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Convergence to equilibrium for the backward Euler scheme and applications. (English) Zbl 1215.65132
This well-written paper is devoted to the stability of discretized gradient flow obtained by backward Euler scheme and $$\theta$$-scheme. In 1960s S. Lojasiewicz [cf. e.g. in: Equ. Derivees partielles, Paris 1962, Colloques internat. Centre nat. Rech. sci. 117, 87–89 (1963; Zbl 0234.57007)] proved that if $$F$$ is real analytic, then any bounded solution of the gradient flow $U'(t)=-\nabla F(U(t)),\quad t\geq 0,\tag{1}$ converges to a critical point of $$F$$ as $$t$$ tends to infinity. As the main result of this paper, it is shown that the same convergence to equilibrium is established for numerical solutions of (1) obtained by backward Euler method and $$\theta$$-method for $$\theta \in [1/2,1]$$. Similarly to the continuous-time case, the key of the proof is the so-called Lojasiewicz inequality, which also allows to obtain the optimal convergence rate to the equilibrium. In addition, extension to an infinite dimensional problem as well as applications to semilinear parabolic equations such as the Allen-Cahn and Cahn-Hilliard equations are discussed in detail.

##### MSC:
 65L07 Numerical investigation of stability of solutions 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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