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The finite element method with non-uniform mesh sizes applied to the exterior Helmholtz problem

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Summary

The finite element method with non-uniform mesh sizes is employed to approximately solve Helmholtz type equations in unbounded domains. The given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary. This approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases. This results in a great reduction in the number of equations to be solved. It is proved that optimal error estimates hold inL 2,H 1 andL ∞, provided that certain relationships hold between the frequency, mesh size and outer radius.

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References

  1. Atkinson, F.V.: On Sommerfeld's “Radiation Condition”. The Philosophical Magazine. Vol.40, 645–651 (1949)

    Google Scholar 

  2. Aziz, A.K., Kellogg, R.B.: Finite element analysis of a scattering problem. Math. Comput. (in press, 1981)

  3. Babuska, I.: The finite element method for infinite domains I. Math. Comput.26, 1–11 (1972)

    Google Scholar 

  4. Babuska, I.: Finite element method for domains with corners. Computing6, 264–273 (1970)

    Google Scholar 

  5. Bayliss, A., Gunzberger, M., Turkel, E.: Boundary conditions for the numerical solution of elliptic equations in exterior regions. ICASE Report 80-1 (1979)

  6. Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam: North-Holland 1978

    Google Scholar 

  7. Eidus, D.M.: The principle of limiting absorption. Amer. Math. Soc. Transl.47, 156–192 (1965)

    Google Scholar 

  8. Eisenstat, S.C., Schultz, M.H.: Computational aspects of the finite element method. The mathematical foundations of the finite element method, pp. 505–524 (A.K. Aziz, ed.). New York: Academic Press 1972

    Google Scholar 

  9. Giroire, J., Nedelec, J.C.: Numerical solutions of an exterior Neumann problem using a double layer potential. Math. Comput.32, 973–990 (1978)

    Google Scholar 

  10. Goldstein, C.I.: The finite element method with non-uniform mesh sizes for unbounded domains. Math. Comput.36, 381–404 (1981)

    Google Scholar 

  11. Goldstein, C.I.: Numerical methods for Helmholtz type equations in unbounded regions. Mathematical methods and applications of scattering theory. (J.A. DeSanto, A.W. Saenz, W.W. Zachary, eds.), pp. 26–33. Lecture Notes in Physics. Vol. 130. Berlin-Heidelberg-New York: Springer 1980

    Google Scholar 

  12. Goldstein, C.I.: Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries III. Trans. Amer. Math. Soc.143, 283–301 (1969)

    Google Scholar 

  13. Kellogg, O.D.: Foundations of potential theory. New York: Frederick Ungar Publishing Company 1929

    Google Scholar 

  14. Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications. Vol. I, Grundlehren der math. Wissenschaften, 183. Band. Berlin-Heidelberg-New York: Springer 1972

    Google Scholar 

  15. Marin, S.: A finite element method for problems involving the Helmholtz equation in two dimensional exterior regions. Thesis. Carnegie-Mellon University

  16. Nitsche, J.A., Schatz, A.H.: Interior estimates for Ritz-Galerkins methods. Math. Comput.28, 937–958 (1974)

    Google Scholar 

  17. Schatz, A.H., Wahlbin, L.B.: Maximum norm estimates in the finite element method on plane polygonal domains. Part 1. Math. Comput.32, 73–109 (1978)

    Google Scholar 

  18. Schatz, A.H., Wahlbin, L.B.: Maximum norm estimates in the finite element method on plane polygonal domains. Part 2, Refinements. Math. Comput.33, 465–492 (1979)

    Google Scholar 

  19. Schechter, M.: General boundary value problems for elliptic partial differential equations. Comm. Pure Appl. Math.12, 457–486 (1959)

    Google Scholar 

  20. Thatcher, R.W.: The use of infinite grid refinements at singularities in the solution of Laplace's equation. Numer. Math.25, 163–178 (1976)

    Google Scholar 

  21. Ying, L.: The infinite similar element method for calculating stress intensity factors. Scientia Sinica, Vol.XXI, 19–43 (1978)

    Google Scholar 

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  1. Applied Mathematics Department, Brookhaven National Laboratory, 11973, Upton, New York, USA

    C. I. Goldstein

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  1. C. I. Goldstein
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Goldstein, C.I. The finite element method with non-uniform mesh sizes applied to the exterior Helmholtz problem. Numer. Math. 38, 61–82 (1982). https://doi.org/10.1007/BF01395809

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