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**A multiscale finite element method for the Helmholtz equation.**
*(English)*
Zbl 0937.65119

Summary: It is well known that when the standard Galerkin method is applied to the Helmholtz equation it exhibits an error in the wavenumber and the solution does not, therefore, preserve the phase characteristics of the exact solution. Improvements on the Galerkin method, including Galerkin least squares (GLS) methods, have been proposed. However, these approaches rely on a dispersion analysis of the underlying difference stencils in order to reduce the error in the solution. In this paper we propose a multiscale finite element for the Helmholtz equation. The method employs a multiscale variational formulation which leads to a subgrid model in which subgrid scales are incorporated analytically through appropriate Green’s functions. It is shown that entirely new and accurate methods emerge and that GLS methods can be obtained as special cases of the more general subgrid model.

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

65N15 | Error bounds for boundary value problems involving PDEs |

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

### Keywords:

superconvergence; multiscale finite element method; Galerkin least squares methods; Galerkin method; Helmholtz equation
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\textit{A. A. Oberai} and \textit{P. M. Pinsky}, Comput. Methods Appl. Mech. Eng. 154, No. 3--4, 281--297 (1998; Zbl 0937.65119)

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### References:

[1] | Babuška, I.; Ihlenburg, F.; Paik, E.T.; Sauter, S.A., A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution, Comput. methods appl. mech. engrg., 128, 325-359, (1995) · Zbl 0863.73055 |

[2] | Harari, I.; Hughes, T.J.R., Galerkin/least-squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains, Comput. methods appl. mech. engrg., 98, 411-454, (1992) · Zbl 0762.76053 |

[3] | Hughes, T.J.R., Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. methods appl. mech. engrg., 129, 387-401, (1995) · Zbl 0866.76044 |

[4] | T.J.R. Hughes, G.R. Feijóo, L. Mazzei and J.-B. Quincy, The variational multiscale method—a paradigm for computational mechanics, in preparation. |

[5] | A.A. Oberai and P.M. Pinsky, Finite element methods for the Helmholtz equation based on global multiscale variational forms, in preparation. · Zbl 0937.65119 |

[6] | Roach, G.F., Green’s functions, (1982), Cambridge University Press New York · Zbl 0478.34001 |

[7] | 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 |

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