Superconvergent DPG methods for second order elliptic problems

TL;DR

This paper introduces superconvergent DPG methods for second order elliptic problems, achieving higher accuracy through polynomial degree increase or postprocessing, with analysis of test norms and numerical validation.

Contribution

It provides the first theoretical explanation for the optimal $L^2$ approximation of the scalar variable in DPG methods and analyzes superconvergence under various test norms.

Findings

Superconvergence achieved by degree elevation or postprocessing.

Quasi-optimal test norm yields higher convergence rates with convection.

DPG method provides best $L^2$ approximation of scalar variable.

Abstract

We consider DPG methods with optimal test functions and broken test spaces based on ultra-weak formulations of general second order elliptic problems. Under some assumptions on the regularity of solutions of the model problem and its adjoint, superconvergence for the scalar field variable is achieved by either increasing the polynomial degree in the corresponding approximation space by one or by a local postprocessing. We provide a uniform analysis that allows to treat different test norms. Particularly, we show that in the presence of convection only the quasi-optimal test norm leads to higher convergence rates, whereas other norms considered do not. Moreover, we also prove that our DPG method delivers the best $L^2$ approximation of the scalar field variable up to higher order terms, which is the first theoretical explanation of an observation made previously by different authors. Numerical studies that support our theoretical findings are presented.