Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



An adaptive finite element method for two-phase Stefan problems in two space dimensions. I. Stability and error estimates

Authors: R. H. Nochetto, M. Paolini and C. Verdi
Journal: Math. Comp. 57 (1991), 73-108, S1
MSC: Primary 65N50; Secondary 65N12, 65N30, 80A22
MathSciNet review: 1079028
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A simple and efficient adaptive local mesh refinement algorithm is devised and analyzed for two-phase Stefan problems in 2D. A typical triangulation is coarse away from the discrete interface, where discretization parameters satisfy a parabolic relation, whereas it is locally refined in the vicinity of the discrete interface so that the relation becomes hyperbolic. Several numerical tests are performed on the computed temperature to extract information about its first and second derivatives as well as to predict discrete free boundary locations. Mesh selection is based upon equidistributing pointwise interpolation errors between consecutive meshes and imposing that discrete interfaces belong to the so-called refined region. Consecutive meshes are not compatible in that they are not produced by enrichment or coarsening procedures but rather regenerated. A general theory for interpolation between noncompatible meshes is set up in $ {L^p}$-based norms. The resulting scheme is stable in various Sobolev norms and necessitates fewer spatial degrees of freedom than previous practical methods on quasi-uniform meshes, namely $ O({\tau ^{ - 3/2}})$ as opposed to $ O({\tau ^{ - 2}})$, to achieve the same global asymptotic accuracy; here $ \tau > 0$ is the (uniform) time step. A rate of convergence of essentially $ O({\tau ^{1/2}})$ is derived in the natural energy spaces provided the total number of mesh changes is restricted to $ O({\tau ^{ - 1/2}})$, which in turn is compatible with the mesh selection procedure. An auxiliary quasi-optimal pointwise error estimate for the Laplace operator is proved as well. Numerical results illustrate the scheme's efficiency in approximating both solutions and interfaces.

References [Enhancements On Off] (What's this?)

  • [1] I. Babuška, O. C. Zienkiewicz, J. Gago, and E. R. de A. Oliveira, Accuracy estimates and adaptive refinements infinite element computations, Wiley, Chichester, 1986. MR 879442 (87j:65004)
  • [2] P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
  • [3] J. F. Ciavaldini, Analyse numérique d'un problème de Stefan à deux phases par une méthode d'éléments finis, SIAM J. Numer. Anal. 12 (1975), 464-487. MR 0391741 (52:12561)
  • [4] C. M. Elliott, Error analysis of the enthalpy method for the Stefan problem, IMA J. Numer. Anal. 7 (1987), 61-71. MR 967835 (90a:65222)
  • [5] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems I: A linear problem, preprint 31, University of Göteborg, 1988. MR 929542 (89c:65119)
  • [6] A. Friedman, The Stefan problem in several space variables, Trans. Amer. Math. Soc. 133 (1968), 51-87. MR 0227625 (37:3209)
  • [7] P. Grisvard, Elliptic problems in non-smooth domains, Pitman, Boston, 1985.
  • [8] J. W. Jerome and M. Rose, Error estimates for the multidimensional two-phase Stefan problem, Math. Comp. 39 (1982), 377-414. MR 669635 (84h:65097)
  • [9] R. Löhner, Some useful data structures for the generation of unstructured meshes, Comm. Appl. Numer. Methods 4 (1988), 123-135. MR 937765 (89b:65225)
  • [10] E. Magenes, Problemi di Stefan bifase in più variabili spaziali, Matematiche 36 (1981), 65-108. MR 736797 (85f:35198)
  • [11] E. Magenes, C. Verdi, and A. Visintin, Theoretical and numerical results on the two-phase Stefan problem, SIAM J. Numer. Anal. 26 (1989), 1425-1438. MR 1025097 (90i:35293)
  • [12] R. H. Nochetto, Error estimates for two-phase Stefan problems in several space variables, I: Linear boundary conditions, Calcolo 22 (1985), 457-499. MR 859087 (88a:65122a)
  • [13] -, Error estimates for multidimensional singular parabolic problems, Japan J. Appl. Math. 4 (1987), 111-138. MR 899207 (89c:65107)
  • [14] -, Numerical methods for free boundary problems, Free Boundary Problems: Theory and Applications II (K.-H. Hoffmann and J. Sprekels, eds.), Res. Notes in Math. 186, Pitman, London, 1990, pp. 555-566.
  • [15] R. H. Nochetto, M. Paolini, and C. Verdi, Local mesh refinements for two-phase Stefan problems in two space variables, Mathematical Models for Phase Change Problems (J. F. Rodrigues, ed.), Birkhäuser Verlag, Basel, 1989, pp. 261-286. MR 1038074 (91b:80019)
  • [16] -, An adaptive finite element method for two-phase Stefan problems in two space dimensions. Part II: Implementation and numerical experiments, SIAM J. Sci. Statist. Comput. 1991 (to appear). MR 1114983 (92f:65138)
  • [17] R.H. Nochetto and C. Verdi, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal. 25 (1988), 784-814. MR 954786 (89m:65102)
  • [18] M. Paolini, G. Sacchi and C. Verdi, Finite element approximations of singular parabolic problems, Internat. J. Numer. Methods. Engrg. 26 (1988), 1989-2007. MR 955582 (89j:76023)
  • [19] M. Paolini and C. Verdi, An automatic triangular mesh generator for planar domains, Riv. Inform. 20 (1990), 251-267.
  • [20] J. C. W. Rogers and A. E. Berger, Some properties of the nonlinear semigroup for the problem $ {u_t} - \Delta f(u) = 0$, Nonlinear Anal. 8 (1984), 909-939. MR 753767 (86b:35075)
  • [21] A. H. Schatz and L. B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. Part 1, Math. Comp. 32 (1978), 73-109. MR 0502065 (58:19233a)
  • [22] -, Maximum norm estimates in the finite element method on plane polygonal domains. Part 2, Refinements, Math. Comp. 33 (1979), 465-492. MR 0502067 (58:19233b)
  • [23] -, On the quasi-optimality in $ {L^\infty }$ of the $ H_0^1$-projection into finite element spaces, Math. Comp. 38 (1982), 1-22.
  • [24] C. Verdi, Optimal error estimates for an approximation of degenerate parabolic problems, Numer. Funct. Anal. Optim. 9 (1987), 657-670. MR 895990 (88m:65165)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N50, 65N12, 65N30, 80A22

Retrieve articles in all journals with MSC: 65N50, 65N12, 65N30, 80A22

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society