A quasi optimal Petrov–Galerkin method for Helmholtz problem

Loula, Abimael F. D.; Fernandes, Daniel T.

doi:10.1002/nme.2677

Cited by 20 publications

(27 citation statements)

References 29 publications

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“…Moreover, for low-order approximations dense meshes must be enhanced with stabilized formulations to control dispersion errors [44][45][46]. In spite of the improved e ciency of high-order approximations [47,48] DOF when 100 nodes are used for parameters ✓ and !.…”

Section: The Proper Generalized Decomposition Methods (Pgd)mentioning

confidence: 99%

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

Modesto

Zlotnik

Huerta

2015

Computer Methods in Applied Mechanics and Engineering

116

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Solving the Helmholtz equation for a large number of input data in an heterogeneous media and unbounded domain still represents a challenge. This is due to the particular nature of the Helmholtz operator and the sensibility of the solution to small variations of the data. Here a reduced order model is used to determine the scattered solution everywhere in the domain for any incoming wave direction and frequency. Moreover, this is applied to a real engineering problem: water agitation inside real harbors for low to mid-high frequencies.The Proper Generalized Decomposition (PGD) model reduction approach is used to obtain a separable representation of the solution at any point and for any incoming wave direction and frequency. Here, its applicability to such a problem is discussed and demonstrated. More precisely, the contributions of the paper include the PGD implementation into a Perfectly Matched Layer framework to model the unbounded domain, and the separability of the operator which is addressed here using an e cient higher-order projection scheme.Then, the performance of the PGD in this framework is discussed and improved using the higher-order projection and a Petrov-Galerkin approach to construct the separated basis. Moreover, the e ciency of the higherorder projection scheme is demonstrated and compared with the higher-order singular value decomposition.

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Section: The Proper Generalized Decomposition Methods (Pgd)mentioning

confidence: 99%

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

Modesto

Zlotnik

Huerta

2015

Computer Methods in Applied Mechanics and Engineering

116

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“…The QOPG finite element method, for the Helmholtz problem we present in [30], is also closely related to the present finite difference method in the sense that it is based on a minimization of a least-squares residual of discrete plane waves at a macroelement level (stencil) or, more precisely, at the level n = 1 with A i = A 1 i . QOPG is also capable of reducing phase error on uniform, non-uniform and unstructured meshes, and to naturally impose non-essential boundary conditions (Robin or Neumann) but not to deal with A i = A n i , with n>1.…”

Section: Remarkmentioning

confidence: 98%

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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“…The mixed formulation and the discontinuity of the functional spaces is needed to derive an easy, practical, and inexpensive way to compute the optimal test space. Compared to other PG approaches (e.g., [5] or [25]), the method may be difficult to implement within existing classical FEM codes, but fits perfectly within the framework of hybrid methods like the original DPG method developed in [6]. The essential difference is in the computation of optimal test functions, an operation performed purely on the element level using a simple preprocessing routine.…”

Section: Introductionmentioning

confidence: 97%

A class of discontinuous Petrov–Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D

Zitelli

Muga

Demkowicz

et al. 2011

Journal of Computational Physics

127

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a b s t r a c tThe phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for onedimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov-Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L 2 -norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed.

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A quasi optimal Petrov–Galerkin method for Helmholtz problem

Cited by 20 publications

References 29 publications

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

Quasi optimal finite difference method for Helmholtz problem on unstructured grids

A class of discontinuous Petrov–Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D

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