A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains

Author:
Charles I. Goldstein

Journal:
Math. Comp. **39** (1982), 309-324

MSC:
Primary 65N30; Secondary 65N15, 78A50

MathSciNet review:
669632

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Abstract: A finite element method is described for solving Helmholtz type boundary value problems in unbounded regions, including those with infinite boundaries. Typical examples include the propagation of acoustic or electromagnetic waves in waveguides. The radiation condition at infinity is based on separation of variables and differs from the classical Sommerfeld radiation condition. It is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain. The behavior of the solution near infinity is incorporated in a nonlocal boundary condition. This problem is given a weak or variational formulation, and the finite element method is then applied. It is proved that optimal error estimates hold.

**[1]**C. I. Goldstein, "Scattering theory in waveguides,"*Scattering Theory in Mathematical Physics*(J. A. LaVita and J. P. Marchand, editors), Reidel, Dordrecht, 1974, pp. 35-52.**[2]**George J. Fix and Samuel P. Marin,*Variational methods for underwater acoustic problems*, J. Comput. Phys.**28**(1978), no. 2, 253–270. MR**0502898****[3]**D. M. Eidus, "The principle of limiting absorption,"*Amer. Math. Soc. Transl.*(2), v. 47, 1965, pp. 157-191.**[4]**Charles Irwin Goldstein,*Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries. I*, Trans. Amer. Math. Soc.**135**(1969), 1–31. MR**0234140**, 10.1090/S0002-9947-1969-0234140-4**[5]**Charles Goldstein,*Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries. III*, Trans. Amer. Math. Soc.**143**(1969), 283–301. MR**0609010**, 10.1090/S0002-9947-1969-0609010-X**[6]**Jean-Claude Guillot and Calvin H. Wilcox,*Steady-state wave propagation in simple and compound acoustic waveguides*, Math. Z.**160**(1978), no. 1, 89–102. MR**496589**, 10.1007/BF01182333**[7]**A. Bayliss, M. Gunzberger & E. Turkel,*Boundary Conditions for the Numerical Solution of Elliptic Equations in Exterior Regions*, ICASE Report 80-1, 1979.**[8]**C. I. Goldstein,*The finite element method with nonuniform mesh sizes for unbounded domains*, Math. Comp.**36**(1981), no. 154, 387–404. MR**606503**, 10.1090/S0025-5718-1981-0606503-5**[9]**C. I. Goldstein,*The finite element method with nonuniform mesh sizes applied to the exterior Helmholtz problem*, Numer. Math.**38**(1981/82), no. 1, 61–82. MR**634753**, 10.1007/BF01395809**[10]**F. Brezzi, C. Johnson, and J.-C. Nédélec,*On the coupling of boundary integral and finite element methods*, Proceedings of the Fourth Symposium on Basic Problems of Numerical Mathematics (Plzeň, 1978) Charles Univ., Prague, 1978, pp. 103–114. MR**566158****[11]**S. Marin,*A Finite Element Method for Problems Involving the Helmholtz Equation in Two Dimensional Exterior Regions*, Thesis, Carnegie-Mellon University, Pittsburgh, Pa., 1978.**[12]**A. K. Aziz and R. Bruce Kellogg,*Finite element analysis of a scattering problem*, Math. Comp.**37**(1981), no. 156, 261–272. MR**628694**, 10.1090/S0025-5718-1981-0628694-2**[13]**J. H. Bramble and J. E. Pasciak,*Iterative techniques for time dependent Stokes problems*, Comput. Math. Appl.**33**(1997), no. 1-2, 13–30. Approximation theory and applications. MR**1442058**, 10.1016/S0898-1221(96)00216-7**[14]**J.-L. Lions and E. Magenes,*Non-homogeneous boundary value problems and applications. Vol. I*, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR**0350177****[15]**D. S. Jones,*The eigenvalues of ∇²𝑢+𝜆𝑢=0 when the boundary conditions are given on semi-infinite domains*, Proc. Cambridge Philos. Soc.**49**(1953), 668–684. MR**0058086****[16]**James H. Bramble and Vidar Thomée,*Discrete time Galerkin methods for a parabolic boundary value problem*, Ann. Mat. Pura Appl. (4)**101**(1974), 115–152. MR**0388805****[17]**D. S. Jones,*The theory of electromagnetism*, International Series of Monographs on Pure and Applied Mathematics, Vol. 47. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR**0161555****[18]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174**

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1982-0669632-7

Article copyright:
© Copyright 1982
American Mathematical Society