Optimal rates of convergence for degenerate parabolic problems in two dimensions

SIAM Journal on Numerical Analysis

Volumn 33, Issue 1, 1996, Pages 56-67

  Rulla, Jim ^a   Walkington, Noel J ^b  

^a University of Arkansas   (United States)

^b CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

Error estimates; Semigroup; Stefan problem

Indexed keywords

EID: 0002124686     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/0733004     Document Type: Article

References (21)

1. 1 existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions, Appl. Math. Optim., 17 (1988), pp. 203-224.
2. t - Δf(u) = 0, RAIRO Numer. Anal., 13 (1979), pp. 297-312.
3. H. BREZIS, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Notas Mat. 50, North-Holland, Amsterdam, 1973.
4. H. BREZIS AND A. PAZY, Convergence and approximations of semigroups of nonlinear operators in Banach spaces, J. Fund. Anal., 9 (1972), pp. 63-74.
5. 1, J. Math. Soc. Japan, 25 (1973), pp. 565-590.
6. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
7. M. G. CRANDALL, An introduction to evolution equations governed by accretive operators, in Dynamical Systems, Vol. 1, An International Symposium, Academic Press, New York, 1976.
8. M. G. CRANDALL AND T. M. LIGGETT, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), pp. 265-293.
9. J. F. EPPERSON, Finite element methods for a class of nonlinear evolution equations, SIAM J. Numer. Anal., 21 (1984), pp. 1066-1079.
10. J. W. JEROME AND M. E. ROSE, Error estimates for the multidimensional two-phase Stefan problem, Math. Comp., 160 (1982), pp. 377-414.
11. R. H. NOCHETTO, Error estimates for multidimensional singular parabolic problems, Japan J. Appl. Math., 4 (1987), pp. 111-138.
12. R. H. NOCHETTO AND C. VERDI, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal., 25 (1988), pp. 784-814.
13. J. C. W. ROGERS, A. E. BERGER, AND M. CIMENT, The alternating phase truncation method for numerical solution of a Stefan problem, SIAM J. Numer. Anal., 16 (1979), pp. 563-587.
14. L. I. RUBINSTEIN, The Stefan Problem, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1971.
15. J. RULLA, Error Analysis for Implicit Approximations to Solutions to Cauchy Problems, SIAM J. Numer. Anal., 33 (1996), pp.68-87.
16. A. H. SCHATZ, V. THOMEE, AND L. B. WAHLBIN, Maximum norm stability and error estimates in parabolic finite element equations, Comm. Pure Appl., Math, 33 (1980), pp. 265-304.
17. 0 projection into finite element spaces, Math. Comp., 38 (1982), pp. 1-22.
18. R. E. SHOWALTER, Hilbert Space Methods for Partial Differential Equations, Pitman, Boston, MA, 1979.
19. R. E. WHITE, An enthalpy formulation of the Stefan problem, SIAM J. Numer. Anal., 19 (1982), pp. 1129-1157.
20. K. YOSIDA, Functional analysis, Grundlehren Math. Wiss. 123, Springer-Verlag, Berlin, New York, 1980.
21. M. ZLAMAL, A finite element solution of the nonlinear heat equation, RAIRO Numer. Anal., 14 (1980), pp. 203-216.