×

Optimal error estimates for semidiscrete phase relaxation models. (English) Zbl 0874.65069

The authors examine the combined effect of phase relaxation with small parameter \(\varepsilon\) and time discretization for a heat transfer equation with temperature-dependent convection and reaction. Three methods are considered: semi-implicit, semi-explicit and extrapolation. New strong stability estimates for all three schemes are derived which play important roles in the error analysis. It is proved that in special norm the error is \(O(\tau)\), \(O(\frac{\tau}{\sqrt\varepsilon})\), \(O(\tau)\) respectively, where \(\tau\) is the time-step.
Reviewer: V.Makarov (Kiev)

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] T. ARGOGAST, M. F. WHEELER and N. Y. ZHANG, A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media, SIAM J. Numer. Anal, (to appear). Zbl0856.76033 MR1403565 · Zbl 0856.76033 · doi:10.1137/S0036142994266728
[2] C. BAIOCCHI, 1989, Discretization of evolution inequalities, in Partial Differential Equations and the Calculus of Variations, F. Colombini, A. Marino, M. Modica and S. Spagnolo eds, Birkäuser, Boston, pp. 59-92. Zbl0677.65068 MR1034002 · Zbl 0677.65068
[3] J. W. BARRETT and P. KNABNER, Finite element approximation of transport of reactive solutes in porous media. Part 2 : Error estimates for equilibrium adsorption processes (to appear). Zbl0904.76039 · Zbl 0904.76039 · doi:10.1137/S0036142993258191
[4] H. BREZIS, 1971, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis, E. Zarantonello ed., Academic Press, New York, pp. 101-156. Zbl0278.47033 MR394323 · Zbl 0278.47033
[5] M. G. CRANDALL and T. M. LIGGETT, 1971, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93,pp. 265-298. Zbl0226.47038 MR287357 · Zbl 0226.47038 · doi:10.2307/2373376
[6] X. JIANG and R. H. NOCHETTO, A P1-P1 finite element method for a phase relaxation model. Part I : Quasi-uniform mesh (to appear). Zbl0972.65067 MR1619875 · Zbl 0972.65067 · doi:10.1137/S0036142996297783
[7] [7] E. MAGENES, R. H. NOCHETTO and C. VERDI, 1987, Energy error estimates for a linear scheme to approximate nonlinear parabolic problems, RAIRO Model. Math. Anal. Numer., 21, pp, 655-678. Zbl0635.65123 MR921832 · Zbl 0635.65123
[8] R. H. NOCHETTO, 1987, Error estimates for multidimensional singular parabolic problems, Japan J. Appl Math., 4, pp.111-138. Zbl0657.65132 MR899207 · Zbl 0657.65132 · doi:10.1007/BF03167758
[9] R. H. NOCHETTO, 1991, Finite element methods for parabolic free boundary problems, in Advances in Numerical Analysis, Vol I: Nonlinear Partial Differential Equations and Dynamical Systems, W. Light ed., 1990 Lancaster Summer School Proceedings, Oxford University Press, pp. 34-88. Zbl0733.65089 MR1138471 · Zbl 0733.65089
[10] R. H. NOCHETTO, M. PAOLINI and C. VERDI, 1994, Continuous and semidiscrete travelling waves for a phase relaxation model, European J. Appl. Math,, 5, pp. 177-199. Zbl0812.35166 MR1285038 · Zbl 0812.35166 · doi:10.1017/S095679250000139X
[11] R. H. NOCHETTO and C. VERDI, 1988, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal., 25, pp. 784-814. Zbl0655.65131 MR954786 · Zbl 0655.65131 · doi:10.1137/0725046
[12] J. RULLA, 1996, Error analysis for implicit approximations to solutions to Cauchy problems, SIAM J. Numer. Anal., 33, pp. 68-87. Zbl0855.65102 MR1377244 · Zbl 0855.65102 · doi:10.1137/0733005
[13] G. SAVARE, Weak solutions and maximal regularity for abstract evolution inequalities (to appear). Zbl0858.35073 MR1411975 · Zbl 0858.35073
[14] C. VERDI, 1994, Numerical aspects of parabolic free boundary and hysteresis problems, in Phase Transitions and Hysteresis Phenomena, A. Vismtin (ed.). Springer-Verlag, Berlin, pp. 213-284. Zbl0819.35155 MR1321834 · Zbl 0819.35155
[15] C. VERDI and A. VISINTIN, 1987, Numerical analysis of the multidimensional Stefan problem with supercooling and superheating, Boll Unione Mat. Ital. I-B, 7, pp. 795-814. Zbl0629.65130 MR916294 · Zbl 0629.65130
[16] [16] C. VERDI and A. VISINTIN, 1988, Error estimates for a semi-explicit numerical scheme for Stefan-type problems, Numer. Math. 52, pp. 165-185. Zbl0617.65125 MR923709 · Zbl 0617.65125 · doi:10.1007/BF01398688
[17] A. VISINTIN, 1985, Stefan problem with phase relaxation, IMA J. Appl. Math., 34,p. 225-245. Zbl0585.35053 MR804824 · Zbl 0585.35053 · doi:10.1093/imamat/34.3.225
[18] A. VISINTIN, 1985, Supercooling and superheating effects in phase transitions,IMA J. Appl. Math., 35, pp 233-256. Zbl0615.35090 MR839201 · Zbl 0615.35090 · doi:10.1093/imamat/35.2.223
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.