Wavenumber explicit analysis of a DPG method for the multidimensional Helmholtz equation

Demkowicz, Leszek; Gopalakrishnan, Jay; Muga, Ignacio; Zitelli, Jeffrey

doi:10.1016/j.cma.2011.11.024

Cited by 76 publications

(54 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A new variational formulation of the Helmholtz equation as a first-order system was recently introduced in [26]. This "discontinuous Petrov Galerkin (DPG)" method can be thought of as a least-squares method in a nonstandard inner product.…”

Section: Trefftz-discontinuous Galerkin Methodsmentioning

confidence: 99%

“…This "discontinuous Petrov Galerkin (DPG)" method can be thought of as a least-squares method in a nonstandard inner product. Using k-explicit bounds on the solution of the Helmholtz interior impedance problem, a fully k-explicit analysis of the "theoretical" version of this method is given in [26], whereas a k-explicit analysis of the "practical" version is still lacking. For this latter version, the matrix of the Galerkin discretization is only positive-semidefinite rather than positive-definite.…”

Section: Trefftz-discontinuous Galerkin Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Is the Helmholtz Equation Really Sign-Indefinite?

Moiola¹,

Spence²

2014

SIAM Rev.

View full text Add to dashboard Cite

show abstract

Section: Trefftz-discontinuous Galerkin Methodsmentioning

confidence: 99%

Section: Trefftz-discontinuous Galerkin Methodsmentioning

confidence: 99%

Is the Helmholtz Equation Really Sign-Indefinite?

Moiola¹,

Spence²

2014

SIAM Rev.

View full text Add to dashboard Cite

show abstract

“…More precisely, given that the exact solution u in a space V with norm · V and the finite element solution u h in a discrete space V h ⊂ V , the pollution error may be defined as follows (cf. [18,31]). Assume that an estimate of the following form holds:…”

mentioning

confidence: 97%

“…For preasymptotic and asymptotic error analyses of other methods including discontinuous Galerkin methods and spectral methods, we refer to [14, etc. ], [18,26,27,36,43,49].…”

mentioning

confidence: 98%

Preasymptotic Error Analysis of High Order Interior Penalty Discontinuous Galerkin Methods for the Helmholtz Equation with High Wave Number

Zhu

2015

J Sci Comput

View full text Add to dashboard Cite

A preasymptotic error analysis of the interior penalty discontinuous Galerkin (IPDG) method of high order for Helmholtz equation with the first order absorbing boundary condition in two and three dimensions is proposed. We derive the H 1 -and L 2 -error estimates with explicit dependence on the wave number k. In particular, it is shown that if k(kh) 2 p is sufficiently small, then the pollution errors of IPDG method in H 1 -norm are bounded by O(k(kh) 2 p ), which coincides with the phase error of the finite element method obtained by existent dispersion analyses on Cartesian grids, where h is the mesh size, p is the order of the approximation space and is fixed. Numerical tests are provided to verify the theoretical findings and to illustrate great capability of the symmetric IPDG method in reducing the pollution effect.

show abstract

“…Similar results were obtained using the continuous interior penalty finite element method in a recent work by Wu [31] and numerical investigations showed that the pollution error could be greatly reduced by choosing the penalty parameter appropriately. For wave-number-explicit error analyses of other methods including spectral methods and discontinuous Petrov-Galerkin methods, we refer to [13,14,25,29,34].…”

Section: Introductionmentioning

confidence: 99%

Linear continuous interior penalty finite element method for Helmholtz equation With High Wave Number: One-Dimensional Analysis

Burman

Zhu

2016

Numer. Methods Partial Differential Eq.

View full text Add to dashboard Cite

This paper addresses the properties of Continuous Interior Penalty (CIP) finite element solutions for the Helmholtz equation. The h-version of the CIP finite element method with piecewise linear approximation is applied to a one-dimensional model problem. We first show discrete well posedness and convergence results, using the imaginary part of the stabilization operator, for the complex Helmholtz equation. Then we consider a method with real valued penalty parameter and prove an error estimate of the discrete solution in the H 1 -norm, as the sum of best approximation error plus a pollution term that is the order of the phase difference. It is proved that the pollution effect can be eliminated by selecting the penalty parameter appropriately. As a result of this analysis, thorough and rigorous understanding of the error behavior throughout the range of convergence is gained. Numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior. In particular, we give numerical evidence that the optimal penalty parameter obtained in one-dimensional case also works very well for the CIP-FEM on two-dimensional Cartesian grids.

show abstract

Wavenumber explicit analysis of a DPG method for the multidimensional Helmholtz equation

Cited by 76 publications

References 14 publications

Is the Helmholtz Equation Really Sign-Indefinite?

Is the Helmholtz Equation Really Sign-Indefinite?

Preasymptotic Error Analysis of High Order Interior Penalty Discontinuous Galerkin Methods for the Helmholtz Equation with High Wave Number

Linear continuous interior penalty finite element method for Helmholtz equation With High Wave Number: One-Dimensional Analysis

Contact Info

Product

Resources

About