A DOUBLY SPLITTING SCHEME FOR THE CAGINALP SYSTEM WITH SINGULAR POTENTIALS AND DYNAMIC BOUNDARY CONDITIONS

Franck Davhys Reval Langa , Morgan Pierre (1)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 7348]

We propose a time semi-discrete scheme for the Caginalp phase-field system with singular potentials and dynamic boundary conditions. The scheme is based on a time splitting which decouples the equations and on a convex splitting of the energy associated to the problem. The scheme is unconditionally uniquely solvable and the energy is nonincreasing if the time step is small enough. The discrete solution is shown to converge to the energy solution of the problem as the time step tends to 0. The proof involves a multivalued operator and a monotonicity argument. This approach allows us to compute numerically singular solutions to the problem.

Mots clés: maximal monotone operator; multivalued operator; dynamic boundary conditions; Caginalp phase-field system; convex splitting; logarithmic potential

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]

Franck Davhys Reval Langa, Morgan Pierre. A DOUBLY SPLITTING SCHEME FOR THE CAGINALP SYSTEM WITH SINGULAR POTENTIALS AND DYNAMIC BOUNDARY CONDITIONS. 2019. ⟨hal-02310210⟩