Existence and Approximate Solutions of a Model for Ostwald Ripening

Abstract

We analyze a fully discrete finite element scheme based on the backward Euler method for a system consisting of a Cahn–Hilliard and several Allen–Cahn type equations closely related to that proposed by Fan et al. for modeling Ostwald ripening in a two-phase system. As a consequence of the analysis, we prove the existence and uniqueness of solutions of the discrete problems, as well as their convergence to the solution of the original system. Error estimates are also presented.

Keywords:

• Allen–Cahn equation
• Cahn–Hilliard equation
• Existence and uniqueness
• Full discretization
• Phase transitions
• Ostwald ripening

AMS Subject Classification:

• 35K55
• 65M60

ACKNOWLEDGMENT

The first author was partially supported by FAPESP, Brazil, grant 98/15946-5.