Superconvergence in a DPG method for an ultra-weak formulation

TL;DR

This paper demonstrates superconvergence properties of a DPG method applied to an ultra-weak formulation of reaction-diffusion problems, showing improved convergence rates and a simple postprocessing technique validated through numerical experiments.

Contribution

It introduces a refined analysis of convergence rates, a novel postprocessing method, and extends superconvergence results to broader problem settings.

Findings

Higher convergence rates for scalar variables with increased polynomial order

A simple elementwise postprocessing yields superconvergence

Numerical experiments confirm the theoretical results beyond the model problem

Abstract

In this work we study a DPG method for an ultra-weak variational formulation of a reaction-diffusion problem. We improve existing a priori convergence results by sharpening an approximation result for the numerical flux. By duality arguments we show that higher convergence rates for the scalar field variable are obtained if the polynomial order of the corresponding approximation space is increased by one. Furthermore, we introduce a simple elementwise postprocessing of the solution and prove superconvergence. Numerical experiments indicate that the obtained results are valid beyond the underlying model problem.