×

A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. (English) Zbl 0863.73055

Summary: When using the Galerkin FEM for solving the Helmholtz equation in two dimensions, the error of the corresponding solution differs substantially from the error of the best approximation, and this effect increases with higher wave number \(k\). In this paper, we will design a generalized finite element method for the Helmholtz equation such that the pollution effect is minimal.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74J20 Wave scattering in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Babuška, I.; Osborn, J. E., Generalized Finite Element Methods: Their performance and their relation to mixed methods, SIAM J. Numer. Anal., 20, 3, 510-536 (1983) · Zbl 0528.65046
[2] Babuška, I. M.; Sauter, S. A., Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers, (Technical Report BN-1172 (1994), IPST, University of Maryland: IPST, University of Maryland College Park) · Zbl 0894.65050
[3] Hackbusch, W., Elliptic Differential Equations (1992), Springer Verlag: Springer Verlag Berlin
[4] Harari, I.; Hughes, T. J.R., Finite element methods for the Helmholtz equation in an exterior domain: model problems, Comput. Methods Appl. Mech. Engrg., 87, 59-96 (1991) · Zbl 0760.76047
[5] Ihlenburg, F.; Babuška, I., Finite element solution to the Helmholtz equation with high wave number. Part I: The \(h\)-version of the FEM, Comput. Math. Appl., 30, 9, 9-37 (1995) · Zbl 0838.65108
[6] F. Ihlenburg and I. Babuška, Dispersion analysis and error estimation of Galerkin finite element methods for the numerical computation of waves, Int. J. Numer. Methods Engrg., in press.; F. Ihlenburg and I. Babuška, Dispersion analysis and error estimation of Galerkin finite element methods for the numerical computation of waves, Int. J. Numer. Methods Engrg., in press.
[7] F. Ihlenburg and I. Babuška, Finite element solution to the Helmholtz equation with high wave number. Part II: The \(hp\); F. Ihlenburg and I. Babuška, Finite element solution to the Helmholtz equation with high wave number. Part II: The \(hp\)
[8] Keller, J. B.; Givoli, D., Exact non-reflecting boundary conditions, J. Comp. Phys., 82, 172-192 (1989) · Zbl 0671.65094
[9] Thompson, L. L.; Pinsky, P. M., A Galerkin least squares finite element method for the two-dimensional Helmholtz equation, Int. J. Numer. Methods Engrg., 38, 371-397 (1995) · Zbl 0844.76060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.