A nearly optimal Galerkin projected residual finite element method for Helmholtz problem

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

doi:10.1016/j.cma.2007.11.001

Cited by 8 publications

(15 citation statements)

References 19 publications

(41 reference statements)

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“…Alternate methods that achieve this objective were proposed earlier within a variational setting and with similar implementation/computational cost, viz. the RBFEM 26, the DGB method 32, the GPR method 34 etc. Can this path to obtain the QSFEM be simplified?…”

Section: Discussionmentioning

confidence: 99%

“…Nodally exact Ritz discretization of the 1D diffusion–absorption/production equations via variational finite calculus (FIC) and modified equation methods using a single stabilization parameter were presented in 33. The Galerkin‐projected residual (GPR) method for the Helmholtz equation was presented in 34. A survey of finite element methods for time‐harmonic acoustics is done in 35.…”

Section: Introductionmentioning

confidence: 99%

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A fourth‐order compact scheme for the Helmholtz equation: Alpha‐interpolation of FEM and FDM stencils

Nadukandi

Oñate

García

2010

Numerical Meth Engineering

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We propose a fourth-order compact scheme on structured meshes for the Helmholtz equation given by R():= f (x)+D+n 2 = 0. The scheme consists of taking the alpha-interpolation of the Galerkin finite element method and the classical central finite difference method. In 1D, this scheme is identical to the alpha-interpolation method (J. Comput. Appl. Math. 1982; 8(1):15-19) and in 2D making the choice = 0.5 we recover the generalized fourth-order compact Padé approximation (J. 128:325-359) for the analysis of this scheme and its performance on square meshes is compared with that of the quasi-stabilized FEM (Comput. Meth. Appl. Mech. Engrg 1995; 128:325-359). In particular, we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate O((n ) 4 ), where n, represent the wavenumber and the mesh size, respectively. An expression for the parameter is given that minimizes the maximum relative phase error in a sense that will be explained in Section 4.5. Convergence studies of the error in the L 2 norm, the H 1 semi-norm and the l ∞ Euclidean norm are done and the pollution effect is found to be small. 19As the current problem admits a variational principle, naturally, discretization methods based on variational formulations viz. the Galerkin and the Trefftz-Galerkin-type methods have been preferred to other methods. The Galerkin-type methods are domain-based wherein the integral statement involves only the weak form of the governing differential equation and the sub-space of test-functions are assumed to satisfy a priori the kinematic compatibility and essential boundary conditions. The Trefftz-Galerkin-type methods are boundary-based and are formulated using the reciprocal principle wherein the integral statement involves only the kinematic compatibility and essential boundary conditions of the problem and the sub-space of test-functions are assumed to satisfy a priori the governing differential equation [1,2].In the context of the Galerkin-type methods, the finite element method (FEM) is a powerful technique to systematically generate subspaces of test-functions (classically piecewise polynomial spaces). Some of the earlier works on the use of FEM for the numerical solution of the Helmholtz equation can be found in [3-10] and the references cited therein. In [5,7] error estimates were given for the asymptotic (n 2 assumed sufficiently small) and pre-asymptotic (n assumed sufficiently small) cases, respectively. It was shown that for the discrete problem the LBB ‡ constant can be expressed as h = min{| h m −n 2 |/ h m } [6]. Thus, for the continuous problem (visualized as → 0) the LBB constant can be expressed as = min{| m −n 2 |/ m }, which in an average sense implies that is inversely proportional to the wavenumber n, i.e. ∝ n −1 [6,7]. Thus for high wavenumbers and for the case of degeneracy (n → h m ), the LBB constant for the discrete problem tends to be small which in turn leads to a loss of stability. The loss of stability...

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A fourth‐order compact scheme for the Helmholtz equation: Alpha‐interpolation of FEM and FDM stencils

Nadukandi

Oñate

García

2010

Numerical Meth Engineering

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“…From definition (20) we identify two types of bubbles. One with support restricted to a single element, when the nodes l and j do not belong to the same element edge, and the other with support on two adjacent elements when both nodes l and j belong to the same element edge.…”

Section: The Macroelement Bubblesmentioning

confidence: 99%

“…The simplest and most popular, which is in the origin of basically all residual stabilization methods as in [7,16,19,20], consists in adding to the Galerkin formulation extra residual forms in the interior of the elements as introduced by Brooks and Hughes for convection-diffusion problems with the SUPG method [21]. The alternative stabilization strategy considered here is based on the concept of optimal or nearly optimal weighting functions as in [13,[22][23][24][25].…”

Section: Petrov-galerkin Formulationmentioning

confidence: 99%

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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“…do Carmo et al (2008) proposed the Galerkin projected residual (GPR) method for the Helmholtz equation. This work consists of projecting the residual into a subspace, defined for each element, built by systematically exploring some a priori criteria; this procedure showed to be robust for mid and high wave numbers.…”

Section: Introductionmentioning

confidence: 99%

The size function for a HDG method applied to the Helmholtz problem

et al. 2019

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In this work, we propose a hybridized Continuous/Discontinuous Galerkin formulation for the Helmholtz problem that works with a continuous trace space. Additionally, it is done a static condensation analysis where we obtain a global system that is smaller than global systems generated by other current hybrid methods. Therefore, we conjecture that the computational effort of our formulation is lower than that of the classic DG methods. Furthermore, static condensation is proved as being a well-posed problem from a certain degree of mesh refinement. As result, we present the computational time for the method with continuous trace space compared with the computational time of a continuous Galerkin approach for a fixed polynomial approximation and different mesh refinements. Numerical results show the robustness and accuracy of our formulation as well as the potentiality of the present method.

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A nearly optimal Galerkin projected residual finite element method for Helmholtz problem

Cited by 8 publications

References 19 publications

A fourth‐order compact scheme for the Helmholtz equation: Alpha‐interpolation of FEM and FDM stencils

A fourth‐order compact scheme for the Helmholtz equation: Alpha‐interpolation of FEM and FDM stencils

A quasi optimal Petrov–Galerkin method for Helmholtz problem

The size function for a HDG method applied to the Helmholtz problem

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