The DPG-star method

Abstract

This article introduces the DPG-star (from now on, denoted DPG${}^{\ast }$) 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${}^{\ast }$ 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${}^{\ast }$ 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${}^{\ast }$ 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${}^{\ast }$ and DPG analyses is that the convergence rates of the former are limited by the regularity of an extraneous Lagrange multiplier variable.

References in corpus (5)

• High-order polygonal discontinuous Petrov-Galerkin (PolyDPG) methods using ultraweak formulations
• Discrete least-squares finite element methods
• On perfectly matched layers for discontinuous Petrov-Galerkin methods
• Goal-oriented adaptive mesh refinement for non-symmetric functional settings
• Superconvergence in a DPG method for an ultra-weak formulation

Cited by in corpus (5)

• The Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement for Compressible Viscous Flows
• A Least-Squares Formulation of the Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement
• A Deep Double Ritz Method (D${}^2$RM) for solving Partial Differential Equations using Neural Networks
• Goal-Oriented Error Estimation for the Automatic Variationally Stable FE Method for Convection-Dominated Diffusion Problems
• Optimal approximation spaces for discontinuous Petrov-Galerkin finite element methods