The approximation of the exact boundary conditions at an artificial boundary for linear elastic equations and its applications
Authors:
Hou De Han and Xiao Nan Wu
Journal:
Math. Comp. 59 (1992), 2137
MSC:
Primary 35J25; Secondary 35A35, 65N30, 73C02, 73V05
MathSciNet review:
1134732
Fulltext PDF Free Access
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Abstract: The exterior boundary value problems of linear elastic equations are considered. A sequence of approximations to the exact boundary conditions at an artificial boundary is given. Then the original problem is reduced to a boundary value problem on a bounded domain. Furthermore, a finite element approximation of this problem and optimal error estimates are obtained. Finally, a numerical example shows the effectiveness of this method.
 [1]
Ivo
Babuška and A.
K. Aziz, Survey lectures on the mathematical foundations of the
finite element method, The mathematical foundations of the finite
element method with applications to partial differential equations (Proc.
Sympos., Univ. Maryland, Baltimore, Md., 1972), Academic Press, New York,
1972, pp. 1–359. With the collaboration of G. Fix and R. B.
Kellogg. MR
0421106 (54 #9111)
 [2]
Philippe
G. Ciarlet, The finite element method for elliptic problems,
NorthHolland Publishing Co., Amsterdam, 1978. Studies in Mathematics and
its Applications, Vol. 4. MR 0520174
(58 #25001)
 [3]
Charles
I. Goldstein, A finite element method for solving
Helmholtz type equations in waveguides and other unbounded
domains, Math. Comp. 39
(1982), no. 160, 309–324. MR 669632
(84e:65112), http://dx.doi.org/10.1090/S00255718198206696327
 [4]
Kang
Feng, Asymptotic radiation conditions for reduced wave
equation, J. Comput. Math. 2 (1984), no. 2,
130–138. MR
901405 (89f:65134)
 [5]
Thomas
Hagstrom and H.
B. Keller, Exact boundary conditions at an artificial boundary for
partial differential equations in cylinders, SIAM J. Math. Anal.
17 (1986), no. 2, 322–341. MR 826697
(87g:35022), http://dx.doi.org/10.1137/0517026
 [6]
T.
M. Hagstrom and H.
B. Keller, Asymptotic boundary conditions and
numerical methods for nonlinear elliptic problems on unbounded
domains, Math. Comp. 48
(1987), no. 178, 449–470. MR 878684
(88d:65173), http://dx.doi.org/10.1090/S00255718198708786845
 [7]
Hou
De Han and Xiao
Nan Wu, Approximation of infinite boundary condition and its
application to finite element methods, J. Comput. Math.
3 (1985), no. 2, 179–192. MR 854359
(87k:65134)
 [8]
A. Jepson and H. B. Keller, Boundary value problems on semiinfinite intervals. I. Linear problems, Numer. Math. (to appear).
 [9]
Herbert
B. Keller, Numerical solution of two point boundary value
problems, Society for Industrial and Applied Mathematics,
Philadelphia, Pa., 1976. Regional Conference Series in Applied Mathematics,
No. 24. MR
0433897 (55 #6868)
 [10]
M.
Lenoir and A.
Tounsi, The localized finite element method and its application to
the twodimensional seakeeping problem, SIAM J. Numer. Anal.
25 (1988), no. 4, 729–752. MR 954784
(89k:65138), http://dx.doi.org/10.1137/0725044
 [11]
Marianela
Lentini and Herbert
B. Keller, Boundary value problems on semiinfinite intervals and
their numerical solution, SIAM J. Numer. Anal. 17
(1980), no. 4, 577–604. MR 584732
(81j:65092), http://dx.doi.org/10.1137/0717049
 [12]
S.
G. Mikhlin, The problem of the minimum of a quadratic
functional, Translated by A. Feinstein. HoldenDay Series in
Mathematical Physics, HoldenDay Inc., San Francisco, Calif., 1965. MR 0171196
(30 #1427)
 [13]
N.
I. Muskhelishvili, Some basic problems of the mathematical theory
of elasticity. Fundamental equations, plane theory of elasticity, torsion
and bending, Translated from the Russian by J. R. M. Radok, P.
Noordhoff Ltd., Groningen, 1963. MR 0176648
(31 #920)
 [14]
J.L.
Lions, Optimal control of systems governed by partial differential
equations., Translated from the French by S. K. Mitter. Die
Grundlehren der mathematischen Wissenschaften, Band 170, SpringerVerlag,
New York, 1971. MR 0271512
(42 #6395)
 [1]
 I. Babuška and A. K. Aziz, Survey lecture on the mathematical foundations of the finite element method, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, ed.), Academic Press, New York, 1972, pp. 5359. MR 0421106 (54:9111)
 [2]
 P. G. Ciarlet, The finite element method for elliptic problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [3]
 C. I. Goldstein, A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains, Math. Comp. 39 (1982), 309324. MR 669632 (84e:65112)
 [4]
 K. Feng, Asymptotic radiation conditions for reduced wave equation, J. Comput. Math. 2 (1984), 130138. MR 901405 (89f:65134)
 [5]
 T. M. Hagstrom and H. B. Keller, Exact boundary conditions at an artificial boundary for partial differential equations in cylinders, SIAM J. Math. Anal. 17 (1986), 322341. MR 826697 (87g:35022)
 [6]
 , Asymptotic boundary conditions and numerical methods for nonlinear elliptic problems on unbounded domains, Math. Comp. 48 (1987), 449470. MR 878684 (88d:65173)
 [7]
 H. Han and X. Wu, Approximation of infinite boundary condition and its application to finite element methods, J. Comput. Math. 3 (1985), 179192. MR 854359 (87k:65134)
 [8]
 A. Jepson and H. B. Keller, Boundary value problems on semiinfinite intervals. I. Linear problems, Numer. Math. (to appear).
 [9]
 H. B. Keller, Numerical solution of two point boundary value problems, No. 24, CBMS/NSF Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia, PA, 1976. MR 0433897 (55:6868)
 [10]
 M. Lenoir and A. Tounsi, The localized finite element method and its application to the twodimensional seakeeping problem, SIAM J. Numer. Anal. 25 (1988), 729752. MR 954784 (89k:65138)
 [11]
 M. Lentini and H. B. Keller, Boundary value problems on semiinfinite intervals and their numerical solution, SIAM J. Numer. Anal. 17 (1980), 577604. MR 584732 (81j:65092)
 [12]
 S. G. Mikhlin, The problem of the minimum of a quadratic functional, HoldenDay, San Francisco, London, Amsterdam, 1965. MR 0171196 (30:1427)
 [13]
 N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity, Noordhoff, Groningen, 1963. MR 0176648 (31:920)
 [14]
 J. L. Lions, Optimal control of systems governed by partial differential equations, SpringerVerlag, Berlin and Heidelberg, 1971. MR 0271512 (42:6395)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199211347320
PII:
S 00255718(1992)11347320
Keywords:
Unbounded domains,
artificial boundaries,
approximate boundary conditions at an artificial boundary
Article copyright:
© Copyright 1992 American Mathematical Society
