Dispersive and dissipative errors in the DPG method with scaled norms for Helmholtz equation

TL;DR

This paper investigates how scaling the test norm in the DPG method affects dispersion and dissipation errors when solving the Helmholtz equation, revealing complex wavenumbers and artificial dissipation.

Contribution

It introduces a scaled norm in the DPG method, analyzes its dispersion properties, and compares its performance with standard least-squares methods for the Helmholtz equation.

Findings

Optimal scaling improves solution accuracy.

Discrete wavenumbers become complex, indicating dissipation.

DPG exhibits artificial dissipation explained by complex wavenumbers.

Abstract

We consider the discontinuous Petrov-Galerkin (DPG) method, wher the test space is normed by a modified graph norm. The modificatio scales one of the terms in the graph norm by an arbitrary positive scaling parameter. Studying the application of the method to the Helmholtz equation, we find that better results are obtained, under some circumstances, as the scaling parameter approaches a limiting value. We perform a dispersion analysis on the multiple interacting stencils that form the DPG method. The analysis shows that the discrete wavenumbers of the method are complex, explaining the numerically observed artificial dissipation in the computed wave approximations. Since the DPG method is a nonstandard least-squares Galerkin method, we compare its performance with a standard least-squares method.