Analysis of a Discontinuous Galerkin Finite Element discretization of a degenerate Cahn-Hilliard equation with a single-well potential
Wednesday 5th July 2017
This work concerns the construction and the convergence analysis of a Discontinuous Galerkin Finite Element approximation of a Cahn-Hilliard type equation with degenerate mobility and single-well singular potential of Lennard-Jones type. This equation has been introduced in literature as a diffuse interface model for the evolution of solid tumors. Differently from the Cahn-Hilliard equation analyzed in the literature, in this model the singularity of the potential does not compensate the degeneracy of the mobility at zero by constraining the solution to be strictly positive. In previous works a finite element approximation with continuous elements of the problem has been developed by the author and co- authors. In the latter case, the positivity of the solution is enforced through a discrete variational inequality, which is solved only on active nodes of the triangulation where the degenerate operator can be inverted. Moreover, a lumping approximation of the L2 scalar product is introduced in the formulation in order to select the solutions with a moving support with finite speed of velocity from the unphysical solutions with fixed support. As a consequence of this approximation, the order of convergence of the method is lowered down with respect to the case of the classical Cahn-Hilliard equation with constant mobility. In the present discretization with discontinuous elements, the concept of active nodes is delocalized to the concept of active elements of the triangulation and no lumping approximation of the mass products is needed to select the physical solutions. The well posedness of the discrete formulation is shown, together with the convergence to the weak solution. Different algorithms to solve the discrete variational inequality, based on iterative solvers of the associated complementarity system, are derived and implemented. Simulation results in two space dimensions are reported in order to test the validity of the proposed algorithms, in which the dynamics of the spinodal decomposition and the evolution behaviour in the coarsening regime are studied. Similar results to the ones obtained in standard phase ordering dynamics are found, which highlight nucleation and pattern formation phenomena and the evolution of single domains to steady state with constant curvature. Since the present formulation does not depend on the particular form of the potential, but it’s based on the fact that the singularity set of the potential and the degeneracy set of the mobility do not coincide, it can be applied also to the degenerate CH equation with smooth potential.