Commuting Smoothed Projectors in Weighted Norms with an Application to Axisymmetric Maxwell Equations

Gopalakrishnan, Jay; Oh, Min‐Cheol

doi:10.1007/s10915-011-9513-3

Cited by 13 publications

(15 citation statements)

References 23 publications

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“…This was expounded in [2] in the context of the standard method for the Helmholtz equation, but it also well recognized for Maxwell (see e.g. [14,Fig. 6]) and other wave phenomena.…”

Section: The Main Results and Discussionmentioning

confidence: 93%

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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“…This was expounded in [2] in the context of the standard method for the Helmholtz equation, but it also well recognized for Maxwell (see e.g. [14,Fig. 6]) and other wave phenomena.…”

Section: The Main Results and Discussionmentioning

confidence: 93%

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

Demkowicz¹,

Gopalakrishnan²,

Muga³

et al. 2011

Self Cite

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“…One can then prove, as indicated in [27] (or see more details in [15,Lemma 4.2]), that the operators norms I − R i h L 2 (Ω) = O(δ). Hence, choosing δ sufficiently small, the operator R i h restricted to the finite element subspace is invertible.…”

Section: 2mentioning

confidence: 95%

Partial expansion of a Lipschitz domain and some applications

Gopalakrishnan

Qiu

2012

Front. Math. China

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We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C 1 curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to prove a regular decomposition of standard vector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of low-regularity projectors into finite element spaces with partial boundary conditions is also indicated.2010 Mathematics Subject Classification. 65L60, 65N30, 46E35, 52B10, 26A16.

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“…In this section, we define appropriate weighted Sobolev spaces that will be used in the sequel and establish some of their properties; the corresponding proofs can be found in [13,31,41] (see also [24,30,35] for further general results on weighted Sobolev spaces, and some applications, respectively).…”

Section: Preliminaries and Axisymmetric Formulationmentioning

confidence: 99%

Stabilized mixed approximation of axisymmetric Brinkman flows

et al. 2015

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This paper is devoted to the numerical analysis of an augmented finite element approximation of the axisymmetric Brinkman equations. Stabilization of the variational formulation is achieved by adding suitable Galerkin least-squares terms, allowing us to transform the original problem into a formulation better suited for performing its stability analysis. The sought quantities (here velocity, vorticity, and pressure) are approximated by Raviart−Thomas elements of arbitrary order k ≥ 0, piecewise continuous polynomials of degree k + 1, and piecewise polynomials of degree k, respectively. The wellposedness of the resulting continuous and discrete variational problems is rigorously derived by virtue of the classical Babuška-Brezzi theory. We further establish a priori error estimates in the natural norms, and we provide a few numerical tests illustrating the behavior of the proposed augmented scheme and confirming our theoretical findings regarding optimal convergence of the approximate solutions.Mathematics Subject Classification. 65N30, 65N12, 76D07, 65N15, 65J20.

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Commuting Smoothed Projectors in Weighted Norms with an Application to Axisymmetric Maxwell Equations

Cited by 13 publications

References 23 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

Partial expansion of a Lipschitz domain and some applications

Stabilized mixed approximation of axisymmetric Brinkman flows

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