Error Estimates for Full Discretization of a Model for Ostwald Ripening

Abstract

We analyze certain finite element schemes for a family of systems consisting of a Cahn–Hilliard equation coupled with several Allen–Cahn type equations, which are related to a model proposed by Fan and Chen for the evolution of Ostwald ripening in two-phase material systems. We obtain error bounds both for a semidiscrete (in time) scheme and a fully discrete scheme.

Keywords:

• Allen–Cahn equation
• Cahn–Hilliard equation
• Error bounds
• Full discretization
• Phase transitions
• Ostwald ripening

AMS Subject Classification:

• 65M15
• 35K55

Notes

¹The estimates obtained by these formal computations can be rigorously proved by considering suitable spectral Galerkin approximations c _m and θ _im (that is, by using the eigenfunctions of the Laplacian as a basis) for which the formal computations can be justified; the estimates then follow by taking the limit and using the lower semicontinuity.