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An error estimate for a finite-element scheme for a phase field model. (English) Zbl 0801.65091
The paper deals with Caginalp’s phase-field model, consisting of two coupled nonlinear parabolic equations of the type $$\partial u/ \partial t + \partial \varphi/ \partial t - \Delta u = f$$ and $$\partial \varphi/ \partial t - \Delta \varphi = g(\varphi) + 2u$$ with $$g$$ being the gradient of a two-well potential. The paper proposes a fully discrete finite element scheme, using the backward Euler formula in time and linear finite elements with a numerical integration in space. An optimal convergence rate in the $$L^ \infty(0,T; L^ 2 (\Omega))$$ norm is demonstrated without any condition on the (here omitted) physical coefficients appearing in the system. Some numerical examples are briefly reported, too.

MSC:
 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35K55 Nonlinear parabolic equations
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