Breaking spaces and forms for the DPG method and applications including Maxwell equations

Carstensen, Carsten; Demkowicz, Leszek; Gopalakrishnan, Jay

doi:10.1016/j.camwa.2016.05.004

Cited by 145 publications

(185 citation statements)

References 30 publications

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“…×Û, whereÛ can be interpreted as a space of Lagrange multipliers due to the broken nature of the test space. Similarly, the operator B DPG in a DPG method can be decomposed into two terms [19]:…”

Section: 3mentioning

confidence: 99%

“…This is known as the (first-order) "strong variational formulation" of the Poisson equation [19]. A DLS discretization can be found by using the same finite-dimensional spaces U h as in the FOSLS method above, as well as an L…”

Section: Robustness Observe That If B(umentioning

confidence: 99%

“…Only later, when presenting a computer implementation of the method, a tunable discretization of the test space is proposed with the assumption that all properties of the idealized method-such as stability-will be inherited once the test space discretization is rich enough. In some scenarios, these assumptions can be proven [45,18,19,35,40,39] or analyzed [56], but for difficult problems, they can only be speculated. We will not devote our attention in this article to such implications of different discretizations, but rather deliberate the structure of these equations mainly from a numerical linear algebra perspective.…”

Section: Introductionmentioning

confidence: 99%

“…There appear to be no published manuscripts which derive or consider the possible advantages inherent in solving the overdetermined system of equations. We note that in specific instances where the test space is L 2 in at least one component-and therefore, for most alternative variational formulations [19,48]-the implementation of DPG permits some ambiguity. For instance, as developed in [48] and contrary to the DLS approach, one can choose not to discretize the L 2 -Riesz map and so construct a least-squares (strong variational formulations) or least-squares hybrid (all other variational formulations) discretization.…”

Section: Introductionmentioning

confidence: 99%

“…In order to explore less trivial variational formulations, consider the broken primal formulation of the Poisson equation in (5.3) (with α = 0) [28]. In this setting, we seek a [19,48] for definitions of mesh-trace Sobolev spaces) such that…”

mentioning

confidence: 99%

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Discrete least-squares finite element methods

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Fuentes

et al. 2017

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ABSTRACT. A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz map on the test space. The resulting linear system is overdetermined. Two different approaches for solving the system are suggested (although others are discussed): (1) solving the associated normal equation with linear solvers for symmetric positive-definite systems (e.g. Cholesky factorization); and (2) solving the overdetermined system with orthogonalization algorithms (e.g. QR factorization). The finite element assembly algorithm for each of these approaches is described in detail. The normal equation approach is usually faster for direct solvers and requires less storage. The second approach reduces the condition number of the system by a power of two and is less sensitive to round-off error. The rectangular stiffness matrix of second approach is demonstrated to have condition number O(h −1 ) for a variety of formulations of Poisson's equation. The stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least-squares finite element methods and it is proved that the two are identical in some cases. An example with Poisson's equation indicates that the solutions of these two different linear systems can be nearly indistinguishable (if round-off error is not an issue) and rapidly converge to each other. The orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems. Experiments with Poisson's equation in single-precision arithmetic as well as the linear acoustics problem near resonance in double-precision arithmetic verify this conclusion. The methodology described here was developed as an outgrowth of the discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan [29]. The strength of DPG is highlighted throughout, however, the majority of theory presented is general. Extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments.

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Section: 3mentioning

confidence: 99%

Section: Robustness Observe That If B(umentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Discrete least-squares finite element methods

Keith

Petrides

Fuentes

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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Discontinuous P etrov– G alerkin ( DPG ) Method

Demkowicz

Gopalakrishnan

2017

Encyclopedia of Computational Mechanics Second Edition

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This chapter reviews the fundamentals of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions. The main idea admits three different interpretations: a Petrov–Galerkin method with (optimal) test functions that realize the supremum in the inf–sup condition; a minimum‐residual method with residual measured in a dual norm; and a mixed formulation where one solves simultaneously for the Riesz representation of the residual. The methodology can be applied to any well‐posed variational problem, but it is especially effective in context of discontinuous (broken) test spaces. We discuss how one can “break” test functions in any variational formulation, and use the convection‐dominated diffusion model problem to illustrate challenges related to the choice of an appropriate test norm.

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

Kercher

Corrigan

Kessler

2021

Numerical Methods in Fluids

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The moving discontinuous Galerkin finite element method with interface condition enforcement is applied to the case of viscous flows. This method uses a weak formulation that separately enforces the conservation law, constitutive law, and the corresponding interface conditions in order to provide the means to detect interfaces or underresolved flow features. To satisfy the resulting overdetermined weak formulation, the discrete domain geometry is introduced as a variable, so that the method implicitly fits a priori unknown interfaces and moves the grid to resolve sharp, but smooth, gradients, achieving a form of anisotropic curvilinear r‐adaptivity. This approach avoids introducing low‐order errors that arise using shock capturing, artificial dissipation, or limiting. The utility of this approach is demonstrated with its application to a series of test problems culminating with the compressible Navier–Stokes solution to a Mach 5 viscous bow shock for a Reynolds number of 105 in two‐dimensional space. Time accurate solutions of unsteady problems are obtained via a space‐time formulation, in which the unsteady problem is formulated as a higher dimensional steady space‐time problem. The method is shown to accurately resolve and transport viscous structures without relying on numerical dissipation for stabilization.

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Breaking spaces and forms for the DPG method and applications including Maxwell equations

Cited by 145 publications

References 30 publications

Discrete least-squares finite element methods

Discrete least-squares finite element methods

Discontinuous P etrov– G alerkin ( DPG ) Method

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

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