A robust DPG method for large domains

TL;DR

This paper enhances the robustness of the Discontinuous Petrov-Galerkin (DPG) method for large domain problems by employing scaled test norms, demonstrated through Poisson and plate bending models with numerical validation.

Contribution

It introduces a scaled test norm approach to restore DPG robustness on large domains, including discrete variants with approximated test functions.

Findings

Robustness of DPG improved with scaled test norms

Numerical experiments confirm effectiveness on complex domains

Applicable to Poisson and Kirchhoff--Love models

Abstract

We observe a dramatic lack of robustness of the DPG method when solving problems on large domains and where stability is based on a Poincar\'e-type inequality. We show how robustness can be re-established by using appropriately scaled test norms. As model cases we study the Poisson problem and the Kirchhoff--Love plate bending model, and also include fully discrete variants where optimal test functions are approximated. Numerical experiments for both model problems, including an-isotropic domains and mixed boundary conditions, confirm our findings.