Adaptive pseudospectral solution of a diffuse interface model

A diffuse interface type model, using an energy-based variational formulation with a free energy that is a function of the density and its gradients is presented. All of the boundary terms are retained and related to external surface forces, which can be of particular interest when considering the fluid-fluid-solid region. The numerical solution of these types of problems can be troublesome if a thin transition layer is desired. Here, Chebyshev pseudospectral methods with mesh adaptation for the solution of diffuse interface type problems are studied. A mesh adaptation algorithm based in the equidistribution principle following a continuation process is derived. In order to achieve high precision for problems exhibiting thin transition layers, a modified version of the arc-length monitor function is proposed which yields a sufficiently smooth coordinate transformation. At every step of the continuation process, a fixed number of iterations is implemented, so that the equidistribution equations are not solved completely at each step, which saves a considerable amount of computational effort. Numerical results for the static phase field model exhibiting thin transition layers are presented.