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On the stability of a family of finite element methods for hyperbolic problems


Author: Gerard R. Richter
Journal: Math. Comp. 71 (2002), 527-535
MSC (2000): Primary 65M60, 65M12
DOI: https://doi.org/10.1090/S0025-5718-01-01334-5
Published electronically: May 22, 2001
MathSciNet review: 1885613
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Abstract:

We consider a family of tensor product finite element methods for hyperbolic equations in $R^{N}$, $N\ge 2$, which are explicit and generate a continuous approximate solution. The base case $N=2$ (an extension of the box scheme to higher order) is due to Winther, who proved stability and optimal order convergence. By means of a simple counterexample, we show that, for linear approximation with $N \ge 3$, the corresponding methods are unstable.


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  • 1. P. Balland and E. Süli, Analysis of the cell-vertex finite volume method for hyperbolic problems with variable coefficients, SIAM J. Numer. Anal. 34 (1997), 1127-1151. MR 98i:65089
  • 2. P. I. Crumpton and G. J. Shaw, Cell vertex finite volume discretizations in three dimensions, Internat. J. Numer. Meth. Fluids 14 (1992), 505-527. MR 92k:76060
  • 3. R. S. Falk and G. R. Richter, Analysis of a continuous finite element method for hyperbolic equations, SIAM J. Numer. Anal. 24 (1987), 257-278. MR 88d:65133
  • 4. R. S. Falk and G. R. Richter, Local error estimates for a finite element method for hyperbolic and convection-diffusion equations, SIAM J. Numer. Anal. 29 (1992), 730-754. MR 93e:65137
  • 5. R. S. Falk and G. R. Richter, Explicit finite element methods for symmetric hyperbolic systems, SIAM J. Numer. Anal. 36 (1999), 935-952. MR 2000b:65181
  • 6. C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46 (1986), 1-26. MR 88b:65109
  • 7. P. Lesaint and P. A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical Aspects of Finite Elements in Partial Differential Equations, (C. deBoor, ed.), Academic Press, 1974, pp. 89-123. MR 58:31918
  • 8. K. W. Morton, M. Stynes, and E. Süli, Analysis of a cell-vertex finite volume method for convection-diffusion problems, Math. Comp. 66 (1997), 1389-1406. MR 98a:65103
  • 9. W. H. Reed and T. R. Hill, Triangular mesh methods for solving the neutron transport equation, Los Alamos Scientific Laboratory Report LA-UR-73-479..
  • 10. B. Wendroff, On centered difference approximations for hyperbolic systems, J. Soc. Indust. Appl. Math. 8 (1960), 549-555. MR 22:7259
  • 11. R. Winther, A stable finite element method for initial-boundary value problems for first-order hyperbolic systems, Math. Comp. 36 (1981), 65-86. MR 81m:65181

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Additional Information

Gerard R. Richter
Affiliation: Department of Computer Science, Rutgers University, New Brunswick, New Jersey 08903
Email: richter@cs.rutgers.edu

DOI: https://doi.org/10.1090/S0025-5718-01-01334-5
Keywords: Finite elements, hyperbolic, explicit
Received by editor(s): December 8, 1999
Received by editor(s) in revised form: August 8, 2000
Published electronically: May 22, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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