Error estimates for the multidimensional two-phase Stefan problem

Authors:
Joseph W. Jerome and Michael E. Rose

Journal:
Math. Comp. **39** (1982), 377-414

MSC:
Primary 65M60; Secondary 65M05, 65M10

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669635-2

MathSciNet review:
669635

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we derive rates of convergence for regularizations of the multidimensional two-phase Stefan problem and use the regularized problems to define backward-difference in time and ${C^0}$ piecewise-linear in space Galerkin approximations. We find an ${L^2}$ rate of convergence of order $\sqrt \varepsilon$ in the $\varepsilon$-regularization and an ${L^2}$ rate of convergence of order $({h^2}/\varepsilon + \Delta t/\sqrt \varepsilon )$ in the Galerkin estimates which leads to the natural choices $\varepsilon \sim {h^{4/3}}$, $\Delta t \sim {h^{4/3}}$, and a resulting $O({h^{2/3}})\;{L^2}$ rate of convergence of the numerical scheme to the solution of the differential equation. An essentially $O(h)$ rate is demonstrated when $\varepsilon = 0$ and $\Delta t \sim {h^2}$ in our Galerkin scheme under a boundedness hypothesis on the Galerkin approximations. The latter result is consistent with computational experience.

- S. Agmon, A. Douglis, and L. Nirenberg,
*Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I*, Comm. Pure Appl. Math.**12**(1959), 623–727. MR**125307**, DOI https://doi.org/10.1002/cpa.3160120405
O. B. Andersland & D. M. Anderson (eds.), - J. H. Bramble, A. H. Schatz, V. Thomée, and L. B. Wahlbin,
*Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations*, SIAM J. Numer. Anal.**14**(1977), no. 2, 218–241. MR**448926**, DOI https://doi.org/10.1137/0714015 - H. Brézis,
*Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert*, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR**0348562** - Haïm Brézis and Walter A. Strauss,
*Semi-linear second-order elliptic equations in $L^{1}$*, J. Math. Soc. Japan**25**(1973), 565–590. MR**336050**, DOI https://doi.org/10.2969/jmsj/02540565 - B. M. Budak, E. N. Solov′eva, and A. B. Uspenskiĭ,
*A difference method with smoothing of coefficients for the solution of the Stefan problem*, Ž. Vyčisl. Mat i Mat. Fiz.**5**(1965), no. 5, 828–840 (Russian). MR**199969** - L. A. Caffarelli and L. C. Evans,
*Continuity of the temperature in the two-phase Stefan problem*, Arch. Rational Mech. Anal.**81**(1983), no. 3, 199–220. MR**683353**, DOI https://doi.org/10.1007/BF00250800 - Philippe G. Ciarlet,
*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174** - J. F. Ciavaldini,
*Analyse numerique d’un problème de Stefan à deux phases par une methode d’éléments finis*, SIAM J. Numer. Anal.**12**(1975), 464–487 (French, with English summary). MR**391741**, DOI https://doi.org/10.1137/0712037 - Alain Damlamian,
*Some results on the multi-phase Stefan problem*, Comm. Partial Differential Equations**2**(1977), no. 10, 1017–1044. MR**487015**, DOI https://doi.org/10.1080/03605307708820053
E. Di Benedetto, - Jim Douglas Jr., Todd Dupont, and Lars Wahlbin,
*Optimal $L_{\infty }$ error estimates for Galerkin approximations to solutions of two-point boundary value problems*, Math. Comp.**29**(1975), 475–483. MR**371077**, DOI https://doi.org/10.1090/S0025-5718-1975-0371077-0
N. Dunford & J. Schwartz, - Einar Hille and Ralph S. Phillips,
*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR**0089373** - Joseph W. Jerome,
*Nonlinear equations of evolution and a generalized Stefan problem*, J. Differential Equations**26**(1977), no. 2, 240–261. MR**481543**, DOI https://doi.org/10.1016/0022-0396%2877%2990193-0 - Joseph W. Jerome,
*Existence and approximation of weak solutions of nonlinear Dirichlet problems with discontinuous coefficients*, SIAM J. Math. Anal.**9**(1978), no. 4, 730–742. MR**498348**, DOI https://doi.org/10.1137/0509052 - Claes Johnson and Vidar Thomée,
*Error estimates for some mixed finite element methods for parabolic type problems*, RAIRO Anal. Numér.**15**(1981), no. 1, 41–78 (English, with French summary). MR**610597**
S. Kamenomostskaja, "On the Stefan problem," - Tosio Kato,
*Linear evolution equations of “hyperbolic” type. II*, J. Math. Soc. Japan**25**(1973), 648–666. MR**326483**, DOI https://doi.org/10.2969/jmsj/02540648
S. N. Kruzhkov, "First order quasilinear equations in several independent variables," - J.-L. Lions,
*Quelques méthodes de résolution des problèmes aux limites non linéaires*, Dunod; Gauthier-Villars, Paris, 1969 (French). MR**0259693** - Gunter H. Meyer,
*Multidimensional Stefan problems*, SIAM J. Numer. Anal.**10**(1973), 522–538. MR**331807**, DOI https://doi.org/10.1137/0710047 - J. A. Nitsche,
*$L_{\infty }$-convergence of finite element approximation*, Journées “Éléments Finis” (Rennes, 1975) Univ. Rennes, Rennes, 1975, pp. 18. MR**568857** - Rolf Rannacher,
*Zur $L^{\infty }$-Konvergenz linearer finiter Elemente beim Dirichlet-Problem*, Math. Z.**149**(1976), no. 1, 69–77 (German). MR**488859**, DOI https://doi.org/10.1007/BF01301633 - Michael E. Rose,
*Numerical methods for flows through porous media. I*, Math. Comp.**40**(1983), no. 162, 435–467. MR**689465**, DOI https://doi.org/10.1090/S0025-5718-1983-0689465-6 - Milton E. Rose,
*A method for calculating solutions of parabolic equations with a free boundary*, Math. Comput.**14**(1960), 249–256. MR**0115283**, DOI https://doi.org/10.1090/S0025-5718-1960-0115283-8 - L. I. Rubenšteĭn,
*The Stefan problem*, American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by A. D. Solomon; Translations of Mathematical Monographs, Vol. 27. MR**0351348** - A. A. Samarskiĭ and B. D. Moiseenko,
*An efficient scheme for the through computation in a many dimensional Stefan problem*, Ž. Vyčisl. Mat i Mat. Fiz.**5**(1965), 816–827 (Russian). MR**203960** - A. H. Schatz and L. B. Wahlbin,
*On the quasi-optimality in $L_{\infty }$ of the $\dot H^{1}$-projection into finite element spaces*, Math. Comp.**38**(1982), no. 157, 1–22. MR**637283**, DOI https://doi.org/10.1090/S0025-5718-1982-0637283-6 - Ridgway Scott,
*Optimal $L^{\infty }$ estimates for the finite element method on irregular meshes*, Math. Comp.**30**(1976), no. 136, 681–697. MR**436617**, DOI https://doi.org/10.1090/S0025-5718-1976-0436617-2 - S. L. Sobolev,
*Applications of functional analysis in mathematical physics*, Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by F. E. Browder. MR**0165337** - Alan Solomon,
*Some remarks on the Stefan problem*, Math. Comp.**20**(1966), 347–360. MR**202391**, DOI https://doi.org/10.1090/S0025-5718-1966-0202391-1 - Gilbert Strang,
*Approximation in the finite element method*, Numer. Math.**19**(1972), 81–98. MR**305547**, DOI https://doi.org/10.1007/BF01395933 - Gilbert Strang and George J. Fix,
*An analysis of the finite element method*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR**0443377**
J. A. Wheeler, Jr.,

