Energy error estimates for a linear scheme to approximate nonlinear parabolic problems. (English) Zbl 0635.65123

The authors estimate the rate of convergence of a semidiscretization scheme used in the nonlinear parabolic problem \(u_ t+Ab(u)=f(b(u))\) where A is a linear elliptic operator and b(s) \((s\in R^ 1)\) is a Lipschitz continuous nondecreasing function. The used approximation scheme is based on Chernoff’s formula, studied in the theory of nonlinear semigroups of contractions. On each time step it leads to the solution of a linear elliptic problem. The obtained energy type error estimates are discussed for both degenerate and non-degenerate equations. The obtained results can be applied to the Stefan problem and to the porous medium equations. A variational technique is used.
Reviewer: J.Kačur


65N40 Method of lines for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
80A17 Thermodynamics of continua
76S05 Flows in porous media; filtration; seepage
Full Text: DOI EuDML


[1] Ph. BENILAN, Solutions intégrales d’équations d’évolution dans un espace de Banach, C. R. Acad. Sci. Paris, A-274 (1972), 47-50. Zbl0246.47068 MR300164 · Zbl 0246.47068
[2] [2] A. E. BERGER, H BREZIS & J. C. W. ROGERS, A numerical method for solving the problem u t - \Delta f ( u ) = 0 R.A.I.R.O. Anal. Numér., 13 (1979), 297-312. Zbl0426.65052 MR555381 · Zbl 0426.65052
[3] A. BOSSAVIT, A. DAMLAMIAN & M. FREMOND Eds., Free boundary problems: applications and theory, vol. III, Research Notes in Math. 120, Pitman, Boston (1985). Zbl0578.35003 MR863154 · Zbl 0578.35003
[4] H. BREZIS, On some degenerate non-linear parabolic equations, in Non-linear functional analysis (F. E. Browder Ed.), A.M.S. XVIII 1 (1970), 28-38. Zbl0231.47034 MR273468 · Zbl 0231.47034
[5] H. BREZIS & A. PAZY, Convergence and approximation of semigroups of non-linear operators in Banach spaces, J. Funct. Anal., 9 (1972), 63-74. Zbl0231.47036 MR293452 · Zbl 0231.47036 · doi:10.1016/0022-1236(72)90014-6
[6] J. F. CIAVALDINI, Analyse numérique d’un problème de Stefan à deux phases par une méthode d’éléments finis, SIAM J. Numer. Anal., 12 (1975), 464-487. Zbl0272.65101 MR391741 · Zbl 0272.65101 · doi:10.1137/0712037
[7] M. G. CRANDALL& T. M. LIGGETT, Generation of semi-groups of non-linear transformations on general Banach spaces, Amer J. Math., 93 (1971), 265-298. Zbl0226.47038 MR287357 · Zbl 0226.47038 · doi:10.2307/2373376
[8] J. DOUGLAS Jr.& T. DUPONT, Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 7 (1970), 575-626. Zbl0224.35048 MR277126 · Zbl 0224.35048 · doi:10.1137/0707048
[9] J. DOUGLAS Jr. & T. DUPONT, Alternating-direction Galerkin methods on rectangles, in Numerical solutions of partial differential equations, vol. II (B. Hubbard Ed.), Academic Press, New York (1971), 133-214. Zbl0239.65088 MR273830 · Zbl 0239.65088
[10] J. DOUGLAS Jr., T. DUPONT & R. E. EWING, Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem, SIAM J. Numer. Anal., 16 (1979), 503-522. Zbl0411.65064 MR530483 · Zbl 0411.65064 · doi:10.1137/0716039
[11] C. M. ELLIOTT, Error analysis of the enthalpy method for the Stefan problem, IMA J. Numer. Anal., 7 (1987), 61-71. Zbl0638.65088 MR967835 · Zbl 0638.65088 · doi:10.1093/imanum/7.1.61
[12] R. E. EWING, Efficient multistep procedures for nonlinear parabolic problems with nonlinear Neumann boundary conditions, Calcolo, 19 (1982), 231-252. Zbl0522.65072 MR695388 · Zbl 0522.65072 · doi:10.1007/BF02575804
[13] J. W. JEROME, Approximation of nonlinear evolution systems, Academic Press, New York (1983). Zbl0512.35001 MR690582 · Zbl 0512.35001
