A time-stepping DPG scheme for the heat equation

TL;DR

This paper presents a novel discontinuous Petrov-Galerkin scheme with optimal test functions for the heat equation, combining backward Euler time stepping with an ultra-weak formulation, and provides stability, error estimates, and numerical validation.

Contribution

It introduces a new DPG method for the heat equation using an ultra-weak formulation and analyzes its stability and convergence properties.

Findings

The scheme is stable for the field variables.

Error estimates are derived showing convergence.

Numerical experiments support theoretical results.

Abstract

We introduce and analyze a discontinuous Petrov-Galerkin method with optimal test functions for the heat equation. The scheme is based on the backward Euler time stepping and uses an ultra-weak variational formulation at each time step. We prove the stability of the method for the field variables (the original unknown and its gradient weighted by the square root of the time step) and derive a C\'ea-type error estimate. For low-order approximation spaces this implies certain convergence orders when time steps are not too small in comparison with mesh sizes. Some numerical experiments are reported to support our theoretical results.