference series.

Compared results with the method of particular solutions for validation.

Abstract

We apply second order finite difference to calculate the lowest eigenvalues of the Helmholtz equation, for complicated non-tensor domains in the plane, using different grids which sample exactly the border of the domain. We show that the results obtained applying Richardson and Pad\'e-Richardson extrapolation to a set of finite difference eigenvalues corresponding to different grids allows to obtain extremely precise values. When possible we have assessed the precision of our extrapolations comparing them with the highly precise results obtained using the method of particular solutions. Our empirical findings suggest an asymptotic nature of the FD series. In all the cases studied, we are able to report numerical results which are more precise than those available in the literature.