A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

Nadukandi, Prashanth; Oñate, Eugenio; García, Julio Martínez

doi:10.1002/nme.3291

Cited by 4 publications

(6 citation statements)

References 25 publications

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“…Fortunately, this idea which was born to treat this shortcoming in the CDR problem, has opened door to a class of higher-order compact Petrov-Galerkin FEM effective for the Helmholtz problem. The design of such a Petrov-Galerkin FEM and its applications to the Helmholtz equation is the subject matter of the paper [64].…”

Section: Discussionmentioning

confidence: 99%

A high-resolution Petrov–Galerkin method for the convection–diffusion–reaction problem. Part II—A multidimensional extension

Nadukandi

Oñate

García

2012

Computer Methods in Applied Mechanics and Engineering

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Section: Discussionmentioning

confidence: 99%

A high-resolution Petrov–Galerkin method for the convection–diffusion–reaction problem. Part II—A multidimensional extension

Nadukandi

Oñate

García

2012

Computer Methods in Applied Mechanics and Engineering

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“…This additional structure throws light on the extension of this stencil to unstructured meshes. In 70, a new Petrov–Galerkin method involving two parameters viz. α 1 , α 2 is presented which yields this nonstandard compact stencil on rectangular meshes.…”

Section: Discussionmentioning

confidence: 99%

“…Precisely, if it is possible to design a Petrov–Galerkin method that would yield the FDM stencil on a structured mesh, then this scheme can be extended to unstructured meshes in a straightforward manner. We show that indeed it is possible to design such a Petrov–Galerkin method using just the lowest‐order block finite elements 70.…”

Section: Analysis In 2dmentioning

confidence: 94%

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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“…The latter approach enriching test spaces is based on a PetrovGalerkin setting, see e.g. [4,11,28]. The Petrov-Galerkin concept is also the basis for the variational multiscale method (VMS) by Hughes et al, e.g.…”

Section: Introductionmentioning

confidence: 99%

Revisiting generalized FEM: a Petrov–Galerkin enrichment based FEM interpolation for Helmholtz problem

Kovtunenko

Kunisch

2018

Calcolo

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A concept of Petrov-Galerkin enrichment which is appropriate for highly accurate and stable interpolation of variational solutions is introduced. In the finite element context, the setting refers to standard trial functions for the solution, while the test space will be enriched. The FEM interpolation procedure that we propose will be justified by local wavelets with vanishing moments based on Gegenbauer polynomials. For the reference Helmholtz equation, the continuous piecewise polynomial test functions are enriched using dispersion analysis on uniform meshes in 2d and 3d. From a-priori and a-posteriori numerical analysis it follows that the Petrov-Galerkin based enrichment approximates the exact interpolate solution of the Helmholtz equation with at least seventh order of accuracy.

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A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

Cited by 4 publications

References 25 publications

A high-resolution Petrov–Galerkin method for the convection–diffusion–reaction problem. Part II—A multidimensional extension

A high-resolution Petrov–Galerkin method for the convection–diffusion–reaction problem. Part II—A multidimensional extension

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

Revisiting generalized FEM: a Petrov–Galerkin enrichment based FEM interpolation for Helmholtz problem

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