A discontinuous finite element formulation for Helmholtz equation

Alvarez, Gustavo Benitez; Loula, Abimael F. D.; Carmo, Eduardo Gomes Dutra do; Rochinha, Fernando A.

doi:10.1016/j.cma.2005.07.013

Cited by 45 publications

(42 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One advantage of the IPDG method is that it contains several parameters which can be tuned for a particular purpose. In [2], it is shown that it is possible to reduce the pollution error of the IPDG method by choosing appropriate parameters σ and iγ 0,e . Recall that all choice of σ but one lead to nonsymmetric formulations.…”

Section: Error Estimates For Schemementioning

confidence: 99%

See 1 more Smart Citation

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

Feng¹,

Wu²

2009

SIAM J. Numer. Anal.

163

236

View full text Add to dashboard Cite

Abstract. This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence well-posed) without any mesh constraint. For each fixed wave number k, optimal order (with respect to h) error estimate in the broken H 1 -norm and sub-optimal order estimate in the L 2 -norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime (i.e., k 2 h 1). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to k) in the error bounds. The novelties of the proposed interior penalty discontinuous Galerkin methods include: first, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a non-Hermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To the end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [23,24,35].

show abstract

Section: Error Estimates For Schemementioning

confidence: 99%

“…[1,2,5,6,12,16,21,23,27,29,33,37,36,39,42] and the reference therein). It is well known that in every coordinate direction, one must put some minimal number of grid points in each wave length = 2π/k in order to resolve the wave, that is, the mesh size h must satisfy the constraint hk 1.…”

mentioning

confidence: 99%

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

Feng¹,

Wu²

2009

SIAM J. Numer. Anal.

163

236

View full text Add to dashboard Cite

show abstract

“…Moreover, the use of standard adaptive mesh refinement techniques based on a posteriori error estimators is marred by the pollution effect [5,27]. Recently, Discontinuous Galerkin (DG) methods [14,24,34] have been increasingly applied to wave propagation problems in general [13] and the Helmholtz equation in particular [2,3,18,19,20,21] including hybridized DG approximations [23]. An a posteriori error analysis of DG methods for standard second order elliptic boundary value problems has been performed in [1,8,10,26,31,35], and a convergence analysis has been 1 provided in [9,25,32].…”

mentioning

confidence: 99%

Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the Helmholtz equation

Hoppe

Sharma

2012

IMA Journal of Numerical Analysis

View full text Add to dashboard Cite

Abstract. We are concerned with a convergence analysis of an adaptive Interior Penalty Discontinuous Galerkin (IPDG) method for the numerical solution of acoustic wave propagation problems as described by the Helmholtz equation. The mesh adaptivity relies on a residual-type a posteriori error estimator that does not only control the approximation error but also the consistency error caused by the nonconformity of the approach. As in the case of IPDG for standard second order elliptic boundary value problems, the convergence analysis is based on the reliability of the estimator, an estimator reduction property, and a quasi-orthogonality result. However, in contrast to the standard case, special attention has to be paid to a proper treatment of the lower order term in the equation containing the wavenumber which is taken care of by an Aubin-Nitsche type argument for the associated conforming finite element approximation. Numerical results are given for an interior Dirichlet problem and a screen problem illustrating the performance of the adaptive IPDG method.

show abstract

“…Recently, we introduced discontinuous finite element formulations for Helmholtz equation depending on two stabilization parameters [12,14]. Numerical experiments show the good performance and potential of this formulation to reduce the pollution effect.…”

Section: Introductionmentioning

confidence: 96%

“…Numerical experiments show the good performance and potential of this formulation to reduce the pollution effect. Completely discontinuous formulation, as presented by Alvarez et al [12], may lead to high computational cost since the degrees of freedom associated with the discontinuity cannot be eliminated. Moreover, the two parameters of this formulation ( and ) are determined through numerical experiments.…”

Section: Introductionmentioning

confidence: 98%

A locally discontinuous enriched finite element formulation for acoustics

Rochinha

Alvarez

Carmo

et al. 2006

Commun. Numer. Meth. Engng.

View full text Add to dashboard Cite

SUMMARYIn (Comput. Methods Appl. Mech. Eng. 2006, in press) we introduced a discontinuous Galerkin finite element method for Helmholtz equation in which continuity is relaxed locally in the interior of the element. The shape functions associated with interior nodes of the element are bilinear discontinuous bubbles, and the corresponding degrees of freedom can be eliminated at element level by static condensation yielding a global finite element method with the same connectivity of classical C 0 Galerkin finite element approximations. Stability is provided by the discontinuous bubbles with appropriate choice of the stabilization parameters related to the weak enforcement of continuity inside each element. In the present work, departing from the stencil obtained by condensation of the bubble degrees of freedom, we build a new strategy for determining the optimal values of these parameters aiming at matching the exact wave number in two different directions. Stability and accuracy of the proposed formulation are demonstrated in several numerical examples.

show abstract

A discontinuous finite element formulation for Helmholtz equation

Cited by 45 publications

References 40 publications

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the Helmholtz equation

A locally discontinuous enriched finite element formulation for acoustics

Contact Info

Product

Resources

About