Analysis of the DPG Method for the Poisson Equation

Demkowicz, Leszek; Gopalakrishnan, Jay

doi:10.1137/100809799

Cited by 132 publications

(167 citation statements)

References 20 publications

(40 reference statements)

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“…In hindsight, we understand that this is the essence of the analysis not only in this paper, but also in [6].…”

Section: The Analysis In Hindsightmentioning

confidence: 73%

“…The details of the derivation are very similar to the case of the Poisson equation [6], so we omit them and simply present the DPG weak formulation, after a foreword on notations. Let Note that in the latter definition, complex conjugations are absent, so to match conjugate linearity of other terms, we will often use notations like w,¯ ∂Ω h and w, ∂Ω h , whose meanings are self-explanatory.…”

Section: Application To the Helmholtz Equationmentioning

confidence: 99%

“…This choice is dictated by our previous computational experience [6,8,9], whereby it was clear that using a higher ∆p did not result in any error improvements, while using a lower ∆p could result in a non-injectiveT .…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…In this paper, we will provide a different way of theoretical analysis that proves (1) for higher dimensions with C(ω) independent of ω. The new analytical technique is built on the approach developed in [6].…”

Section: Introductionmentioning

confidence: 99%

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Wavenumber Explicit Analysis for a DPG Method for the Multidimensional Helmholtz Equation

Demkowicz¹,

Gopalakrishnan²,

Muga³

et al. 2011

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Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. Abstract. We study the properties of a novel discontinuous Petrov Galerkin (DPG) method for acoustic wave propagation. The method yields Hermitian positive definite matrices and has good pre-asymptotic stability properties. Numerically, we find that the method exhibits negligible phase errors (otherwise known as pollution errors) even in the lowest order case. Theoretically, we are able to prove error estimates that explicitly show the dependencies with respect to the wavenumber ω, the mesh size h, and the polynomial degree p. But the current state of the theory does not fully explain the remarkably good numerical phase errors. Theoretically, comparisons are made with several other recent works that gave wave number explicit estimates. Numerically, comparisons are made with the standard finite element method and its recent modification for wave propagation with clever quadratures. The new DPG method is designed following the previously established principles of optimal test functions. In addition to the nonstandard test functions, in this work, we also use a nonstandard wave number dependent norm on both the test and trial space to obtain our error estimates.

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“…In hindsight, we understand that this is the essence of the analysis not only in this paper, but also in [6].…”

Section: The Analysis In Hindsightmentioning

confidence: 73%

Section: Application To the Helmholtz Equationmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Wavenumber Explicit Analysis for a DPG Method for the Multidimensional Helmholtz Equation

Demkowicz¹,

Gopalakrishnan²,

Muga³

et al. 2011

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“…In this research topic, we focus on the phenomenon indicated as volume locking which can lead to failure of a numerical method when constraints, as related to incompressible materials, are not accounted properly. In this contribution we firstly introduce the primal discontinuous Petrov-Galerkin (dPG) Finite Element Method (FEM) for linear elasticity based on the novel dPG discretization method proposed recently in [1][2][3]. Therefore, a further primary variable is introduced resulting in a mixed FEM and the interpolation functions for the test space are chosen to be discontinuous.…”

Section: Introductionmentioning

confidence: 99%

A primal discontinuous Petrov‐Galerkin Finite Element Method for linear elasticity

Steiner

Wriggers

2017

Proc Appl Math and Mech

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Volume-locking is a significant problem when using the standard Finite Element Method (FEM) since this phenomenon can lead to failure of a numerical method. To circumvent constraints as related in (nearly) incompressible materials the novel discontinuous Petrov-Galerkin (dPG) FEM is introduced for linear elasticity. A first result of the implemented primal dPG FEM is shown for a patch test. Further tests on the performance of the proposed method are work in progress and are not part of this contribution.

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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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Analysis of the DPG Method for the Poisson Equation

Cited by 132 publications

References 20 publications

Wavenumber Explicit Analysis for a DPG Method for the Multidimensional Helmholtz Equation

Wavenumber Explicit Analysis for a DPG Method for the Multidimensional Helmholtz Equation

A primal discontinuous Petrov‐Galerkin Finite Element Method for linear elasticity

Discontinuous P etrov– G alerkin ( DPG ) Method

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