AIMS Mathematics, 2018, 3(1): 66-95. doi: 10.3934/Math.2018.1.66

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

1 Technische Universität Chemnitz, Faculty of Mathematics, Reichenhainer Strasse 41, 09126 Chemnitz, Germany
2 Institute of Applied Mathematics, Middle East Technical University, 06800 Ankara, Turkey

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.
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