Symmetric interior penalty Galerkin method for fractional-in-space phase-field equations

Abstract

Fractional differential equations are becoming increasingly popular as a modelling tool todescribe a wide range of non-classical phenomena with spatial heterogeneities throughout the appliedsciences and engineering. However, the non-local nature of the fractional operators causes essentialdifficulties and challenges for numerical approximations. We here investigate the numerical solution offractional-in-space phase-field models such as Allen-Cahn and Cahn-Hilliard equations via the contourintegral method (CIM) for computing the fractional power of a matrix times a vector. Time discretizationis performed by the first-and second-order implicit-explicit schemes with an adaptive time-stepsize approach, whereas spatial discretization is performed by a symmetric interior penalty Galerkin(SIPG) method. Several numerical examples are presented to illustrate the effect of the fractionalpower.

Keywords

• Allen-Cahn/Cahn-Hilliard equations| fractional diffusion| contour integral method| implicit-explicit methods| discontinuous Galerkin methods