The DPG-star method

TL;DR

The paper introduces the DPG-star (DPG$^*$) finite element method, a dual approach to the DPG method, offering new perspectives for solving boundary value problems with detailed analysis and numerical validation.

Contribution

It develops the DPG$^*$ method as a dual to DPG, including theoretical analysis, error control, and numerical experiments, expanding the framework of finite element methods.

Findings

DPG$^*$ is a dual to the DPG method.

Error analysis and error control are established for DPG$^*$.

Numerical experiments demonstrate the method's features and convergence behavior.

Abstract

This article introduces the DPG-star (from now on, denoted DPG$^*$) finite element method. It is a method that is in some sense dual to the discontinuous Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG$^*$ methodology is a means to solve an underdetermined discretization. These two viewpoints are developed by embedding the same operator equation into two different saddle-point problems. The analyses of the two problems have many common elements. Comparison to other methods in the literature round out the newly garnered perspective. Notably, DPG$^*$ and DPG methods can be seen as generalizations of $\mathcal{L}\mathcal{L}^\ast$ and least-squares methods, respectively. A priori error analysis and a posteriori error control for the DPG$^*$ method are considered in detail. Reports of several numerical experiments are provided which demonstrate the essential features of the new method. A notable difference between the results from the DPG$^*$ and DPG analyses is that the convergence rates of the former are limited by the regularity of an extraneous Lagrange multiplier variable.