[14] J. W. JEROME & M. E. ROSE, Error estimates for the multidimensional two-phase Stefan problem, Math. Comp., 39 (1982), 377-414. Zbl0505.65060 MR669635 · Zbl 0505.65060 · doi:10.2307/2007320
[15] J. L. LIONS & E. MAGENES, Non-homogeneous boundary value problems and applications, vol. I, Springer-Verlag, Berlin (1972). Zbl0223.35039 MR350177 · Zbl 0223.35039
[16] M. LUSKIN, A Galerkin method for nonlinear parabolic equations with nonlinear boundary conditions, SIAM J. Numer. AnaL, 16 (1979), 284-299. Zbl0405.65059 MR526490 · Zbl 0405.65059 · doi:10.1137/0716021
[17] E. MAGENES, Problemi di Stefan bifase in piu variabili spaziali, V S.A.F.A., Catania, Le Matematiche, XXXVI (1981), 65-108. Zbl0545.35096 · Zbl 0545.35096
[18] E. MAGENES & C. VERDI, On the semigroup approach to the two-phase Stefan problem with nonlinear flux conditions, in [3], 28-39. Zbl0593.35092 MR863159 · Zbl 0593.35092
[19] G. H. MEYER, Multidimensional Stefan problems, SIAM J. Numer. Anal., 10 (1973), 522-538. Zbl0256.65054 MR331807 · Zbl 0256.65054 · doi:10.1137/0710047
[20] R. H. NOCHETTO, Error estimates for two-phase Stefan problems in several space variables, I: linear boundary conditions, Calcolo, 22 (1985), 457-499. Zbl0606.65084 MR859087 · Zbl 0606.65084 · doi:10.1007/BF02575898
[21] R. H. NOCHETTO, Error estimates for two-phase Stefan problems in several space variables, II: nonlinear flux conditions, Calcolo, 22 (1985), 501-534. Zbl0606.65085 MR859088 · Zbl 0606.65085 · doi:10.1007/BF02575899
[22] R. H. NOCHETTO, Error estimates for multidimensional Stefan problems with general boundary conditions, in [3], 50-60. Zbl0593.35094 MR863161 · Zbl 0593.35094
[23] R. H. NOCHETTO, Error estimates for multidimensional singular parabolic problems, Japan J. Appl. Math., 4 (1987), 111-138. Zbl0657.65132 MR899207 · Zbl 0657.65132 · doi:10.1007/BF03167758
[24] R. H. NOCHETTO& C. VERDI, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal., to appear. Zbl0655.65131 MR954786 · Zbl 0655.65131 · doi:10.1137/0725046
[25] J. C. W. ROGERS, A. E. BERGER & M. CIMENT, The alternating phase truncation method for numerical solution of a Stefan problem, SIAM J. Numer. Anal., 16 (1979), 563-587. Zbl0418.65051 MR537272 · Zbl 0418.65051 · doi:10.1137/0716043
[26] M. E. ROSE, Numerical methods for flows through porous media I, Math. Comp., 40 (1983), 435-467. Zbl0518.76078 MR689465 · Zbl 0518.76078 · doi:10.2307/2007525
[27] V. THOMEE, Galerkin finite element methods for parabolic problems, Lecture Notes in Math. 1054, Springer-Verlag, Berlin (1984). Zbl0528.65052 MR744045 · Zbl 0528.65052 · doi:10.1007/BFb0071790
[28] C. VERDI, On the numerical approach to a two-phase Stefan problem with nonlinear flux, Calcolo, 22 (1985), 351-381. Zbl0612.65084 MR860658 · Zbl 0612.65084 · doi:10.1007/BF02600382
[29] [29] C. VERDI & A. VISINTIN, Error estimates for a semi-explicit numerical scheme for Stefan-type problems, submitted to Numer. Math. Zbl0617.65125 MR923709 · Zbl 0617.65125 · doi:10.1007/BF01398688
[30] A. VISINTIN, Stefan problem with phase relaxation, IMA J. Appl. Math., 34 (1985), 225-245. Zbl0585.35053 MR804824 · Zbl 0585.35053 · doi:10.1093/imamat/34.3.225
[31] M. F. WHEELER, A priori L 2 -error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 10 (1973), 723-759. Zbl0232.35060 MR351124 · Zbl 0232.35060 · doi:10.1137/0710062
[32] R. E. WHITE, An enthalpy formulation of the Stefan problem, SIAM J. Numer. Anal., 19 (1982), 1129-1157. Zbl0501.65058 MR679656 · Zbl 0501.65058 · doi:10.1137/0719082
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