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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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