*Geotechnical Engineering for Cold Regions*, McGraw-Hill, New York, 1978. D. M. Anderson & N. R. Morgenstern, "Physics, chemistry and mechanics of frozen ground," in Proc. North American Permafrost Second International Conf., Nat. Acad. Sciences, Washington, D. C., 1973, pp. 257-295.

*Continuity of Weak Solutions to Certain Singular Parabolic Equations*, MRC Tech. Report 2124, Madison, Wisc., 1980.

*Linear Operators*, Vol. I, Wiley, New York, 1957. 15. A. Friedman, "The Stefan problem in several space variables,"

*Trans. Amer. Math. Soc.*, v. 133, 1968, pp. 51-87.

*Mat. Sb.*, v. 53, 1961, pp. 489-514. (Russian)

*Math. USSR Sb.*, v. 10, 1970, pp. 217-243. O. Ladyzhenskaya, V. Solonnikov & N. Ural’ceva,

*Linear and Quasilinear Equations of Parabolic Type*, Transl. Math. Monographs, vol. 23, Amer. Math. Soc., Providence, R. I., 1968. A. Lazaridis, "A numerical solution of the multidimensional solidification (or melting) problem,"

*Internat. J. Heat Mass Transfer*, v. 13, 1970, pp. 1459-1477.

*Simulation of Heat Transfer from a Warm Pipeline Buried in Permafrost*, Proc. 74th National Meeting AIChE, March 1973. J. A. Wheeler, Jr., "Permafrost thermal design for the trans-Alaska pipeline," in

*Moving Boundary Problems*(Wilson, Solomon, Boggs, eds.), Academic Press, New York, 1978, pp. 267-284. J. A. Wheeler, Jr., Personal communication.

Retrieve articles in *Mathematics of Computation*
with MSC:
65M60,
65M05,
65M10

Retrieve articles in all journals with MSC: 65M60, 65M05, 65M10

Additional Information

Article copyright:
© Copyright 1982
American Mathematical Society