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 Free Access

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 -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 as opposed to , to achieve the same global asymptotic accuracy; here is the (uniform) time step. A rate of convergence of essentially is derived in the natural energy spaces provided the total number of mesh changes is restricted to , 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.

**[1]**I. Babuška, O. C. Zienkiewicz, J. Gago, and E. R. de A. Oliveira (eds.),*Accuracy estimates and adaptive refinements in finite element computations*, Wiley Series in Numerical Methods in Engineering, John Wiley & Sons, Ltd., Chichester, 1986. Lectures presented at the international conference held in Lisbon, June 1984; A Wiley-Interscience Publication. MR**879442****[2]**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****[3]**J. F. Ciavaldini,*Analyse numerique d’un problème de Stefan à deux phases par une methode d’éléments finis*, SIAM J. Numer. Anal.**12**(1975), 464–487 (French, with English summary). MR**0391741****[4]**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**, 10.1093/imanum/7.1.61**[5]**Kenneth Eriksson and Claes Johnson,*An adaptive finite element method for linear elliptic problems*, Math. Comp.**50**(1988), no. 182, 361–383. MR**929542**, 10.1090/S0025-5718-1988-0929542-X**[6]**Avner Friedman,*The Stefan problem in several space variables*, Trans. Amer. Math. Soc.**133**(1968), 51–87. MR**0227625**, 10.1090/S0002-9947-1968-0227625-7**[7]**P. Grisvard,*Elliptic problems in non-smooth domains*, Pitman, Boston, 1985.**[8]**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**, 10.1090/S0025-5718-1982-0669635-2**[9]**Rainald Löhner,*Some useful data structures for the generation of unstructured grids*, Comm. Appl. Numer. Methods**4**(1988), no. 1, 123–135. MR**937765**, 10.1002/cnm.1630040116**[10]**Enrico Magenes,*Two-phase Stefan problems in several space variables*, Matematiche (Catania)**36**(1981), no. 1, 65–108 (1983) (Italian). MR**736797****[11]**E. Magenes, C. Verdi, and A. Visintin,*Theoretical and numerical results on the two-phase Stefan problem*, SIAM J. Numer. Anal.**26**(1989), no. 6, 1425–1438. MR**1025097**, 10.1137/0726083**[12]**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**, 10.1007/BF02575898**[13]**Ricardo H. Nochetto,*Error estimates for multidimensional singular parabolic problems*, Japan J. Appl. Math.**4**(1987), no. 1, 111–138. MR**899207**, 10.1007/BF03167758**[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 (Óbidos, 1988) Internat. Ser. Numer. Math., vol. 88, Birkhäuser, Basel, 1989, pp. 261–286. MR**1038074****[16]**R. H. Nochetto, M. Paolini, and C. Verdi,*An adaptive finite element method for two-phase Stefan problems in two space dimensions. II. Implementation and numerical experiments*, SIAM J. Sci. Statist. Comput.**12**(1991), no. 5, 1207–1244. MR**1114983**, 10.1137/0912065**[17]**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**, 10.1137/0725046**[18]**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**, 10.1002/nme.1620260907**[19]**M. Paolini and C. Verdi,*An automatic triangular mesh generator for planar domains*, Riv. Inform.**20**(1990), 251-267.**[20]**Joel C. W. Rogers and Alan E. Berger,*Some properties of the nonlinear semigroup for the problem 𝑢_{𝑡}-Δ𝑓(𝑢)=0*, Nonlinear Anal.**8**(1984), no. 8, 909–939. MR**753767**, 10.1016/0362-546X(84)90111-1**[21]**A. H. Schatz and L. B. Wahlbin,*Maximum norm estimates in the finite element method on plane polygonal domains. I*, Math. Comp.**32**(1978), no. 141, 73–109. MR**0502065**, 10.1090/S0025-5718-1978-0502065-1**[22]**A. H. Schatz and L. B. Wahlbin,*Maximum norm estimates in the finite element method on plane polygonal domains. II. Refinements*, Math. Comp.**33**(1979), no. 146, 465–492. MR**0502067**, 10.1090/S0025-5718-1979-0502067-6**[23]**-,*On the quasi-optimality in**of the*-*projection into finite element spaces*, Math. Comp.**38**(1982), 1-22.**[24]**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**, 10.1080/01630568708816253

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

DOI:
https://doi.org/10.1090/S0025-5718-1991-1079028-X

Article copyright:
© Copyright 1991
American Mathematical Society