  We study the numerical solution of a Cahn-Hilliard/Allen-Cahn system with
strong coupling through state and gradient dependent non-diagonal mobility
matrices. A fully discrete approximation scheme in space and time is proposed
which preserves the underlying gradient flow structure and leads to dissipation
of the free-energy on the discrete level. Existence and uniqueness of the
discrete solution is established and relative energy estimates are used to
prove optimal convergence rates in space and time under minimal smoothness
assumptions. Numerical tests are presented for illustration of the theoretical
results and to demonstrate the viability of the proposed methods.

Numerical Analysis;Numerical Analysis