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A stabilized Galerkin method for a third-order evolutionary problem


Authors: Graeme Fairweather, J. M. Sanz-Serna and I. Christie
Journal: Math. Comp. 55 (1990), 497-507
MSC: Primary 65M60; Secondary 65M15
DOI: https://doi.org/10.1090/S0025-5718-1990-1035932-9
MathSciNet review: 1035932
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Abstract | References | Similar Articles | Additional Information

Abstract: The periodic initial value problem for the partial differential equation $ {u_t} + {u_{xxx}} + \beta {({u^2})_x} + \frac{\gamma }{2}{({u^2})_{xx}} + \varepsilon {u_{xx}} - \delta {u_{tx}} = 0$, $ \varepsilon $, $ \delta > 0$, arises in fluidization models. The numerical integration of the problem is a difficult task in that many "reasonable" finite difference and finite element methods give rise to unstable discretizations. We show how to modify the standard Galerkin technique in order to stabilize it. Optimal-order error estimates are derived and the results of numerical experiments are presented. The stabilization technique suggested in the paper can be interpreted as rewriting the problem in Sobolev form and would also be useful for other equations involving terms of the form $ {u_t} - \delta {u_{tx}}$.


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  • [1] L. Abia, I. Christie, and J. M. Sanz-Serna, Stability of schemes for the numerical treatment of an equation modelling fluidized beds, RAIRO Numer. Anal. Math. Modell. 23 (1989), 125-138. MR 1001327 (90e:76001)
  • [2] D. N. Arnold, J. Douglas, Jr., and V. Thomée, Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable, Math. Comp. 36 (1981), 53-63. MR 595041 (82f:65108)
  • [3] I. Christie and G. Ganser, A numerical study of nonlinear waves arising in a one-dimensional model of a fluidized bed, J. Comput. Phys. 81 (1989), 300-318. MR 994350 (90b:76034)
  • [4] P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
  • [5] J. E. Dendy, Jr., Two methods of Galerkin type achieving optimum $ {L^2}$ rates of convergence for first order hyperbolics, SIAM J. Numer. Anal. 11 (1974), 637-653. MR 0353695 (50:6178)
  • [6] T. Dupont, G. Fairweather, and J. P. Johnson, Three-level Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 11 (1974), 392-410. MR 0403259 (53:7071)
  • [7] G. H. Ganser and D. A. Drew, Nonlinear periodic waves in a two-phase flow model, SIAM J. Appl. Math. 47 (1987), 726-736. MR 898830 (90a:76147)
  • [8] -, Nonlinear analysis of a uniform fluidized bed (submitted).
  • [9] M. T. Nakao, Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension, Numer. Math. 47 (1985), 139-157. MR 797883 (86j:65134)
  • [10] L. B. Wahlbin, A dissipative Galerkin method applied to some quasilinear hyperbolic equations, RAIRO Numer. Anal. R-2 (1974), 109-117. MR 0368447 (51:4688)
  • [11] -, A dissipative Galerkin method for the numerical solution of first order hyperbolic equations, Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, ed), Academic Press, New York, 1974, pp. 147-169. MR 0658322 (58:31929)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1990-1035932-9
Keywords: Stability, Galerkin methods, fluidized beds, Sobolev equations
Article copyright: © Copyright 1990 American Mathematical Society

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