Convergence rates of the DPG method with reduced test space degree

TL;DR

This paper establishes a duality theorem for DPG methods, explaining higher convergence rates in weaker norms, and demonstrates the method's robustness with reduced test space degrees, supported by new theoretical insights.

Contribution

It introduces a duality theorem for DPG methods, analyzes convergence with reduced test space degrees, and develops a non-conforming analysis approach for these scenarios.

Findings

Higher convergence rates explained by duality theorem

DPG remains solvable with odd reduced test space degree

Non-conforming analysis explains observed convergence rates

Abstract

This paper presents a duality theorem of the Aubin-Nitsche type for discontinuous Petrov Galerkin (DPG) methods. This explains the numerically observed higher convergence rates in weaker norms. Considering the specific example of the mild-weak (or primal) DPG method for the Laplace equation, two further results are obtained. First, the DPG method continues to be solvable even when the test space degree is reduced, provided it is odd. Second, a non-conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree.