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Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations


Authors: Ohannes Karakashian and Charalambos Makridakis
Journal: Math. Comp. 74 (2005), 85-102
MSC (2000): Primary 65M60, 65M12
DOI: https://doi.org/10.1090/S0025-5718-04-01654-0
Published electronically: April 20, 2004
MathSciNet review: 2085403
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Abstract: We consider space-time continuous Galerkin methods with mesh modification in time for semilinear second order hyperbolic equations. We show a priori estimates in the energy norm without mesh conditions. Under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates. Estimates of the jump in the Riesz projection in two successive time steps are also derived.


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  • [AM] A.K. Aziz and P. Monk, Continuous finite elements in space and time for the heat equation, Math. Comp. 52 (1989), 255-274. MR 90d:65189
  • [BO] I. Babuska and J. E. Osborn, Analysis of finite element methods for second order boundary value problems using mesh dependent norms, Numer. Math. 34 (1980), 41-62. MR 81g:65143
  • [BB] G. Baker and J. H. Bramble, Semidiscrete and fully discrete approximations for second order hyperbolic equations, RAIRO Anal. Numer. 13 (1979), 75-100. MR 80f:65115
  • [BL] L. Bales and I. Lasiecka, Continuous finite elements in space and time for the nonhomogeneous wave equation, Computers Math. Applic. 27 (1994), 91-102. MR 94k:65138
  • [BS] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994. MR 95f:65001
  • [Ci] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. MR 58:25001
  • [DR] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1975. MR 56:7119
  • [D] T. Dupont, Mesh modification for evolution equations, Math. Comp. 39 (1982), 85-107. MR 84g:65131
  • [E] K. Eriksson, An adaptive finite element method with efficient maximum norm error control for elliptic problems, Math. Models Meth. Appl. Sc. 4 (1994), 313-329. MR 95c:65180
  • [EJ] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems II: Optimal error estimates in $L_{\infty }L_{2}$ and $L_{\infty }L_{\infty }$, SIAM J. Numer. Anal. 32 (1995), 706-740. MR 96c:65162
  • [FP] D. A. French and T. E. Peterson, A continuous space-time finite element method for the wave equation, Math. Comp. 65 (1996), 491-506. MR 96g:65098
  • [F] D. A. French, A space-time finite element method for the wave equation, Comput. Methods Appl. Mech. Engrg. 107 (1993), 145-157. MR 95d:65082
  • [HH] T. J. R. Hughes and G. M. Hulbert, Space-time finite element methods for elastodynamics: Formulations and error estimates, Comput. Methods Appl. Mech. Engrg. 66 (1988), 339-363. MR 89c:73060
  • [J] C. Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 107 (1993), 117-129. MR 95c:65154
  • [KM1] O. Karakashian and Ch. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method, Math. Comp. 67 (1998), 479-499. MR 98i:65078
  • [KM2] O. Karakashian and Ch. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method, SIAM J. Numer. Anal. 36 (1999), 1779-1807. MR 2000h:65139
  • [M] Ch. G. Makridakis, Finite element approximations of nonlinear elastic waves, Math. Comp. 61 (1993), 569-594. MR 94g:73044
  • [SW] A. H. Schatz and L. B. Wahlbin, Interior maximum-norm estimates for finite element methods, Part II, Math. Comp. 64 (1995), 907-928. MR 95j:65143
  • [Thomée] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR 86k:65006
  • [Y] D. Q. Yang, Grid modification for second-order hyperbolic problems, Math. Comp. 64 (1995), 1495-1509. MR 95m:65173

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Additional Information

Ohannes Karakashian
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37966
Email: ohannes@math.utk.edu

Charalambos Makridakis
Affiliation: Department of Applied Mathematics, University of Crete, 714 09 Heraklion, Crete, Greece; IACM-FORTH, 711 10 Heraklion, Crete, Greece
Email: makr@tem.uoc.gr

DOI: https://doi.org/10.1090/S0025-5718-04-01654-0
Received by editor(s): November 20, 2001
Received by editor(s) in revised form: May 7, 2003
Published electronically: April 20, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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