High-order polygonal discontinuous Petrov–Galerkin (PolyDPG) methods using ultraweak formulations

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.

show abstract

“…From the second equation in (49), observe that f = L h v. Therefore, the first equation can be rewritten as…”

Section: 43mentioning

confidence: 99%

“…4 During stiffness matrix assembly, the degrees of freedom of every element edge with a hanging node was constrained by its common edge. Alternatively, because of the ultraweak variational formulation we considered, we could have incorporated each edge independently [45,49].…”

Section: N U M E R I C a L E X P E R I M E N T Smentioning

confidence: 99%

The DPG-star method

Demkowicz

Gopalakrishnan

Keith

2020

Computers & Mathematics with Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…as in [49]. In this test, the standard deviation is σ = √ 10 −3 and the two means are µ 1 = 0.25 and µ 2 = 0.75.…”

Section: 3mentioning

confidence: 99%

“…The difficulty of this problem is the existence of two Gaussian surfaces, where the solution has a fast decay. Here we adopt the same initial mesh as in [49], see Figure 6.5a. It is a polygonal mesh, which does not resolve the Gaussian surfaces.…”

Section: 3mentioning

confidence: 99%

Superconvergent gradient recovery for virtual element methods

Guo

Xie

Zhao

2019

Math. Models Methods Appl. Sci.

View full text Add to dashboard Cite

Virtual element method is a new promising finite element method using general polygonal meshes. Its optimal a priori error estimates are well established in literature. In this paper, we take a different viewpoint. We try to uncover the superconvergent property of virtual element methods by doing some local post-processing only on the degrees of freedom. Using the linear virtual element method as an example, we propose a universal gradient recovery procedure to improve the accuracy of gradient approximation for numerical methods using general polygonal meshes. Its capability of serving as a posteriori error estimators in adaptive computation is also investigated. Compared to the existing residual-type a posteriori error estimators for the virtual element methods, the recovery-type a posteriori error estimator based on the proposed gradient recovery technique is much simpler in implementation and it is asymptotically exact. A series of benchmark tests are presented to numerically illustrate the superconvergence of recovered gradient and validate the asymptotic exactness of the recovery-based a posteriori error estimator.

show abstract

“…Instead, we only lay a simple foundation for their construction in nonsymmetric functional settings. Our focus is on so-called (broken) ultraweak variational formulations which are commonly used with DPG methods and have the special feature that they may be used with general polygonal elements [73]. However, our methods can very easily be applied to any other conceivable variational formulation.…”

Section: Introductionmentioning

confidence: 99%

On perfectly matched layers for discontinuous Petrov–Galerkin methods

2018

Self Cite

View full text Add to dashboard Cite

In this article, several discontinuous Petrov-Galerkin (DPG) methods with perfectly matched layers (PMLs) are derived along with their quasi-optimal graph test norms. Ultimately, two different complex coordinate stretching strategies are considered in these derivations. Unlike with classical formulations used by Bubnov-Galerkin methods, with so-called ultraweak variational formulations, these two strategies in fact deliver different formulations in the PML region. One of the strategies, which is argued to be more physically natural, is employed for numerically solving two-and three-dimensional time-harmonic acoustic, elastic, and electromagnetic wave propagation problems, defined in unbounded domains. Through these numerical experiments, efficacy of the new DPG methods with PMLs is verified.

show abstract

High-order polygonal discontinuous Petrov–Galerkin (PolyDPG) methods using ultraweak formulations

Cited by 35 publications

References 75 publications

The DPG-star method

The DPG-star method

Superconvergent gradient recovery for virtual element methods

On perfectly matched layers for discontinuous Petrov–Galerkin methods

Contact Info

Product

Resources

About