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A note on the approximation of free boundaries by finite element methods. (English) Zbl 0596.65092
The aim of the present paper is to study rates of convergence in measure (and in distance) for the approximate free boundaries (they are defined as suitable level set). The results are applied to the primal and mixed formulation of the obstacle problem and to singular parabolic problems (the two-phase Stefan problem and the porous medium equation) in several variables.
Main result: If an $$L^ p$$-error estimate is known with $$p<\infty$$, then the rates of convergence for the free boundaries are in measure, error estimates in distance are obtained for $$p=\infty$$.
Reviewer: J.Lovíšek

##### MSC:
 65Z05 Applications to the sciences 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35K05 Heat equation 80A17 Thermodynamics of continua 35R35 Free boundary problems for PDEs 76S05 Flows in porous media; filtration; seepage
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##### References:
 [1] D ARONSON, L CAFFARELLI, S KAMIN, How an initially stationary interface begins to move in porous medium flow, SIAM J Math Anal 14, 4 (1983), pp 639-658 Zbl0542.76119 MR704481 · Zbl 0542.76119 · doi:10.1137/0514049 [2] C BAIOCCHI, Estimations d erreur dans L \infty pour les inéquations a obstacle, Mathematical Aspects of F E M , Lectures Notes m Math 606, Springer (1977), pp 27-34 Zbl0374.65053 MR488847 · Zbl 0374.65053 [3] [3] C BAIOCCHI, G POZZI, Error estimates and free-boundary convergence for a finite difference discretization of a parabolic variational inequality, RAIRO Numer Anal 11, 4 (1977), pp 315-340 Zbl0371.65020 MR464607 · Zbl 0371.65020 · eudml:193305 [4] H BREZIS, Seuil de régulante pour certains problèmes unilatéraux, C R Acad Sci Paris 273 (1971), pp 35-37 Zbl0214.10703 MR287366 · Zbl 0214.10703 [5] H BREZIS, D KINDERLEHRER, The smoothness of solutions to nonlinear variational inequahties Indiana Univ Math J 23, 9 (1974), pp 831-844 Zbl0278.49011 MR361436 · Zbl 0278.49011 · doi:10.1512/iumj.1974.23.23069 [6] [6] F BREZZI, L CAFFARELLI, Convergence of the discrete free boundaries for finite element approximations, RAIRO Numer Anal 17 (1983), pp 385-395 Zbl0547.65081 MR713766 · Zbl 0547.65081 · eudml:193422 [7] [7] F BREZZI, W HAGER, P RAVIART, Error estimates for the finite element solution of variational inequahties Part I Primal Theory, Numer Math 28 (1977), pp 431-443 Zbl0369.65030 MR448949 · Zbl 0369.65030 · doi:10.1007/BF01404345 · eudml:132496 [8] [8] F BREZZI, W HAGER, P RAVIART, Error estimates for the finite element solution of variational inequalities Part II Mixed Methods, Numer Math 31 (1978), pp 1-16 Zbl0427.65077 MR508584 · Zbl 0427.65077 · doi:10.1007/BF01396010 · eudml:132563 [9] F BREZZI, G SACCHI, A finite element approximation of a variational inequality related to hydraulics, Calcolo 13, III (1976), pp. 257-274 Zbl0353.76068 MR520171 · Zbl 0353.76068 · doi:10.1007/BF02575934 [10] L CAFFARELLI, A remark on the Hausdorff measure of a free boundary, and the convergence of coincidence sets, Boll U M I (1981), pp. 109-113 Zbl0453.35085 MR607212 · Zbl 0453.35085 [11] L CAFFARELLI, L EVANS, Continuity of the temperature in the two-phase Stefan problems Arch Rational Mech Anal 81, 3 (1983), pp 199-220 Zbl0516.35080 MR683353 · Zbl 0516.35080 · doi:10.1007/BF00250800 [12] L CAFFARELLI, A FRIEDMAN, Regularity of the free boundary for the one dimensional flow of gas in a porous medium, Amer J Math (1979), pp 1193-1218 Zbl0439.76084 MR548877 · Zbl 0439.76084 · doi:10.2307/2374136 [13] L CAFFARELLI, A FRIEDMAN, Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Univ Math J 29 (1980), pp 361-391 Zbl0439.76085 MR570687 · Zbl 0439.76085 · doi:10.1512/iumj.1980.29.29027 [14] L CAFFARELLI, N RIVIÈRE, Asymptotic behavior of free boundaries at their singular points, Ann Math 106 (1977), pp 309-317 Zbl0364.35041 MR463690 · Zbl 0364.35041 · doi:10.2307/1971098 [15] P CIARLET, The finite element method for elliptic problems, North-Holland (1978) Zbl0383.65058 MR520174 · Zbl 0383.65058 [16] P CIARLET, P RAVIART, Maximum principle and uniform convergence for the finite element method, Comput Methods Appl Mech Engrg 2 (1973), pp 17-31 Zbl0251.65069 MR375802 · Zbl 0251.65069 · doi:10.1016/0045-7825(73)90019-4 [17] A DAMLAMIAN, Some results in the multiphase Stefan problem, Comm Partial Differential Equations 2, 10 (1977), pp 1017-1044 Zbl0399.35054 MR487015 · Zbl 0399.35054 · doi:10.1080/03605307708820053 [18] E DI BENEDETTO, Continuity of weak-solutions to certain singular parabolic equations Ann Mat Pura Appl IV, 130 (1982), pp 131-176 Zbl0503.35018 MR663969 · Zbl 0503.35018 · doi:10.1007/BF01761493 [19] A boundary modulus of continuity for a class of singular parabolic equations (to appear) Zbl0606.35044 · Zbl 0606.35044 · doi:10.1016/0022-0396(86)90064-1 [20] A FRIEDMAN, , The Stefan problem in several space variables, Trans Amer Math Soc 133 (1968), pp 51-87 Zbl0162.41903 MR227625 · Zbl 0162.41903 · doi:10.2307/1994932 [21] A FRIEDMAN, Variational Principles and Free-Boundary Problems, John Wiley & Sons (1982) Zbl0564.49002 MR679313 · Zbl 0564.49002 [22] L JEROME, M ROSE, Error estimetes for the multidimensional two-phase Stefan problem, Math Comp 39, 160 (1982), pp 377-414 Zbl0505.65060 MR669635 · Zbl 0505.65060 · doi:10.2307/2007320 [23] B KNERR, The porous medium equation in one dimension, Trans Amer Math Soc 234 (1977), pp 381-415 Zbl0365.35030 MR492856 · Zbl 0365.35030 · doi:10.2307/1997927 [24] M NIEZGODKA, I PAWLOW, A generalized Stefan problem in several space variables Appl Math Optim 9 (1983), pp 193-224 Zbl0519.35079 MR687720 · Zbl 0519.35079 · doi:10.1007/BF01460125 [25] J NITSCHE, L \infty -convergence of finite element approximations, Mathematical Aspects of F E M, Lectures Notes m Math 606, Springer (1977), pp 261-274 Zbl0362.65088 MR488848 · Zbl 0362.65088 [26] R NOCHETTO, Error estimates for two-phase Stefan problems in several space variables, I linear boundary conditions, II non-linear flux conditions (to appear in Calcolo) Zbl0606.65084 MR859087 · Zbl 0606.65084 · doi:10.1007/BF02575898 [27] R NOCHETTO, Error estimates for multidimensional Stefan problems with general boundary conditions Free boundary problems applications and theory, Vol III (A Bossavitera/ eds ), Res Notes Math 120, Pitman (1985), pp 50-60 Zbl0593.35094 MR863161 · Zbl 0593.35094 [28] R NOCHETTO, A class of non-degenerate two-phase Stefan problems in several space variables, Pubblicazione N^\circ 442 del I A N di Pavia (1984) (to appear in Comm Partial Differential Equations) Zbl0624.35085 MR869101 · Zbl 0624.35085 · doi:10.1080/03605308708820483 [29] P PIETRA, C VERDI, Convergence of the approximate free-boundary for the multidimensional one-phase Stefan problem, Pubblicazione N^\circ 440 del I A N di Pavia (1984) (to appear in Comp Mech Int J ) Zbl0622.65126 · Zbl 0622.65126 · doi:10.1007/BF00277696 [30] R RANNACHER, R SCOTT, Some optimal error estimates for piecewise linear finite element approximations Math Comp 38, 158 (1982), pp 437-445 Zbl0483.65007 MR645661 · Zbl 0483.65007 · doi:10.2307/2007280 [31] M ROSE, Numerical methods for flows through porous media I, Math Comp 40, 162 (1983), pp 435-467 Zbl0518.76078 MR689465 · Zbl 0518.76078 · doi:10.2307/2007525 [32] A VISINTIN, Sur le problème de Stefan avec flux non lineaire, Boll U M I , Anal Funz e Appl, V, 18 C, 1 (1981), pp 63-86 Zbl0471.35078 MR631569 · Zbl 0471.35078
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