Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Finite element approximations of nonlinear elastic waves


Author: Charalambos G. Makridakis
Journal: Math. Comp. 61 (1993), 569-594
MSC: Primary 73V05; Secondary 65M60, 73D15, 73D35
DOI: https://doi.org/10.1090/S0025-5718-1993-1195426-X
MathSciNet review: 1195426
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study finite element methods for a class of problems of nonlinear elastodynamics. We discretize the equations in space using Galerkin methods. For the temporal discretization, the construction of our schemes is based on rational approximations of $ \cos x$ and $ {e^x}$. We analyze semidiscrete as well as second- and fourth-order accurate in time fully discrete methods for the approximation of the solution of the problem and prove optimal-order $ {L^2}$ error estimates. For some schemes a Taylor-type technique is used so that only linear systems of equations need be solved at each time step. In the proofs we need various estimates for a nonlinear elliptic projection, the proofs of which are also established in the paper.


References [Enhancements On Off] (What's this?)

  • [1] G. A. Baker and J. H. Bramble, Semidiscrete and fully discrete approximations for second-order hyperbolic equations, RAIRO Anal. Numér. 13 (1979), 75-100. MR 533876 (80f:65115)
  • [2] G. A. Baker, V. A. Dougalis, and S. M. Serbin, High-order two-step approximations for hyperbolic equations, RAIRO Anal. Numér. 13 (1979), 201-226. MR 543933 (81c:65044)
  • [3] L. A. Bales, Semidiscrete and single step fully discrete approximations for second order hyperbolic equations with time-dependent coefficients, Math. Comp. 43 (1984), 383-414. MR 758190 (86g:65179a)
  • [4] -, High-order single-step fully discrete approximations for nonlinear second order hyperbolic equations, Comput. Math. Appl. 12A (1986), 581-604. MR 841989 (87h:65164)
  • [5] L. A. Bales, V. A. Dougalis, and S. M. Serbin, Cosine methods for second-order hyperbolic equations with time-dependent coefficients, Math. Comp. 45 (1985), 65-89. MR 790645 (86j:65112)
  • [6] L. A. Bales and V. A. Dougalis, Cosine methods for nonlinear second-order hyperbolic equations, Math. Comp. 52 (1989), 299-319, S15-S33. MR 955747 (89k:65113)
  • [7] J. H. Bramble and P. H. Sammon, Efficient high-order single-step methods for parabolic problems: Part II, unpublished manuscript.
  • [8] V. C. Chen and W. von Wahl, Das Rand-Anfangswertproblem für quasilineare Wellengleichungen in Sobolevräumen niedriger Ordnung, J. Reine Angew. Math. 337 (1982), 77-112. MR 676043 (84b:35081)
  • [9] P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
  • [10] C. M. Dafermos and W. J. Hrusa, Energy methods for quasilinear hyperbolic initial-boundary value problems. Applications to elastodynamics, Arch. Rational Mech. Anal. 87 (1985), 267-292. MR 768069 (86k:35086)
  • [11] J. E. Dendy, Jr., Galerkin's method for some highly nonlinear problems, SIAM J. Numer. Anal. 14 (1977), 327-347. MR 0433914 (55:6884)
  • [12] M. Dobrowolski and R. Rannacher, Finite element methods for nonlinear elliptic systems of second order, Math. Nachr. 94 (1980), 155-172. MR 582526 (81i:65087)
  • [13] T. Dupont, $ {L^2}$-estimates for Galerkin methods for second-order hyperbolic equations, SIAM J. Numer. Anal. 10 (1973), 880-889. MR 0349045 (50:1539)
  • [14] G. Fichera, Existence theorems in elasticity, Encyclopedia of Physics, vol. VI a/2 (C. Truesdell, ed.), Springer-Verlag, Berlin, Heidelberg, New York, 1972, pp. 347-389.
  • [15] M. E. Gurtin, An introduction to continuum mechanics, Academic Press, New York, 1981. MR 636255 (84c:73001)
  • [16] Ch. G. Makridakis, Galerkin/finite element methods for the equations of elastodynamics, Ph.D. thesis, Univ. of Crete, 1989. (Greek)
  • [17] -, Cosine methods for a class of semilinear second-order wave equations, Comput. Math. Appl. 19 (1990), 19-34. MR 1044632 (91d:65141)
  • [18] -, Finite element approximations of nonlinear elastic waves, Technical Report, Dept. of Mathematics, Univ. of Crete, 1992.
  • [19] R. Rannacher, On finite element approximation of general boundary value problems in nonlinear elasticity, Calcolo 17 (1980), 175-193. MR 615816 (82f:73019)
  • [20] -, Private communication, 1988.
  • [21] R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), 437-470. MR 645661 (83e:65180)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 73V05, 65M60, 73D15, 73D35

Retrieve articles in all journals with MSC: 73V05, 65M60, 73D15, 73D35


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1993-1195426-X
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society