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Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential. (English) Zbl 1241.65075

The authors study finite element error estimates for the Allen-Cahn and the Cahn-Hilliard equations with a logarithmic potential. By using a generalized Gronwall lemma, they establish an abstract error analysis for the Allen-Cahn equation and the Cahn-Hilliard equation in Section 3 and Section 4, respectively. In Section 5, they present computable and quasi-optimal error estimates in \(L^{\infty}(0, T; L^2 (\Omega))\) for the Allen-Cahn problem and in \(L^{\infty} (0, T; H^1 (\Omega)')\) for the Cahn-Hilliard problem by applying the results in Section 3 and 4. The results are proposed upon non-standard finite element methods. In section 6, they use the lowest order continuous finite elements to verify the logarithmic bounds for the time integrated principal eigenvalue \(\Lambda_{CH}\) past topological changes, and to analyze the dependence of the solution on the temperature. The extensive computations are carried out both on uniform and adaptive triangulations of the domain \(\Omega\).
The main contributions of the paper are: (1) the derivation of conditional error estimates for Cahn-Hilliard equations that are robust past topological changes, (2) the treatment of logarithmic potentials for Allen-Cahn and Cahn-Hilliard equations, (3) the derivation of quasi-optimal a posteriori error estimates in weaker norms for non-standard finite element methods, (4) the numerical verification of the conditional a posteriori error estimates, and (5) numerical experiments that indicate partial robustness also with respect to critical transition temperature.

MSC:

