Robust DPG Fortin operators

TL;DR

This paper constructs specialized Fortin operators for the DPG method in various function spaces, ensuring stability and optimal convergence even in challenging reaction-dominated diffusion problems.

Contribution

It introduces new, smaller test spaces with constructed Fortin operators that guarantee uniform stability and convergence in DPG methods across different dimensions and polynomial degrees.

Findings

Test spaces are smaller than previous ones.

Uniform stability achieved with exponential layer inclusion.

Numerical experiments confirm improved stability and error control.

Abstract

At the fully discrete setting, stability of the discontinuous Petrov--Galerkin (DPG) method with optimal test functions requires local test spaces that ensure the existence of Fortin operators. We construct such operators for $H^1$ and $\boldsymbol{H}(\mathrm{div})$ on simplices in any space dimension and arbitrary polynomial degree. The resulting test spaces are smaller than previously analyzed cases. For parameter-dependent norms, we achieve uniform boundedness by the inclusion of exponential layers. As an example, we consider a canonical DPG setting for reaction-dominated diffusion. Our test spaces guarantee uniform stability and quasi-optimal convergence of the scheme. We present numerical experiments that illustrate the loss of stability and error control by the residual for small diffusion coefficient when using standard polynomial test spaces, whereas we observe uniform stability and error control with our construction.