On the constants in hp-finite element trace inverse inequalities

We propose a new finite element method for Helmholtz equation in the situation where an acoustically permeable interface is embedded in the computational domain. A variant of Nitsche's method, different from the standard one, weakly enforces the impedance conditions for transmission through the interface. As opposed to a standard finite-element discretization of the problem, our method seamlessly handles a complex-valued impedance function Z that is allowed to vanish. In the case of a vanishing impedance, the proposed method reduces to the classic Nitsche method to weakly enforce continuity over the interface. We show stability of the method, in terms of a discrete Gårding inequality, for a quite general class of surface impedance functions, provided that possible surface waves are sufficiently resolved by the mesh. Moreover, we prove an a priori error estimate under the assumption that the absolute value of the impedance is bounded away from zero almost everywhere. Numerical experiments illustrate the performance of the method for a number of test cases in 2D and 3D with different interface conditions.

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“…where a is the bilinear form defined in expression (17). Since a λ is symmetric and consistent (by Lemma 3.1),…”

Section: A Priori Error Estimatementioning

confidence: 97%

“…To show a Gårding inequality with respect to this triple, take the real part of bilinear form (17) with q =p to obtain…”

Section: Existence and Uniquenessmentioning

confidence: 99%

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A Nitsche-type method for Helmholtz equation with an embedded acoustically permeable interface

Yedeg

Wadbro

Hansbo

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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“…It is often desired that the constant is as small as possible [33]. Here we just list these results without any further specification of the constants.…”

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confidence: 99%

A local discontinuous Galerkin method for the Burgers–Poisson equation

Liu

Ploymaklam

2014

Numer. Math.

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] as a simplified model for shallow water waves, admits conservation of both momentum and energy as two invariants. The proposed numerical method is high order accurate and preserves two invariants, hence producing solutions with satisfying long time behavior. The L 2 -stability of the scheme for general solutions is a consequence of the energy preserving property. The optimal order of accuracy for polynomial elements of even degree is proven. A series of numerical tests is provided to illustrate both accuracy and capability of the method.

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“…We extend these results to the Maxwell equation (1) for IP-DG and also provide an explicit expression of the DG method originally introduced in [8] as a slightly modified version of [4]. Our results are based on the trace inverse inequality [30] and on an extension of an accurate estimate for the lifting operators [25].…”

Section: Introductionmentioning

confidence: 76%

Optimal Penalty Parameters for Symmetric Discontinuous Galerkin Discretisations of the Time-Harmonic Maxwell Equations

2010

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We provide optimal parameter estimates and a priori error bounds for symmetric discontinuous Galerkin (DG) discretisations of the second-order indefinite time-harmonic Maxwell equations. More specifically, we consider two variations of symmetric DG methods: the interior penalty DG (IP-DG) method and one that makes use of the local lifting operator in the flux formulation. As a novelty, our parameter estimates and error bounds are (i) valid in the pre-asymptotic regime; (ii) solely depend on the geometry and the polynomial order; and (iii) are free of unspecified constants. Such estimates are particularly important in three-dimensional (3D) simulations because in practice many 3D computations occur in the pre-asymptotic regime. Therefore, it is vital that our numerical experiments that accompany the theoretical results are also in 3D. They are carried out on tetrahedral meshes with high-order (p = 1, 2, 3, 4) hierarchic H (curl)-conforming polynomial basis functions.

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On the constants in hp-finite element trace inverse inequalities

Abstract: We derive trace inverse inequalities for hp-finite elements. Utilizing orthogonal polynomials, we show how to derive explicit expressions for the constants when considering triangular and tetrahedral elements. We also discuss how to generalize this technique to the general d-simplex.

Cited by 211 publications

References 7 publications

A Nitsche-type method for Helmholtz equation with an embedded acoustically permeable interface

A Nitsche-type method for Helmholtz equation with an embedded acoustically permeable interface

A local discontinuous Galerkin method for the Burgers–Poisson equation

Optimal Penalty Parameters for Symmetric Discontinuous Galerkin Discretisations of the Time-Harmonic Maxwell Equations

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