CONVERGENCE TO EQUILIBRIUM FOR THE BACKWARD EULER SCHEME AND APPLICATIONS

We prove that, under natural assumptions, the solution of the backward Euler scheme applied to a gradient flow converges to an equilibrium, as time goes to infinity. Optimal convergence rates are also obtained. As in the continuous case, the proof relies on the well known Lojasiewicz inequality. We extend these results to the $\theta$-scheme with $\theta\in [1/2, 1]$, and to the semilinear heat equation. Applications to semilinear parabolic equations, such as the Allen-Cahn or Cahn-Hilliard equation, are given

[1]  Ralph Chill,et al.  On the Łojasiewicz–Simon gradient inequality , 2003 .

[2]  Charles M. Elliott,et al.  The Cahn-Hilliard Model for the Kinetics of Phase Separation , 1989 .

[3]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[4]  Morgan Pierre.  Uniform convergence for a finite-element discretization of a viscous diffusion equation , 2010 .

[5]  Mohamed Ali Jendoubi,et al.  A Simple Unified Approach to Some Convergence Theorems of L. Simon , 1998 .

[6]  L. Simon.  Asymptotics for a class of non-linear evolution equations, with applications to geometric problems , 1983 .

[7]  J. Bolte,et al.  Characterizations of Lojasiewicz inequalities and applications , 2008, 0802.0826.

[8]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[9]  Local and asymptotic analysis of the flow generated by the Cahn–Hilliard–Gurtin equations , 2006 .

[10]  J. Demailly,et al.  Analyse numérique des équations différentielles , 2006 .

[11]  Charles M. Elliott,et al.  A second order splitting method for the Cahn-Hilliard equation , 1989 .

[12]  Herbert Gajewski,et al.  A descent method for the free energy of multicomponent systems , 2006 .

[13]  Andrew M. Stuart,et al.  Model Problems in Numerical Stability Theory for Initial Value Problems , 1994, SIAM Rev..

[14]  C. M. Elliott,et al.  The viscous Cahn-Hilliard equation , 2002 .

[15]  Andrew M. Stuart,et al.  The viscous Cahn-Hilliard equation. I. Computations , 1995 .

[16]  Robert E. Mahony,et al.  Convergence of the Iterates of Descent Methods for Analytic Cost Functions , 2005, SIAM J. Optim..

[17]  On the convergence of global and bounded solutions of some evolution equations , 2007 .

[18]  Otared Kavian,et al.  Introduction à la théorie des points critiques : et applications aux problèmes elliptiques , 1993 .

[19]  M. Gurtin.  Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance , 1996 .

[20]  Alain Haraux,et al.  Rate of decay to equilibrium in some semilinear parabolic equations , 2003 .

[21]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[22]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[23]  Morgan Pierre,et al.  Stable discretizations of the Cahn-Hilliard-Gurtin equations , 2008 .

[24]  Piotr Rybka,et al.  Convergence of solutions to cahn-hilliard equation , 1999 .

[25]  A. Haraux,et al.  Convergence of Solutions of Second-Order Gradient-Like Systems with Analytic Nonlinearities , 1998 .

[26]  S. Łojasiewicz.  Ensembles semi-analytiques , 1965 .

[27]  Matthias Winter,et al.  Stationary solutions for the Cahn-Hilliard equation , 1998 .