Analysis of backward Euler primal DPG methods

TL;DR

This paper analyzes backward Euler time stepping schemes for primal DPG methods applied to parabolic problems, establishing optimal error estimates and highlighting the importance of elliptic projection operators in the analysis.

Contribution

It provides a rigorous analysis of backward Euler primal DPG methods, including error bounds and the role of elliptic projections for problems with advection and reaction terms.

Findings

Optimal error estimates in natural and $L^2$ norms.

Solution coincides with standard Galerkin scheme for heat equation.

Elliptic projections are crucial for error analysis with advection and reaction.

Abstract

We analyse backward Euler time stepping schemes for the primal DPG formulation of a class of parabolic problems. Optimal error estimates are shown in the natural norm and in the $L^2$ norm of the field variable. For the heat equation the solution of our primal DPG formulation equals the solution of a standard Galerkin scheme and, thus, optimal error bounds are found in the literature. In the presence of advection and reaction terms, however, the latter identity is not valid anymore and the analysis of optimal error bounds requires to resort to elliptic projection operators. It is essential that these operators be projections with respect to the spatial part of the PDE, as in standard Galerkin schemes, and not with respect to the full PDE at a time step, as done previously.