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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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##### References:
 [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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