This work was partially supported with grants by NSF (DMS-1418822, DMS-1624776), AFOSR (FA9550-17-1-0090), and ONR (N00014-15-1-2496).
Finite element method, Inequalities (Mathematics), Discontinuous functions, Galerkin methods, Numerical analysis
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 othermethods in the literature round out the newly garnered perspective. Notably, DPG* and DPG methods can be seen as generalizations of LL* 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.
Demkowicz, Leszek; Gopalakrishnan, Jay; and Keith, Brendan, "The DPG-Star Method" (2018). Portland Institute for Computational Science Publications. 15.