A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains
Author:
Charles I. Goldstein
Journal:
Math. Comp. 39 (1982), 309324
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.
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 [3]
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 C. I. Goldstein, "Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries. III," Trans. Amer. Math. Soc., v. 143, 1969, pp. 283301. MR 0609010 (58:29414)
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 J. C. Guillot & C. H. Wilcox, "Steady state wave propagation in simple and compound acoustic waveguides," Math. Z., v. 160, 1978, pp. 89102. MR 496589 (80c:35023)
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 A. Bayliss, M. Gunzberger & E. Turkel, Boundary Conditions for the Numerical Solution of Elliptic Equations in Exterior Regions, ICASE Report 801, 1979.
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 D. S. Jones, "The eigenvalues of when the boundary conditions are given on semiinfinite domains," Proc. Cambridge Philos. Soc., v. 49, 1953, pp. 668684. MR 0058086 (15:319c)
 [16]
 J. H. Bramble & V. Thomée, "Discrete time Galerkin methods for a parabolic boundary value problem," Ann. Mat. Pura Appl., v. 101, 1974, pp. 115152. MR 0388805 (52:9639)
 [17]
 D. S. Jones, The Theory of Electromagnetism, Pergamon Press, New York, 1964. MR 0161555 (28:4759)
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 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206696327
PII:
S 00255718(1982)06696327
Article copyright:
© Copyright 1982
American Mathematical Society
