Error estimation for FEM in acoustic scattering
Dedicated to the memory of Prof. dr. habil. mz. Marian Kmiecik
Abstract
We consider apriori and aposteriori error estimation for the FEM solution of Helmholtz problems that arise in acoustic scattering. Our focus is on the case of high wavenumber (highly oscillatory solutions) where existing asymptotic estimates had to be generalized to "preasymptotic" statements that are applicable in the range of engineering computations. We refer the key results of an 1D analytic study of error behawior (apriori estimates) and announce new results on aposteriori error estimation. Specifically, we show that the standard local aposteriori error indicators are not, in general, reliable for Helmholtz problems with high wave number, due to considerable numerical pollution in the error. We then discuss a methodology how to (aposteriori) estimate, in addition to the local error, the pollution error. Throughout, the theoretical results will be supplemented by numerical evaluation.
Keywords
References
[1] I. Babuška, F. Ihlenburg, T. Strouboulis, S. Gangaraj . A posteriori error estimation for FEM solution to Helmholtz's equation - part I: The quality of local indicators and estimators; part II: estimation of the pollution error. TICAM Report, 96-32 and 96-33.[2] I. Babuška, F. Ihlenburg, E. T. Paik, S. A. Sauter. A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Comp. Methods Appl. Mech. Eng., 128: 325- 359, 1995.
[3] L. Demkowicz, W. Rachowicz, K. Banaś, J . Kucwaj. 2D hp adaptive package. Technical report, Section of Applied Mathematics, Politechnika Krakowska, Cracow, 1992.
[4] L. Demkowicz, A. Bajer, K. Banaś. Geometrical modeling package. TICAM report 92- 06 , University of Texas at Austin, 1992.
[5] L. Demkowicz, K. Gerdes. Convergence of the Infinite Element Methods for the Helmholtz Equation. TICAM Report 95- 07.
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