Goal-oriented adaptive mesh refinement for non-symmetric functional settings

TL;DR

This paper develops a unified duality theory for Petrov-Galerkin methods and introduces goal-oriented adaptive mesh refinement strategies for DPG and DPG* methods, improving error estimation and demonstrating effectiveness through 3D numerical experiments.

Contribution

It presents a new duality theory and goal-oriented refinement strategies for DPG methods, including the novel DPG* approach, enhancing error estimation and adaptivity.

Findings

Improved reliability estimates for DPG methods.

Derived three classes of refinement indicators for DPG* methods.

Numerical experiments confirm effectiveness in 3D Poisson problems.

Abstract

In this article, a new unified duality theory is developed for Petrov-Galerkin finite element methods. This novel theory is then used to motivate goal-oriented adaptive mesh refinement strategies for use with discontinuous Petrov-Galerkin (DPG) methods. The focus of this article is mainly on broken ultraweak variational formulations of stationary boundary value problems, however, many of the ideas presented within are general enough that they be extended to any such well-posed variational formulation. The proposed goal-oriented adaptive mesh refinement procedures require the construction of refinement indicators for both a primal problem and a dual problem. In the DPG context, the primal problem is simply the system of linear equations coming from a standard DPG method and the dual problem is a similar system of equations, coming from a new method which is dual to DPG. This new method has the same coefficient matrix as the associated DPG method but has a different load. We refer to this new finite element method as a DPG* method. A thorough analysis of DPG* methods, as stand-alone finite element methods, is not given here but will be provided in subsequent articles. For DPG methods, the current theory of a posteriori error estimation is reviewed and the reliability estimate in [13, Theorem 2.1] is improved on. For DPG* methods, three different classes of refinement indicators are derived and several contributions are made towards rigorous a posteriori error estimation. At the closure of the article, results of numerical experiments with Poisson's boundary value problem in a three-dimensional domain are provided. These results clearly demonstrate the utility of the goal-oriented adaptive mesh refinement strategies for quantities of interest with either interior or boundary terms.