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Structure preserving discretization of Allen-Cahn type problems modeling the motion of phase boundaries. (English) Zbl 1471.65134

This article discusses the numerical approximation of a class of Allen-Cahn type problems modeling the motion of phase interfaces.The approach relies on a conforming Galerkin approximation for the space discretization which characterizes smooth solutions of the problem. Well-posedness of the resulting semi-discretization is established and the energy decay along discrete solution trajectories is deduced. The discretization in time is made by an adapted implicit time stepping scheme. Various numerical experiments are included to support the theoretical findings.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
74N20 Dynamics of phase boundaries in solids
74H10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
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