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Finite element solution of the Helmholtz equation with high wave number. I: The \(h\)-version of the FEM. (English) Zbl 0838.65108
This paper examines the quality of the discrete numerical solutions to the Helmholtz equation \(\Delta u + k^2u = f\) where \(k\) is the wave number. These equations arise in problems of wave scattering and fluid-solid-interaction.
It is shown that the relative error in the finite element solution in \(H^1\) seminorm is \[ e_1 \leq C_1kh + C_2 k^3 h^2 \] where \(h\) is the step length of the meshes. The first term on the right hand side of the inequality is the approximation error and the second term is due to numerical pollution.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition 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
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[1] Dautray, R.; Lions, L., ()
[2] Junger, M.C.; Feit, D., Sound, structures and their interaction, (1986), MIT Press Cambridge, MA
[3] Harari, I.; Hughes, T.J.R., Finite element method for the Helmholtz equation in an exterior domain: model problems, Comp. meth. appl. mech. eng., 87, 59-96, (1991) · Zbl 0760.76047
[4] Bayliss, C.I.; Goldstein, E., Turkel, on accuracy conditions for the numerical computation of waves, J. comp. phys., 59, 396-404, (1985) · Zbl 0647.65072
[5] Aziz, A.K.; Kellogg, R.B.; Stephens, A.B., A two point boundary value problem with a rapidly oscillating solution, Numer. math., 53, 107-121, (1988) · Zbl 0645.65041
[6] Douglas, J.; Santos, J.E.; Sheen, D.; Schreiyer, L., Frequency domain treatment of one-dimensional scalar waves, Mathematical models and methods in applied sciences, 3, 2, 171-194, (1993) · Zbl 0783.65070
[7] I. Babuška, F. Ihlenburg and Ch. Makridakis, Analysis and finite element methods for a fluid solid interaction problem in one dimension, Technical Note BN-1183, Institute for Physical Science and Technology, University of Maryland at College Park, (in preparation).
[8] F. Ihlenburg and I. Babuška, Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation, Int. J. Numer. Methods Eng. (to appear).
[9] Babuška, I.; Ihlenburg, F.; Paik, E.; Sauter, S., A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution, () · Zbl 0863.73055
[10] Demkowicz, L., Asymptotic convergence in finite and boundary element methods: part I: theoretical results, Computers math. applic., 27, 12, 69-84, (1994) · Zbl 0807.65058
[11] 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, Comp. meth. appl. mech. eng., 98, 411-454, (1992) · Zbl 0762.76053
[12] Thompson, L.L.; Pinsky, P.M., A Galerkin least squares finite element method for the two-dimensional Helmholtz equation, Int. J. numer. methods eng., 38, 3, 371-397, (1995) · Zbl 0844.76060
[13] Babuška, I.; Sauter, S., Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers, () · Zbl 0894.65050
[14] Achieser, N.I., Vorlesungen über approximations theory, (1953), Akademieverlag Berlin · Zbl 0052.29002
[15] John, F., Partial differential equations, (1982), Springer New York
[16] Babuška, I.; Aziz, A.K., The mathematical foundations of the finite element method, (), 5-359
[17] Strang, G.; Fix, G.J., An analysis of the finite element method, (1973), Prentice Hall Englewood Cliffs, NJ · Zbl 0278.65116
[18] Samarskii, A.A., Introduction to the theory of difference schemes [russian], (1971), Moscow
[19] Ihlenburg, F.; Babuška, I., Finite element solution to the Helmholtz equation with high wavenumber—part I: the h-version of the FEM, ()
[20] Babuška, I.; Strouboulis, T.; Mathur, A.; Upadhyay, C.S., Pollution error in the h-version of the FEM and the local quality of a-posteriori error estimators, () · Zbl 0924.65098
[21] Babuška, I.; Strouboulis, T.; Upadhyay, C.S.; Gangaraj, S.K., A-posteriori estimation and adaptive control of the pollution-error in the h-version of the FEM, () · Zbl 0844.65078
[22] Babuška, I.; Strouboulis, T.; Gangaraj, S.K.; Upadhyay, C.S., Pollution error in the h-version of the FEM and the local quality of recovered derivatives, () · Zbl 0896.73055
[23] Thompson, L.L.; Pinsky, P.M., Complex wavenumber Fourier analysis of the p-version finite element method, Computational mechanics, 13, 255-275, (1994) · Zbl 0789.73076
[24] I. Babuška, I.N. Katz and B.S. Szabó, Finite element analysis in one dimension, In Lecture Notes, Springer-Verlag, (to appear).
[25] Burnett, D.S., A three-dimensional acoustic infinite element based on a prolate spheroidal multipole expansion, J. acoust. soc. am., 96, 5, 2798-2816, (1994)
[26] Harari, I.; Hughes, T.J.R., A cost comparison of boundary element and finite element methods for problems of time-harmonic acoustics, Comp. meth. appl. mech. eng., 97, 77-102, (1992) · Zbl 0775.76095
[27] Ihlenburg, F.; Babuška, I., Finite element solution to the Helmholtz equation with high wavenumber—part II: the h-p-version of the FEM, Technical note BN-73, SIAM J. numer. anal., (1994), (to appear)
[28] Schatz, A., An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. comp., 28, 959-962, (1974) · Zbl 0321.65059
[29] Szabó, B.; Babuška, I., Finite element analysis, (1991), J. Wiley New York
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