Error estimates for the finite element approximation of linear elastic equations in an unbounded domain

Han, Houde; Bao, Weizhu

doi:10.1090/S0025-5718-00-01285-0

Error estimates for the finite element approximation of linear elastic equations in an unbounded domain
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by Houde Han and Weizhu Bao PDF

Math. Comp. 70 (2001), 1437-1459 Request permission

Abstract:

In this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain. The finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error estimates show how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. A numerical example for Navier equations outside a circle in the plane is presented. Numerical results demonstrate the performance of our error estimates.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Bjorn Engquist and Andrew Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977), no. 139, 629–651. MR 436612, DOI 10.1090/S0025-5718-1977-0436612-4
Kang Feng, Asymptotic radiation conditions for reduced wave equation, J. Comput. Math. 2 (1984), no. 2, 130–138. MR 901405
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443, DOI 10.1007/978-3-642-96379-7
Dan Givoli, Numerical methods for problems in infinite domains, Studies in Applied Mechanics, vol. 33, Elsevier Scientific Publishing Co., Amsterdam, 1992. MR 1199563
Dan Givoli and Joseph B. Keller, A finite element method for large domains, Comput. Methods Appl. Mech. Engrg. 76 (1989), no. 1, 41–66. MR 1025619, DOI 10.1016/0045-7825(89)90140-0
D. Givoli and J. B. Keller, Non-reflecting boundary conditions for elastic waves, Wave Motion, 12 (1990), 261-279.
Dan Givoli and Igor Patlashenko, Optimal local non-reflecting boundary conditions, Appl. Numer. Math. 27 (1998), no. 4, 367–384. Absorbing boundary conditions. MR 1644669, DOI 10.1016/S0168-9274(98)00020-8
Dan Givoli, Igor Patlashenko, and Joseph B. Keller, High-order boundary conditions and finite elements for infinite domains, Comput. Methods Appl. Mech. Engrg. 143 (1997), no. 1-2, 13–39. MR 1442387, DOI 10.1016/S0045-7825(96)01150-4
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, DOI 10.1090/S0025-5718-1982-0669632-7
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, DOI 10.1137/0517026
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, DOI 10.1090/S0025-5718-1987-0878684-5
Laurence Halpern and Michelle Schatzman, Artificial boundary conditions for incompressible viscous flows, SIAM J. Math. Anal. 20 (1989), no. 2, 308–353. MR 982662, DOI 10.1137/0520021
Houde Han and Weizhu Bao, An artificial boundary condition for two-dimensional incompressible viscous flows using the method of lines, Internat. J. Numer. Methods Fluids 22 (1996), no. 6, 483–493. MR 1381372, DOI 10.1002/(SICI)1097-0363(19960330)22:6<483::AID-FLD331>3.0.CO;2-5
Hou De Han and Wei Zhu Bao, An artificial boundary condition for the incompressible viscous flows in a no-slip channel, J. Comput. Math. 13 (1995), no. 1, 51–63. MR 1320178
H. Han and W. Bao, The artificial boundary conditions for incompressible materials on an unbounded domain, Numer. Math. 77 (1997), no. 3, 347–363. MR 1469676, DOI 10.1007/s002110050290
Hou De Han, Jin Fu Lu, and Wei Zhu Bao, A discrete artificial boundary condition for steady incompressible viscous flows in a no-slip channel using a fast iterative method, J. Comput. Phys. 114 (1994), no. 2, 201–208. MR 1294925, DOI 10.1006/jcph.1994.1160
Houde Han, Weizhu Bao, and Tao Wang, Numerical simulation for the problem of infinite elastic foundation, Comput. Methods Appl. Mech. Engrg. 147 (1997), no. 3-4, 369–385. MR 1460117, DOI 10.1016/S0045-7825(97)00024-8
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
Hou De Han and Xiao Nan Wu, The approximation of the exact boundary conditions at an artificial boundary for linear elastic equations and its applications, Math. Comp. 59 (1992), no. 199, 21–37. MR 1134732, DOI 10.1090/S0025-5718-1992-1134732-0
J.-L. Lions, Optimal control of systems governed by partial differential equations. , Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971. Translated from the French by S. K. Mitter. MR 0271512, DOI 10.1007/978-3-642-65024-6
S. G. Mikhlin, The problem of the minimum of a quadratic functional, Holden-Day Series in Mathematical Physics, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1965. Translated by A. Feinstein. MR 0171196
F. Nataf, An open boundary condition for the computation of the steady incompressible Navier-Stokes equations, J. Comput. Phys. 85 (1989), no. 1, 104–129. MR 1025412, DOI 10.1016/0021-9991(89)90202-7
A. Sidi, A family of matrix problems, SIAM Review (Problems and Solutions), 40 (1998), 718-723.