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An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy. (English) Zbl 0851.65070
The authors consider the initial-boundary value problem: Find \(\{u(x, t), w(x, t)\}\) such that \[ \partial u/\partial t= \Delta w,\;w= \Phi'(u)- \gamma \Delta u\text{ in } \Omega\times (0, T),\tag{1} \] \[ u(x, 0)= u_0(x),\;\forall x\in \Omega,\;\partial u/\partial\nu= \partial w/\partial \nu= 0\text{ on } \partial \Omega\times (0, T), \] with a logarithmic free energy term \(\Phi: [- 1, +1]\to \mathbb{R}^1\), given function \(u_0(\cdot)\), and given positive constants \(\gamma\) and \(T\). \(\Omega\) is a bounded Lipschitz domain in \(\mathbb{R}^d\) \((d\leq 3)\), and \(\nu\) denotes the outward normal to \(\partial\Omega\). The weak formulation of (1) is the starting point for the finite element approximation to (1) from which a fully discrete scheme can easily be derived by backward Euler time discretization. The authors give an overview over existing results (existence, uniqueness, regularity, regularization techniques, a priori estimates, convergence). The main result (Theorem 1.3) of the paper provides an error bound for the solution \[ U(t)= U^n(t- t_{n- 1})/\Delta t+ U^{n- 1}(t_n- t)/\Delta t \] \((t\in [t_{n- 1}- t_n])\) of the fully discrete scheme using linear finite elements for the spatial discretization: \[ |u- \widehat U|^2_{L^2(0, T; H^1(\Omega))}+ |u- U|^2_{L^\infty(0, T; (H^1(\Omega))')}\leq ch, \] provided that \(\Delta t= Ch\) and \(h\) is sufficiently small, where \(\widehat U(t)= U^n\), \(t\in (t_{n- 1}, t_n)\), \(n\geq 1\). Numerical results for the one-dimensional case \((d= 1)\) complete the paper.
Reviewer: U.Langer (Linz)

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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