65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics
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[1] Alikakos N., Bates P., Chen X.: Convergence of the Cahn–Hilliard equation to the Hele-Shaw model. Arch. Rat. Mech. Anal. 128, 165–205 (1994) · Zbl 0828.35105 · doi:10.1007/BF00375025
[2] Allen S., Cahn J. W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta. Metall. 27, 1084–1095 (1979)
[3] Bartels S.: A posteriori error analysis for time-dependent Ginzburg–Landau type equations. Numer. Math. 99(4), 557–583 (2005) · Zbl 1073.65089 · doi:10.1007/s00211-004-0560-7
[4] Bartels, S.: A lower bound for the spectrum of the linearized Allen–Cahn operator near a singularity. Preprint available at http://www.bartels.ins.uni-bonn.de (2010)
[5] Barrett J.W., Blowey J.F.: An error bound for the finite element approximation of the Cahn–Hilliard equation with logarithmic free energy. Numer. Math. 72(1), 1–20 (1995) · Zbl 0851.65070 · doi:10.1007/s002110050157
[6] Barrett J.W., Blowey J.F.: Finite element approximation of the Cahn–Hilliard equation with concentration dependent mobility. Math Comp 68(226), 487–517 (1999) · Zbl 1126.65321 · doi:10.1090/S0025-5718-99-01015-7
[7] Barrett J.W., Blowey J.F., Garcke H.: Finite element approximation of the Cahn–Hilliard equation with degenerate mobility. SIAM J. Numer. Anal 37(1), 286–318 (1999) (electronic) · Zbl 0947.65109 · doi:10.1137/S0036142997331669
[8] Blowey J.F., Elliott C.M.: The Cahn–Hilliard gradient theory for phase separation with nonsmooth free energy. I. Mathematical analysis. Eur. J. Appl. Math. 2(3), 233–280 (1991) · Zbl 0797.35172 · doi:10.1017/S095679250000053X
[9] Blowey J.F., Elliott C.M.: The Cahn–Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis. Eur. J. Appl. Math. 3(2), 147–179 (1992) · Zbl 0810.35158 · doi:10.1017/S0956792500000759
[10] Bartels S., Müller R.: A posteriori error controlled local resolution of evolving interfaces for generalized Cahn–Hilliard equations. Interfaces Free Bound 12(1), 45–73 (2010) · Zbl 1404.65244 · doi:10.4171/IFB/226
[11] Bartels S., Müller R.: Quasi-optimal and robust aposteriori error estimates in L L 2) for the approximation of Allen–Cahn equations past singularities. Math. Comp. 80, 761–780 (2011) · Zbl 1215.65149
[12] Bartels S., Müller R., Ortner C.: Robust a priori and a posteriori error analysis for the approximation of Allen-Cahn and Ginzburg-Landau equations past topological changes. SIAM J. Numer. Anal. 49(1), 110–134 (2011) · Zbl 1235.65116 · doi:10.1137/090751530
[13] Baňas L., Nürnberg R.: A posteriori estimates for the Cahn–Hilliard equation with obstacle free energy. MM2AN Math. Model. Numer. Anal. 43(5), 1003–1026 (2009) · Zbl 1190.65137 · doi:10.1051/m2an/2009015
[14] Cahn J.W.: On spinodal decomposition. Acta. Metall. 9, 795–801 (1961) · doi:10.1016/0001-6160(61)90182-1
[15] Copetti M.I.M., Elliott C.M.: Numerical analysis of the Cahn–Hilliard equation with a logarithmic free energy. Numer. Math. 63(1), 39–65 (1992) · Zbl 0762.65074 · doi:10.1007/BF01385847
[16] Cahn J.W., Hilliard J.E.: Free energy of a non-uniform system. I. interfacial free energy. J. Chem. Phys. 28, 258–267 (1958) · doi:10.1063/1.1744102
[17] Chen X.: Spectrum for the Allen–Cahn, Cahn–Hilliard, and phase-field equations for generic interfaces. Commun. Partial. Diff. Eqs. 19(8), 1371–1395 (1994) · Zbl 0811.35098 · doi:10.1080/03605309408821057
[18] Caffarelli L.A., Muler N.E.: An L bound for solutions of the Cahn–Hilliard equation. Arch. Rational. Mech. Anal. 133(2), 129–144 (1995) · Zbl 0851.35010 · doi:10.1007/BF00376814
[19] de Mottoni P., Schatzman M.: Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347, 1533–1589 (1995) · Zbl 0840.35010
[20] Elliott C.M., Garcke H.: On the Cahn–Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27(2), 404–423 (1996) · Zbl 0856.35071 · doi:10.1137/S0036141094267662
[21] Elliott C.M., Songmu Z.: On the Cahn–Hilliard equation. Arch. Rational. Mech. Anal. 96(4), 339–357 (1986) · Zbl 0624.35048 · doi:10.1007/BF00251803
[22] Feng X., Prohl A.: Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Numer. Math. 94(1), 33–65 (2003) · Zbl 1029.65093 · doi:10.1007/s00211-002-0413-1
[23] Feng X., Prohl A.: Error analysis of a mixed finite element method for the Cahn–Hilliard equation. Numer. Math. 99(1), 47–84 (2004) · Zbl 1071.65128 · doi:10.1007/s00211-004-0546-5
[24] Feng X., Wu H.: A posteriori error estimates for finite element approximations of the Cahn–Hilliard equation and the Hele-Shaw flow. J. Comput. Math. 26(6), 767–796 (2008) · Zbl 1174.65035
[25] Gräser C., Kornhuber R., Sack U.: On hierarchical error estimators for time-discretized phase field models. In: Kreiss, G., Lötstedt, P., Malqvist, A., Neytcheva, M. (eds) Proceedings of ENUMATH 2009., pp. 397–406. Springer, Heidelberg (2010) · Zbl 1216.65114
[26] Kessler D., Nochetto R.H., Schmidt A.: A posteriori error control for the Allen–Cahn problem: circumventing Gronwall’s inequality. M2N Math. Model. Numer. Anal. 38(1), 129–142 (2004) · Zbl 1075.65117 · doi:10.1051/m2an:2004006
[27] Lakkis O., Makridakis, C.: Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Math. Comp. 75(256):1627–1658 (electronic, 2006) · Zbl 1109.65079
[28] Ladyzhenskaya O.A., Ural’tseva N.N.: Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York (1968)
[29] Makridakis C., Nochetto, R.H.: Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal. 41(4):1585–1594 (electronic, 2003) · Zbl 1052.65088
[30] Nochetto R.H., Verdi C.: Convergence past singularities for a fully discrete approximation of curvature-driven interfaces. SIAM J. Numer. Anal. 34(2), 490–512 (1997) · Zbl 0876.35053 · doi:10.1137/S0036142994269526
[31] Penrose O., Fife P.C.: Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Phys D 43(1), 44–62 (1990) · Zbl 0709.76001 · doi:10.1016/0167-2789(90)90015-H
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