The DPG-star method. (1st June 2020)

• Record Type: Journal Article
• Authors: Demkowicz, Leszek; Gopalakrishnan, Jay; Keith, Brendan
• Is Part Of: Computers & mathematics with applications. Volume 79:issue 11(2020)
• Page Start: 3092
• Page End: 3116
• Publication Date: 2020-06-01
• Subjects: DPG method -- Ultraweak formulation -- A priori -- A posteriori -- Duality; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2020.01.012
• Languages: English
• ISSNs: 0898-1221

Abstract

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 L L ∗ 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.