An efficient linear scheme to approximate parabolic free boundary problems: error estimates and implementation

Authors:
Ricardo H. Nochetto and Claudio Verdi

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 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.

**[1]**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****[2]**Alan E. Berger, Haïm Brézis, and Joel C. W. Rogers,*A numerical method for solving the problem 𝑢_{𝑡}-Δ𝑓(𝑢)=0*, RAIRO Anal. Numér.**13**(1979), no. 4, 297–312 (English, with French summary). MR**555381****[3]**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****[4]**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****[5]**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****[6]**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**0375802**, https://doi.org/10.1016/0045-7825(73)90019-4**[7]**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****[8]**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**, https://doi.org/10.1093/imanum/7.1.61**[9]**Joseph W. Jerome,*Approximation of nonlinear evolution systems*, Mathematics in Science and Engineering, vol. 164, Academic Press, Inc., Orlando, FL, 1983. MR**690582****[10]**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**, https://doi.org/10.1090/S0025-5718-1982-0669635-2**[11]**Enrico Magenes,*Two-phase Stefan problems in several space variables*, Matematiche (Catania)**36**(1981), no. 1, 65–108 (1983) (Italian). MR**736797****[12]**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****[13]**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****[14]**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****[15]**T. A. Manteuffel,*An incomplete factorization technique for positive definite linear systems*, Math. Comp.**34**(1980), no. 150, 473–497. MR**559197**, https://doi.org/10.1090/S0025-5718-1980-0559197-0**[16]**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**, https://doi.org/10.1007/BF02575898

Ricardo H. Nochetto,*Error estimates for two-phase Stefan problems in several space variables. II. Nonlinear flux conditions*, Calcolo**22**(1985), no. 4, 501–534 (1986). MR**859088**, https://doi.org/10.1007/BF02575899**[17]**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****[18]**Ricardo H. Nochetto,*Error estimates for multidimensional singular parabolic problems*, Japan J. Appl. Math.**4**(1987), no. 1, 111–138. MR**899207**, https://doi.org/10.1007/BF03167758**[19]**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****[20]**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**, https://doi.org/10.1137/0725046**[21]**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**, https://doi.org/10.1002/nme.1620260907**[22]**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****[23]**Michael E. Rose,*Numerical methods for flows through porous media. I*, Math. Comp.**40**(1983), no. 162, 435–467. MR**689465**, https://doi.org/10.1090/S0025-5718-1983-0689465-6**[24]**C. Verdi,*On the numerical approach to a two-phase Stefan problem with nonlinear flux*, Calcolo**22**(1985), no. 3, 351–381. MR**860658**, https://doi.org/10.1007/BF02600382**[25]**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**, https://doi.org/10.1080/01630568708816253**[26]**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**, https://doi.org/10.1007/BF01398688**[27]**Augusto Visintin,*Stefan problem with phase relaxation*, IMA J. Appl. Math.**34**(1985), no. 3, 225–245. MR**804824**, https://doi.org/10.1093/imamat/34.3.225

Retrieve articles in *Mathematics of Computation*
with MSC:
65N15,
65N30

Retrieve articles in all journals with MSC: 65N15, 65N30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0942142-0

Article copyright:
© Copyright 1988
American Mathematical Society