A Cahn-Hilliard type equation with degenerate mobility and single-well potential. Part I: convergence analysis of a continuous Galerkin finite element discretization.

16/2016

A Cahn-Hilliard type equation with degenerate mobility and single-well potential. Part I: convergence analysis of a continuous Galerkin finite element discretization.

Wednesday 6th April 2016

Agosti, A.; Antonietti, P.F.; Ciarletta, P.; Grasselli, M.; Verani, M.

We consider a Cahn-Hilliard type equation with degenerate mobility and single-well potential of Lennard-Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. We formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution together with the convergence to the weak solution. We present simulation results in one and two space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion.