The DPG-star method

Abstract

A b s t r ac t . 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 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.

Error representation of the time-marching DPG scheme

“…In other words, ||ψ|| V is both an upper bound and a lower bound of ||z − z h || U . It follows directly form (12) and ( 46)…”

“…It is well known that the DPG method can be interpreted as a minimum residual method and also as a mixed problem. In the latter, selecting an enriched test space, the method delivers a stable solution and a built-in error representation usually employed to perform adaptivity [6,12,15,36].…”

The moving discontinuous Galerkin finite element method with interface condition enforcement for compressible viscous flows

“…In the case of least‐squares (LS) methods, the residual is a natural indicator of the discretization error 56,57 . In particular, for the discontinuous Petrov–Galerkin (DPG) method introduced by Demkowicz and Gopalakrishnan, 58‐64 in which the ultra‐weak formulation corresponds to the best approximation in the polynomial space, a posteriori grid adaptation, for both the grid resolution h and the polynomial degree p , is driven by the built‐in error representation function in the form of the Riesz representation of the residual 65 . In addition to such a posteriori hp ‐adaptivity strategies, MDG‐ICE achieves a form of in situ r ‐adaptivity 66‐72 where the resolution of the flow is continuously improved through repositioning of the grid points.…”

A priori error analysis of high-order LL* (FOSLL*) finite element methods

“…The reason for this work is to clear up an apparent confusion in the literature about the numerical accuracy of minimum norm methods. Certain authors have pointed out that their particular minimum norm method has poor accuracy at high polynomial orders [27,52]. However, given how well-established LL * methods are in the literature, other scientists have expressed an instinctive skepticism toward these assertions.…”

“…This may come about during the mathematical analysis of a minimum residual method [25,31] or when using a minimum residual method in an optimization setting; for instance, when designing error estimators based on an extrinsic quantity of interest [39,52,50,20]. Such methods may also be derived entirely on their own [16,47,44,13,45,9,40,10,27,11]. In all of the settings above, the question of accuracy is critical.…”