Stability analysis and best approximation error estimates of discontinuous time-stepping schemes for the Allen–Cahn equation

Chrysafinos, Konstantinos¹

¹

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 2, pp. 551-583.

Résumé

Fully-discrete approximations of the Allen–Cahn equation are considered. In particular, we consider schemes of arbitrary order based on a discontinuous Galerkin (in time) approach combined with standard conforming finite elements (in space). We prove that these schemes are unconditionally stable under minimal regularity assumptions on the given data. We also prove best approximation a-priori error estimates, with constants depending polynomially upon (1/ε) by circumventing Gronwall Lemma arguments. The key feature of our approach is a carefully constructed duality argument, combined with a boot-strap technique.

Détail

DOI : 10.1051/m2an/2018071

Classification : Primary 65M12, 65M60
Keywords: Allen–Cahn equations, best approximation error estimates, discontinuous time-stepping schemes

Bibliographie

[1] G. Akrivis and C. Makridakis, Galerkin time-stepping methods for nonlinear parabolic equations. ESAIM: M2AN 38 (2004) 261–289. | Zbl | MR | mathdoc-id | DOI

[2] S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its applications to antifase domain coarsening. Acta Metall. 27 (1979) 1084–1095. | DOI

[3] N. Alikakos and G. Fusco, The spectrum of the Cahn–Hilliard operator for generic interface in higher space dimensions. Indiana Univ. Math. J. 42 (1993) 637–674. | Zbl | MR | DOI

[4] A.C. Aristotelous and O.A. Karakashian and S.M. Wise, Adaptive, second-order in time, primitive-variable discontinuous Galerkin schemes for a Cahn–Hilliard equation with a mass source. IMA J. Numer. Anal. 35 (2015) 1167–1198. | Zbl | MR | DOI

[5] S. Bartels, Numerical Methods for Nonlinear Partial Differential Equations. In Vol. 47 of Springer Series in Computational Mathematics. (2015). | Zbl | MR | DOI

[6] J. Barrett and J. Blowley, Finite element approximation of an Allen–Cahn/Cahn–Hilliard system. IMA J. Numer. Anal. 22 (2002) 11–71. | Zbl | MR | DOI

[7] J. Barrett and J. Blowley, An error bound for the finite element approximation of the Cahn–Hilliard equation with logarithmic free energy. Numer. Math. 72 (1995) 1–20. | Zbl | MR | DOI

[8] S. Bartels and R. Müller, Quasi-optimal and robust a posteriori error estimates in $L^{\infty }\left(L^2\right)$ for the approximation of the Allen–Cahn equations past singularities. Math. Comput. 80 (2011) 761–780. | Zbl | MR | DOI

[9] S. Bartels, R. Müller and C. Ortner, Robust a priori and a posteriori error analysis for the approximation of the Allen–Cahn and Ginzburg–Landau equations past topological changes. SIAM J. Numer. Anal. 49 (2011) 110–134. | Zbl | MR | DOI

[10] X. Chen, Spectrum for the Allen–Cahn. Cahn-Hiliard, and phase field equations for generic interfaces. Comm. Partial Differ. Eqs. 19 (1994) 1371–1395. | Zbl | MR | DOI

[11] X. Chen Ch. Elliot, A. Gardiner and J. Zhao, Convergence of numerical solutions to Allen–Cahn. Appl. Anal.. 69 (1998) 47–56. | Zbl | MR

[12] K. Chrysafinos and N.J. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 44 (2006) 349–366. | Zbl | MR | DOI

[13] K. Chrysafinos and N.J. Walkington, Discontinous Galerkin approximations of the Stokes and Navier–Stokes problem. Math. Comput. 79 (2010) 2135–2167. | Zbl | MR | DOI

[14] P.G. Ciarlet, The finite element method for elliptic problems. SIAM Classics Appl. Math. 40 (2002) 1–511. | Zbl | MR

[15] M. Delfour, W. Hager and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations. Math. Comput. 36 (1981) 455–473. | Zbl | MR | DOI

[16] C. Elliot and D. French, Numerical studies of the Cahn–Hilliard equation for phsae seperation. IMA J. Appl. Math. 38 (1987) 97–128. | Zbl | MR | DOI

[17] P. De Mottoni and M. Schatzman, Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347 (1995) 1533–1589. | Zbl | MR | DOI

[18] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. . SIAM J. Numer. Anal. 28 (1991) 43–77. | Zbl | MR | DOI

[19] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. II. Optimal error estimates in L_∞(L²) and L_∞(L_∞). SIAM J. Numer. Anal. 32 (1995) 706–740. | Zbl | MR | DOI

