An efficient linear scheme to approximate parabolic free boundary problems: error estimates and implementation
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- by Ricardo H. Nochetto and Claudio Verdi PDF
- Math. Comp. 51 (1988), 27-53 Request permission
Abstract:
This paper deals with a fully discrete scheme to approximate multidimensional singular parabolic problems; two-phase Stefan problems and porous medium equations are included. The algorithm consists of approximating at each time step a linear elliptic partial differential equation by piecewise linear finite elements and then making an element-by-element algebraic correction to account for the nonlinearity. Several energy error estimates are derived for the physical unknowns; a sharp rate of convergence of $O({h^{1/2}})$ is our main result. The crucial point in implementing the scheme is the efficient resolution of linear systems involved. This topic is discussed, and the results of several numerical experiments are shown.References
- O. Axelsson and V. A. Barker, Finite element solution of boundary value problems, Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. Theory and computation. MR 758437
- Alan E. Berger, Haïm Brézis, and Joel C. W. Rogers, A numerical method for solving the problem $u_{t}-\Delta f(u)=0$, RAIRO Anal. Numér. 13 (1979), no. 4, 297–312 (English, with French summary). MR 555381, DOI 10.1051/m2an/1979130402971
- A. Bossavit, A. Damlamian, and M. Frémond (eds.), Free boundary problems: applications and theory. Vol. III, Research Notes in Mathematics, vol. 120, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 863154
- H. Brézis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Functional Analysis 9 (1972), 63–74. MR 0293452, DOI 10.1016/0022-1236(72)90014-6
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- P. G. Ciarlet and P.-A. Raviart, Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Engrg. 2 (1973), 17–31. MR 375802, DOI 10.1016/0045-7825(73)90019-4
- Jim Douglas Jr. and Todd Dupont, Alternating-direction Galerkin methods on rectangles, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 133–214. MR 0273830
- C. M. Elliott, Error analysis of the enthalpy method for the Stefan problem, IMA J. Numer. Anal. 7 (1987), no. 1, 61–71. MR 967835, DOI 10.1093/imanum/7.1.61
- Joseph W. Jerome, Approximation of nonlinear evolution systems, Mathematics in Science and Engineering, vol. 164, Academic Press, Inc., Orlando, FL, 1983. MR 690582
- Joseph W. Jerome and Michael E. Rose, Error estimates for the multidimensional two-phase Stefan problem, Math. Comp. 39 (1982), no. 160, 377–414. MR 669635, DOI 10.1090/S0025-5718-1982-0669635-2
- Enrico Magenes, Two-phase Stefan problems in several space variables, Matematiche (Catania) 36 (1981), no. 1, 65–108 (1983) (Italian). MR 736797
- E. Magenes, Remarques sur l’approximation des problèmes paraboliques non linéaires, Analyse mathématique et applications, Gauthier-Villars, Montrouge, 1988, pp. 297–318 (French). MR 956965
- E. Magenes, R. H. Nochetto, and C. Verdi, Energy error estimates for a linear scheme to approximate nonlinear parabolic problems, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 4, 655–678 (English, with French summary). MR 921832, DOI 10.1051/m2an/1987210406551
- A. Magenes and C. Verdi, The semigroup approach to the two-phase Stefan problem with nonlinear flux conditions, Free boundary problems: applications and theory, Vol. III (Maubuisson, 1984) Res. Notes in Math., vol. 120, Pitman, Boston, MA, 1985, pp. 28–39. MR 863159
- T. A. Manteuffel, An incomplete factorization technique for positive definite linear systems, Math. Comp. 34 (1980), no. 150, 473–497. MR 559197, DOI 10.1090/S0025-5718-1980-0559197-0
- Ricardo H. Nochetto, Error estimates for two-phase Stefan problems in several space variables. I. Linear boundary conditions, Calcolo 22 (1985), no. 4, 457–499 (1986). MR 859087, DOI 10.1007/BF02575898
- R. H. Nochetto, Error estimates for multidimensional Stefan problems with general boundary conditions, Free boundary problems: applications and theory, Vol. III (Maubuisson, 1984) Res. Notes in Math., vol. 120, Pitman, Boston, MA, 1985, pp. 50–60. MR 863161
- Ricardo H. Nochetto, Error estimates for multidimensional singular parabolic problems, Japan J. Appl. Math. 4 (1987), no. 1, 111–138. MR 899207, DOI 10.1007/BF03167758
- Ricardo H. Nochetto, A note on the approximation of free boundaries by finite element methods, RAIRO Modél. Math. Anal. Numér. 20 (1986), no. 2, 355–368 (English, with French summary). MR 852686, DOI 10.1051/m2an/1986200203551
- Ricardo H. Nochetto and Claudio Verdi, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal. 25 (1988), no. 4, 784–814. MR 954786, DOI 10.1137/0725046
- M. Paolini, G. Sacchi, and C. Verdi, Finite element approximations of singular parabolic problems, Internat. J. Numer. Methods Engrg. 26 (1988), no. 9, 1989–2007. MR 955582, DOI 10.1002/nme.1620260907
- P.-A. Raviart, The use of numerical integration in finite element methods for solving parabolic equations, Topics in numerical analysis (Proc. Roy. Irish Acad. Conf., University Coll., Dublin, 1972) Academic Press, London, 1973, pp. 233–264. MR 0345428
- Michael E. Rose, Numerical methods for flows through porous media. I, Math. Comp. 40 (1983), no. 162, 435–467. MR 689465, DOI 10.1090/S0025-5718-1983-0689465-6
- C. Verdi, On the numerical approach to a two-phase Stefan problem with nonlinear flux, Calcolo 22 (1985), no. 3, 351–381. MR 860658, DOI 10.1007/BF02600382
- Claudio Verdi, Optimal error estimates for an approximation of degenerate parabolic problems, Numer. Funct. Anal. Optim. 9 (1987), no. 5-6, 657–670. MR 895990, DOI 10.1080/01630568708816253
- Claudio Verdi and Augusto Visintin, Error estimates for a semi-explicit numerical scheme for Stefan-type problems, Numer. Math. 52 (1988), no. 2, 165–185. MR 923709, DOI 10.1007/BF01398688
- Augusto Visintin, Stefan problem with phase relaxation, IMA J. Appl. Math. 34 (1985), no. 3, 225–245. MR 804824, DOI 10.1093/imamat/34.3.225
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Math. Comp. 51 (1988), 27-53
- MSC: Primary 65N15; Secondary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-1988-0942142-0
- MathSciNet review: 942142