Error estimates for two-phase Stefan problems in several space variables. II: Nonlinear flux conditions. (English) Zbl 0606.65085

[For part I see ibid. 22, 457-499 (1985; reviewed above)]
We derive \(L^ 2\)-error-estimates for multidimensional two-phase Stefan problems involving further nonlinearities such as nonlinear flux conditions. We approximate the enthalpy formulation by a regularization procedure combined with a \(C^ 0\)-piecewise linear finite element scheme in space, and the implicit Euler scheme in time. Under some restrictions on the initial datum and on the finite element mesh, we obtain an \(L^ 2\)-rate of convergence of order \(\epsilon^{1/2}\) for regularized problems, and an \(L^ 2\)-rate of convergence of order (h\(| \log h|^ k+\tau +\epsilon)^{1/2}\) for the discrete problems \((k=0,1)\). These estimates lead to the choice \(h\sim \epsilon \sim \tau\) and yield a global \(L^ 2\)-rate essentially of order \(h^{1/2}.\)
The a priori relationship between the approximation parameters allows the discrete problem to satisfy a maximum principle (without a lumping procedure), from which a priori bounds and monotonicity properties of the discrete solutions are shown.


65Z05 Applications to the sciences
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35K05 Heat equation
35R35 Free boundary problems for PDEs
80A17 Thermodynamics of continua


Zbl 0606.65084
Full Text: DOI


[1] J. Bramble, J. Nitsche, A. Schatz,Maximum-norm interior estimates for Ritz-Galerkin methods, Math. Comp. (29),131 (1975), 677–688. · Zbl 0316.65023
[2] L. Caffarelli, L. Evans,Continuity of the temperature in the two-phase Stefan problems, Arch. Rational Mech. Anal., (81),3 (1983), 199–220. · Zbl 0516.35080
[3] J. R. Cannon, E. Di Benedetto,An N-dimensional Stefan problem with non-linear boundary conditions, S.I.A.M.J. Math. Anal. 11 (1980), 632–645. · Zbl 0459.35090
[4] L. Cermak, M. Zlamal,Transformation of dependent variables and the finite element solution of non-linear evolution equations, Internat. J. Numer. Methods Engrg. 15 (1980), 31–40. · Zbl 0444.65078
[5] P. Ciarlet,The finite element method for elliptic problems, (1978) Holland. · Zbl 0383.65058
[6] P. Ciarlet, P. Raviart,Maximum principle and uniform convergence for the finite element method, Comput. Math. Appl. Mech. Engrg. 2 (1973), 17–31. · Zbl 0251.65069
[7] A. Damlamian,Some results in the multiphase Stefan problems, Comm. Partial Differential Equations, (2),10 (1977), 1017–1044. · Zbl 0399.35054
[8] J. Douglas Jr., T. Dupont,Galerkin methods for parabolic equations with non linear boundary conditions, Numer. Math., 20 (1973), 213–237. · Zbl 0234.65096
[9] E. Di Benedetto,Continuity of weak-solutions to certain singular parabolic equations, Ann. Mat. Pura Appl. IV, 130 (1982), 131–176. · Zbl 0503.35018
[10] E. Di Benedetto,A boundary modulus of continuity for a class of singular parabolic equations, (to appear).
[11] A. Fasano, M. Primicerio,Il problema di Stefan con condizioni al contorno non lineari, Ann. Scuola Norm. Sup. Pisa, 26 (1972), 711–737. · Zbl 0245.35048
[12] A. Friedman,The Stefan problem in several space variables, Trans. Amer. Math. Soc., 133 (1968), 51–87. · Zbl 0162.41903
[13] J. Jerome,Approximation of nonlinear evolutions systems, (1983) Academic Press. · Zbl 0512.35001
[14] J. Jerome, M. Rose,Error estimates for the multidimensional two-phase Stefan problem, Math. Comp., (39),160 (1982), 377–414. · Zbl 0505.65060
[15] S. Kamenomostskaya,On the Stefan problems, Mat. Sb., 53 (1961), 489–514. · Zbl 0102.09301
[16] O. Ladyzenskaya, V. Solonnikov, N. Ural’ceva,Linear and Quasilinear Equations of Parabolic Type, Trans. Math. Monographs Amer. Math. Soc. (1968).
[17] M. Luskin,A Galerkin method for non-linear parabolic equations with non-linear boundary conditions, SIAM J. Numer. Anal., 16 (1979), 284–299. · Zbl 0405.65059
[18] E. Magenes,Problemi di Stefan bifase in più variabili spaziali, V S.A.F.A., Catania, Le Matematiche, XXXVI, (1981), 65–108.
[19] E. Magenes, C. Verdi, A. Visintin,Semigroup approach to the Stefan problem with non-linear flux, Rend. Accad. Naz. LXXV, (1983), 24–33. · Zbl 0562.35089
[20] M. Niezgodka, I. Pawlow,A generalized Stefan problem in several space variables, Appl. Math. Optim., 9 (1983), 193–224. · Zbl 0519.35079
[21] M. Niezgodka, I. Pawlow, A. Visintin,Remarks on the paper by A. Visintin “Sur le Problème de Stefan avec flux non lineaire{”, Boll. Un. Mat. Ital. C(5), 18, N. 1 (1981), 87–88.} · Zbl 0478.35084
[22] R. Nochetto,Análisis numérico del problema de Stefan multidimensional a dos fases por el método de Regularización, Tesis en Ciencias Matemáticas, Univ. of Buenos Aires (1983).
[23] R. Nochetto,Error estimates for two-phase Stefan problems in several space variables, I: Linear boundary conditions, Calcolo, 4 (1985), 457–499. · Zbl 0606.65084
[24] R. Nochetto A class of non-degenerate two-phase Stefan problems in several space variables, Preprint n. 442, Istituto di Analisi Numerica del C.N.R., Pavia (1985). · Zbl 0606.65084
[25] J. Ortega, W. Rheinboldt,Iterative solution of nonlinear equations in several variables, (1970) Academic Press. · Zbl 0241.65046
[26] A. Schatz, L. Wahlbin,Interior maximum norm estimates for finite element methods, Math. Comp., 31 (1977), 414–442. · Zbl 0364.65083
[27] R. Scott,Optimal L estimates for the finite element method on irregular meshes, Math. Comp., 30 (1976), 681–697. · Zbl 0349.65060
[28] C. Verdi,On the numerical approach to a two-phase Stefan problem with non-linear flux, to appear in Calcolo. · Zbl 0612.65084
[29] A. Visintin,Sur le problème de Stefan avec flux non lineaire, Boll. Un. Mat. Ital. C(5), 18, N. 1 (1981), 63–86. · Zbl 0471.35078
[30] R. White,An enthalpy formulation of the Stefan problem, SIAM J. Numer. Anal., 19 (1982), 1129–1157. · Zbl 0501.65058
[31] R. White,A numerical solution of the enthalpy formulation of the Stefan problem, SIAM J. Numer. Anal., 19 (1982), 1158–1172. · Zbl 0501.65059
[32] M. Zlamal,A finite element solution of the nonlinear heat equation, RAIRO Anal. Numér., (14),2 (1980), 203–216.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.