Automatically Stable Discontinuous Petrov-Galerkin Methods for Stationary Transport Problems: Quasi-Optimal Test Space Norm

TL;DR

This paper applies the discontinuous Petrov-Galerkin method with a quasi-optimal test space norm to stationary convection-diffusion problems, enhancing robustness and accuracy especially in low diffusion scenarios.

Contribution

It introduces the use of the quasi-optimal test space norm within the DPG framework for stationary transport, demonstrating improved accuracy and robustness over standard norms.

Findings

Quasi-optimal norm improves accuracy on benchmark problems.

Numerical convergence of optimal test functions studied.

Method detailed for algorithmic implementation.

Abstract

We investigate the application of the discontinuous Petrov-Galerkin (DPG) finite element framework to stationary convection-diffusion problems. In particular, we demonstrate how the quasi-optimal test space norm can be utilized to improve the robustness of the DPG method with respect to vanishing diffusion. We numerically compare coarse-mesh accuracy of the approximation when using the quasi-optimal norm, the standard norm, and the weighted norm. Our results show that the quasi-optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions. We address the problems associated to the resolution of the optimal test functions with respect to the quasi-optimal norm by studying their convergence numerically. In order to facilitate understanding of the method, we also include a detailed explanation of the methodology from the algorithmic point of view.