A discontinuous finite element method at element level for Helmholtz equation

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

doi:10.1016/j.cma.2006.07.008

Cited by 17 publications

(16 citation statements)

References 13 publications

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“…This formulation is consistent in the sense that the exact solution u of problem Equations (1)- (4) is also solution of Equations (13) and (14).…”

Section: The Discontinuous Finite Element Methods At Element Level Conmentioning

confidence: 84%

“…For uniform meshes, this strategy leads to an analytical explicit procedure, thus not requiring any extra computational effort. The new methodology introduced in the present article is extensible for non-uniform meshes, which can be considered as an improvement when compared to our previous work [14]. The stencil of the QSFEM is consistently retrieved for the optimal choice of the stabilization parameters.…”

Section: Discussionmentioning

confidence: 94%

“…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: 95%

“…The finite element method we introduced in [14] is also based on a discontinuous finite element formulation [12,13], but with continuity relaxed only on the interiors of elements instead of across the element edges as it was admitted in our previous formulation [12]. Continuity on the interelement boundaries is strongly enforced by using C 0 Lagrangian interpolation globally as usual.…”

Section: Introductionmentioning

confidence: 99%

“…A great effort has been devoted to alleviate pollution effects [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. In particular, the Galerkin least-squares (GLS) method is able to completely eliminate the phase lag in one-dimensional problems [4].…”

Section: Introductionmentioning

confidence: 99%

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A locally discontinuous enriched finite element formulation for acoustics

Rochinha

Alvarez

Carmo

et al. 2006

Commun. Numer. Meth. Engng.

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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.

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“…This formulation is consistent in the sense that the exact solution u of problem Equations (1)- (4) is also solution of Equations (13) and (14).…”

Section: The Discontinuous Finite Element Methods At Element Level Conmentioning

confidence: 84%

Section: Discussionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A locally discontinuous enriched finite element formulation for acoustics

Rochinha

Alvarez

Carmo

et al. 2006

Commun. Numer. Meth. Engng.

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A quasi optimal Petrov–Galerkin method for Helmholtz problem

Loula

Fernandes

2009

Numerical Meth Engineering

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SUMMARYA Petrov-Galerkin finite element formulation is introduced for Helmholtz problem in two dimensions using polynomial weighting functions. At each node of the mesh, a global basis function for the weighting space is obtained adding to the bilinear C 0 Lagrangian weighting function linear combinations of polynomial bubbles defined on a macroelement containing this node. Quasi optimal weighting functions, with the same support of the corresponding global test functions, are obtained after computing the coefficients of these linear combinations attending to optimality criteria. This is done numerically through a simple preprocessing technique naturally applied to non-uniform and unstructured meshes. In particular, for uniform mesh a quasi optimal interior stencil of the same order of the quasi-stabilized finite element method stencil derived by Babuška et al. (Comput. Methods Appl. Mech. Engrg 1995; 128:325-359) is obtained. Numerical results are presented illustrating the great stability and accuracy of this formulation with uniform and non-uniform meshes.

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Quasi optimal finite difference method for Helmholtz problem on unstructured grids

Fernandes

Loula

2009

Numerical Meth Engineering

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SUMMARYA quasi optimal finite difference method (QOFD) is proposed for the Helmholtz problem. The stencils' coefficients are obtained numerically by minimizing a least-squares functional of the local truncation error for plane wave solutions in any direction. In one dimension this approach leads to a nodally exact scheme, with no truncation error, for uniform or non-uniform meshes. In two dimensions, when applied to a uniform cartesian grid, a 9-point sixth-order scheme is derived with the same truncation error of the quasi-stabilized finite element method (QSFEM) introduced by Babuška et al. (Comp. Meth. Appl. Mech. Eng. 1995; 128:325-359). Similarly, a 27-point sixth-order stencil is derived in three dimensions. The QOFD formulation, proposed here, is naturally applied on uniform, non-uniform and unstructured meshes in any dimension. Numerical results are presented showing optimal rates of convergence and reduced pollution effects for large values of the wave number.

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A discontinuous finite element method at element level for Helmholtz equation

Cited by 17 publications

References 13 publications

A locally discontinuous enriched finite element formulation for acoustics

A locally discontinuous enriched finite element formulation for acoustics

A quasi optimal Petrov–Galerkin method for Helmholtz problem

Quasi optimal finite difference method for Helmholtz problem on unstructured grids

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