The least squares spectral element method for the Cahn–Hilliard equation

The problem of numerically resolving an interface separating two different components is a common problem in several scientific and engineering applications. One alternative is to use phase field or diffuse interface methods such as the Cahn–Hilliard (C–H) equation, which introduce a continuous transition region between the two bulk phases. Different numerical schemes to solve the C–H equation have been suggested in the literature. In this work, the least squares spectral element method (LS-SEM) is used to solve the Cahn–Hilliard equation. The LS-SEM is combined with a time–space coupled formulation and a high order continuity approximation by employing C11p-version hierarchical interpolation functions both in space and time. A one-dimensional case of the Cahn–Hilliard equation is solved and the convergence properties of the presented method analyzed. The obtained solution is in accordance with previous results from the literature and the basic properties of the C–H equation (i.e. mass conservation and energy dissipation) are maintained. By using the LS-SEM, a symmetric positive definite problem is always obtained, making it possible to use highly efficient solvers for this kind of problems. The use of dynamic adjustment of number of elements and order of approximation gives the possibility of a dynamic meshing procedure for a better resolution in the areas close to interfaces.