A robust quasi-optimal test norm for a DPG discretization of the convection-diffusion equation

TL;DR

This paper introduces a new quasi-optimal test norm for DPG discretization of the convection-diffusion equation, ensuring robustness and favorable scaling, supported by theoretical proofs and numerical validation.

Contribution

The paper presents a novel quasi-optimal test norm for DPG methods that remains robust with respect to the diffusion parameter, improving stability and accuracy.

Findings

The proposed test norm provides bounds that are robust against the diffusion parameter.

Theoretical analysis confirms the norm's robustness and favorable scaling.

Numerical experiments validate the theoretical predictions.

Abstract

In this work, we propose a new quasi-optimal test norm for a discontinuous Petrov-Galerkin (DPG) discretization of the ultra-weak formulation of the convection-diffusion equation. We prove theoretically that the proposed test norm leads to bounds between the target norm and the energy norm induced by the test norm which are robust with respect to the diffusion parameter in the solution and gradient components and have favorable scalings in the trace components. We conclude with numerical experiments to confirm our theoretical results.