An adaptive div-curl-grad least-squares finite element method for the Cahn-Hilliard equation

This paper develops a new least-squares finite element approach for the efficient approximation of solutions of the Cahn-Hilliard equation. We discretize in time with either implicit backward Euler or Crank-Nicolson and use a regularized Newton iteration for the nonlinearity. The innermost iteration is reformulated as a coupled first-order div-curl-grad system, which is efficiently solved by a least-squares minimization approach using standard Lagrange finite elements for each unknown. Theoretical findings include results on the optimal finite element convergence of the variational problem, convergence of the nonlinear iteration, error bounds for the time integration schemes, and discrete asymptotic mass conservation and energy dissipation in the least-squares sense. The basic iterative approach is expanded to include adaptivity in both time and space. Numerical examples are provided to confirm the theoretical results and to demonstrate effectiveness of the proposed method for a range of test problems.