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), 2753
MSC:
Primary 65N15; Secondary 65N30
MathSciNet review:
942142
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Abstract: This paper deals with a fully discrete scheme to approximate multidimensional singular parabolic problems; twophase 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 elementbyelement 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
(85m:65116)
 [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
(81g:65120)
 [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
(87h:00021)
 [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
(45 #2529)
 [5]
Philippe
G. Ciarlet, The finite element method for elliptic problems,
NorthHolland Publishing Co., AmsterdamNew YorkOxford, 1978. Studies in
Mathematics and its Applications, Vol. 4. MR 0520174
(58 #25001)
 [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
(51 #11992)
 [7]
Jim
Douglas Jr. and Todd
Dupont, Alternatingdirection 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
(42 #8706)
 [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 (90a:65222), http://dx.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
(85g:35064)
 [10]
Joseph
W. Jerome and Michael
E. Rose, Error estimates for the
multidimensional twophase Stefan problem, Math. Comp. 39 (1982), no. 160, 377–414. MR 669635
(84h:65097), http://dx.doi.org/10.1090/S00255718198206696352
 [11]
Enrico
Magenes, Twophase Stefan problems in several space variables,
Matematiche (Catania) 36 (1981), no. 1, 65–108
(1983) (Italian). MR 736797
(85f:35198)
 [12]
E.
Magenes, Remarques sur l’approximation des problèmes
paraboliques non linéaires, Analyse mathématique et
applications, GauthierVillars, Montrouge, 1988, pp. 297–318
(French). MR
956965 (90f:65158)
 [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
(89b:65220)
 [14]
A.
Magenes and C.
Verdi, The semigroup approach to the twophase 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
(82c:65019), http://dx.doi.org/10.1090/S00255718198005591970
 [16]
Ricardo
H. Nochetto, Error estimates for twophase Stefan problems in
several space variables. I. Linear boundary conditions, Calcolo
22 (1985), no. 4, 457–499 (1986). MR 859087
(88a:65122a), http://dx.doi.org/10.1007/BF02575898
 [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
(89c:65107), http://dx.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 (87m:65190)
 [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
(89m:65102), http://dx.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
(89j:76023), http://dx.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
(49 #10164)
 [23]
Michael
E. Rose, Numerical methods for flows through
porous media. I, Math. Comp.
40 (1983), no. 162, 435–467. MR 689465
(85a:65146), http://dx.doi.org/10.1090/S00255718198306894656
 [24]
C.
Verdi, On the numerical approach to a twophase Stefan problem with
nonlinear flux, Calcolo 22 (1985), no. 3,
351–381. MR
860658 (87m:65140), http://dx.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. 56, 657–670. MR 895990
(88m:65165), http://dx.doi.org/10.1080/01630568708816253
 [26]
Claudio
Verdi and Augusto
Visintin, Error estimates for a semiexplicit numerical scheme for
Stefantype problems, Numer. Math. 52 (1988),
no. 2, 165–185. MR 923709
(89a:65155), http://dx.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
(87a:80014), http://dx.doi.org/10.1093/imamat/34.3.225
 [1]
 O. Axelsson & V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation, Academic Press, Orlando, Florida, 1984. MR 758437 (85m:65116)
 [2]
 A. E. Berger, H. Brezis & J. C. W. Rogers, "A numerical method for solving the problem ," RAIRO Anal. Numér., v. 13, 1979, pp. 297312. MR 555381 (81g:65120)
 [3]
 A. Bossavit, A. Damlamian & M. Fremond (Editors), Free Boundary Problems: Applications and Theory, Vol. III, Research Notes in Math., Pitman, Boston, 1985. MR 863154 (87h:00021)
 [4]
 H. Brezis & A. Pazy, "Convergence and approximation of semigroups of nonlinear operators in Banach spaces," J. Funct. Anal., v. 9, 1972, pp. 6374. MR 0293452 (45:2529)
 [5]
 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [6]
 P. G. Ciarlet & P. A. Raviart, "Maximum principle and uniform convergence for the finite element method," Comput. Methods Appl. Mech. Engrg., v. 2, 1973, pp. 1731. MR 0375802 (51:11992)
 [7]
 J. Douglas, Jr. & T. Dupont, "Alternatingdirection Galerkin methods on rectangles," in Numerical Solutions of Partial Differential Equations, Vol. II (B. Hubbard, ed.), Academic Press, New York, 1971, pp. 133214. MR 0273830 (42:8706)
 [8]
 C. M. Elliott, "Error analysis of the enthalpy method for the Stefan problem," IMA J. Numer. Anal., v. 7, 1987, pp. 6171. MR 967835 (90a:65222)
 [9]
 J. W. Jerome, Approximation of Nonlinear Evolution Systems, Academic Press, New York, 1983. MR 690582 (85g:35064)
 [10]
 J. W. Jerome & M. E. Rose, "Error estimates for the multidimensional twophase Stefan problem," Math. Comp., v. 39, 1982, pp. 377414. MR 669635 (84h:65097)
 [11]
 E. Magenes, "Problemi di Stefan bifase in più variabili spaziali," Le Matematiche, v. 36, 1981, pp. 65108. MR 736797 (85f:35198)
 [12]
 E. Magenes, "Remarques sur l'approximation des problèmes non linéaires paraboliques," to appear in the volume dedicated to J. L. Lions, GauthierVillars, Paris, 1988. MR 956965 (90f:65158)
 [13]
 E. Magenes, R. H. Nochetto & C. Verdi, "Energy error estimates for a linear scheme to approximate nonlinear parabolic equations," RAIRO Modél. Math. Anal. Numér., v. 21, 1987, pp. 655678. MR 921832 (89b:65220)
 [14]
 E. Magenes & C. Verdi, "On the semigroup approach to the twophase Stefan problem with nonlinear flux conditions," in Free Boundary Problems: Applications and Theory, Vol. III (A. Bossavit, A. Damlamian and M. Fremond, eds.), Research Notes in Math., Pitman, Boston, 1985, pp. 2839. MR 863159
 [15]
 T. A. Manteuffel, "The shifted incomplete Cholesky factorization," Report Sandia Laboratories 788226 (1978); "An incomplete factorization technique for positive definite linear systems," Math. Comp., v. 34, 1980, pp. 473497. MR 559197 (82c:65019)
 [16]
 R. H. Nochetto, "Error estimates for twophase Stefan problems in several space variables, I: linear boundary conditions," Calclo, v. 22, 1985, pp. 457499; "II: nonlinear flux conditions," Calcolo, v. 22, 1985, pp. 501534. MR 859087 (88a:65122a)
 [17]
 R. H. Nochetto, "Error estimates for multidimensional Stefan problems with general boundary conditions," in Free Boundary Problems: Applications and Theory, Vol. III (A. Bossavit, A. Damlamian and M. Fremond, eds.), Research Notes in Math., Pitman, Boston, 1985, pp. 5060. MR 863161
 [18]
 R. H. Nochetto, "Error estimates for multidimensional singular parabolic problems," Japan. J. Appl. Math., v. 4, 1987, pp. 111138. MR 899207 (89c:65107)
 [19]
 R. H. Nochetto, "A note on the approximation of free boundaries by finite element methods," RAIRO Modél. Math. Anal. Numér., v. 20, 1986, pp. 355368. MR 852686 (87m:65190)
 [20]
 R. H. Nochetto & C. Verdi, "Approximation of degenerate parabolic problems using numerical integration," SIAM J. Numer. Anal., v. 25, 1988. (To appear.) MR 954786 (89m:65102)
 [21]
 M. Paolini, G. Sacchi & C. Verdi, "Finite element approximations of singular parabolic problems," Internat. J. Numer. Methods Engrg. (To appear.) MR 955582 (89j:76023)
 [22]
 P. A. Raviart, "The use of numerical integration in finite element methods for solving parabolic equations," in Topics in Numerical Analysis (J. J. H. Miller, ed.), Academic Press, London, 1973, pp. 233264. MR 0345428 (49:10164)
 [23]
 M. E. Rose, "Numerical methods for flows through porous media. I," Math. Comp., v. 40, 1983, pp. 435467. MR 689465 (85a:65146)
 [24]
 C. Verdi, "On the numerical approach to a twophase Stefan problem with nonlinear flux," Calcolo, v. 22, 1985, pp. 351381. MR 860658 (87m:65140)
 [25]
 C. Verdi, "Optimal error estimates for an approximation of degenerate parabolic problems," Numer. Funct. Anal. Optim., v. 9, 1987, pp. 657670. MR 895990 (88m:65165)
 [26]
 C. Verdi & A. Visintin, "Error estimates for a semiexplicit numerical scheme for Stefantype problems," Numer. Math., v. 52, 1988, pp. 165185. MR 923709 (89a:65155)
 [27]
 A. Visintin, "Stefan problem with phase relaxation," IMA J. Appl. Math., v. 34, 1985, pp. 225245. MR 804824 (87a:80014)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809421420
PII:
S 00255718(1988)09421420
Article copyright:
© Copyright 1988
American Mathematical Society