[20] K. Ericksson and C. Johnson, Adaptive finite element methods for parabolic problems IV: Nonlinear problems. SIAM J. Numer. Anal. 32 (1995) 1729–1749. | Zbl | MR | DOI

[21] D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic equations. RAIRO Model. Math. Anal. Numer. 27 (1993) 35–54. | Zbl | MR | mathdoc-id | DOI

[22] D. Estep, M. Larson and R. Williams, In Vol. 146 of Memoirs of the American Mathematical Society. (2000). viii+109. | MR

[23] L. Evans, Partial Differential Equations. AMS, Providence, RI (1998). | Zbl | MR

[24] X. Feng and Y. Li, Analysis of symmetric interior penalty discontinuous Galerkin methods for the Allen–Cahn equation and the mean curvature flow. IMA J. Numer. Anal. 35 (2015) 1622–1651. | Zbl | MR | DOI

[25] X. Feng and A. Prohl, Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Numer. Math. 94 (2003) 33–65. | Zbl | MR | DOI

[26] X. Feng and H. Wu, A posteriori error estimates and an adaptive finite element method for the Allen–Cahn equation and the mean curvature flow. J. Sci. Comput. 24 (2005) 121–146. | Zbl | MR | DOI

[27] E. Georgoulis and C. Makridakis, On a posteriori error control for the Allen–Cahn problem. Math. Meth. Appl. Sci. 37 (2014) 173–179. | Zbl | MR | DOI

[28] V. Girault and P-A. Raviart, Finite Element Methods for Navier–Stokes. Springer-Verlag, New York, NY (1986). | MR | DOI

[29] J.G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM. J. Num. Anal. 19 (1982) 275–311. | Zbl | MR | DOI

[30] P. Jamet, Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain. SIAM J. Numer. Anal. 15 (1978) 912–928. | Zbl | MR | DOI

[31] L. Ju, J. Zhang, L. Zhu and Q. Du, Fast explicit integration factor methods for semilinear parabolic equations. J. Sci. Comput. 62 (2015) 431–455. | Zbl | MR | DOI

[32] D. Kessler, R. Nochetto and A. Schmidt, A posteriori error control for the Allen–Cahn problem circumventing Gronwall’s inequality. ESAIM: M2AN 38 (2004) 129–142. | Zbl | MR | mathdoc-id | DOI

[33] P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, edited by C. De Boor. In: Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, New York, NY (1974) 89–123. | Zbl | MR | DOI

[34] D. Leykekham and B. Vexler, Discrete maximal parabolic regularity for Galerkin finite element methods. Numer. Math. 135 (2017) 923–952. | Zbl | MR | DOI

[35] M. Luskin and R. Rannacher, On the smoothing property of the galerkin method for parabolic equations. SIAM J. Numer. Anal. 19 (1982) 93–113. | Zbl | MR | DOI

[36] D. Meidner and B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part I: problems without control constraints. SIAM J. Control. Optim. 47 (2008) 1150–1177. | Zbl | MR | DOI

[37] J. Shen and X. Yang, Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete Contin. Dyn. Syst. 28 (2010) 1669–1691. | Zbl | MR | DOI

[38] I. Smears, Robust and efficient preconditioners for the discontinuous Galerkin time-stepping method. IMA J. Numer. Anal. 37 (2016) 1961–1985. | Zbl | MR

[39] R. Temam, Infinite Difemsional Dynamical Systems in Mechanics and Physics. In Vol. 68 of Applied Mathematical Sciences. Springer-Verlag, Berlin (1997). | Zbl | MR

[40] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Spinger-Verlag, Berlin (1997). | Zbl | MR | DOI

[41] N.J. Walkington, Compactness properties of the DG and CG time stepping schemes for parabolic equations. SIAM J. Numer. Anal. 47 (2010) 4680–4710. | Zbl | MR | DOI

[42] J. Yang, Q. Du and W. Zhang, Uniform Lp-bound of the AllenCahn equation and its numerical discretization. Inter. J. Numer. Anal. Model. 15 (2018) 213–228. | MR | Zbl

[43] E. Zeidler, Nonlinear Functional Analysis and its Applications, II/B Nonlinear Monotone Operators. Springer-Verlag, New York, NY (1990). | Zbl | MR

[44] J. Zhang and Q. Du, Numerical studies of discrete approximation to the Allen–Cahn equation in sharp interface limit. SIAM J. Sci. Comput. 31 (2009) 3042–3063. | Zbl | MR | DOI