Finite element methods for the simulation of wave propagation in two- dimensional inhomogeneous elastic media. (English) Zbl 0592.73039

The author defines certain continuous and discrete-time standard Galerkin procedures and obtains optimal order convergence results in \(L^ 2(\Omega)\) and \(H^ 1(\Omega)\). Efficient time-stepping procedures which are analogous to the so-called Laplace-modified and alternating direction methods are also provided. These Galerkin procedures are applicable to approximate solutions for the wave equation in two-dimensional, inhomogeneous elastic media.
Reviewer: M.A.Ibiejugba


74J99 Waves in solid mechanics
74E05 Inhomogeneity in solid mechanics
74S99 Numerical and other methods in solid mechanics
Full Text: DOI


[1] R. A. Adams,Sobolev Spaces, Academic Press, New York, (1975).
[2] O. Axelsson,On preconditioning and convergence acceleration in sparse matrix problems, CERN European Organization for Nuclear Research, Geneva (1974). · Zbl 0354.65020
[3] O. Axelsson,On the computational complexity of some matrix iterative algorithms, Report 74.06, Department of Computer Science, Chalmers University of Technology, Göteborg, (1974). · Zbl 0354.65020
[4] J. E. Dendy, Jr.,An analysis of some Galerkin schemes for the solution of nonlinear time-dependent problems, SIAM J. Numer. Anal.12 (1975), 541–565. · Zbl 0338.65052 · doi:10.1137/0712042
[5] J. Douglas, Jr.–T. Dupont,Alternating-direction methods on rectangles, Numerical Solution of Partial Differential Equations II, B. Hubbard ed., Academic Press, New York, (1971), 133–214.
[6] J. Douglas, Jr.–T. Dupont,Preconditioned conjugate gradient iteration applied to Galerkin methods for a mildly nonlinear Dirichlet problem, Sparse Matrix Computations, J. R. Bunch and D. J. Rose eds., Academic Press, New York, (1976), 333–348. · Zbl 0346.65020
[7] J. Douglas, Jr.,Effective time-stepping methods for the numerical solution of nonlinear parabolic problems, The Mathematics of Finite Elements and Applications III, J. R. Whiteman ed., Academic Press, New York, (1979), 289–304.
[8] J. Douglas, Jr.–T. Dupont–R. E. Ewing,Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem, SIAM J. Numer. Anal.,16 (1979), 503–522. · Zbl 0411.65064 · doi:10.1137/0716039
[9] T. Dupont,L 2-estimates for Galerkin methods for second order hyperbolic equations, SIAM J. Numer. Anal.,10 (1973), 880–889. · doi:10.1137/0710073
[10] G. Duvaut–J. L. Lions,Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976. · Zbl 0331.35002
[11] G. Fichera,Existence theorems in Elasticity-Boundary value problems of elasticity with unilateral constraints, Encyclopedia of Physics, S. Flüge, ed., vol. VI a/2: Mechanics of Solids II, C. Truesdell, ed., Springer-Verlag, Berlin, (1972), 347–424.
[12] J. Lysmer–R. L. Kuhlemeyer,Finite dynamic model for infinite media, Eng. Mech. Division, Proc. Amer. Soc. Civil Eng.,95 (1969), 859–877.
[13] J. A. Nitsche,Private communication.
[14] J. A. Nitsche,Finite element approximation for solving the elastic problem, 2nd Int. Symp. on Computing Methods in Applied Sciences and Engineering, Dec. 15–19, 1975, Versailles, France, Comp. Meth. in Appl. Sc. and Eng.,134 (1976), 156–167.
[15] J. A. Nitsche,On Korn’s second inequality, preprint, Institute für Angenwandte Mathematik, Albert Ludwig Universität, Herman-Herder Str. 10, 7800, Freiburg i. Br., West Germany.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.