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

Martin Stoll; Hamdullah Yücel; Martin Stoll; Hamdullah Yücel

doi:10.3934/Math.2018.1.66

AIMS Mathematics

2018, Volume 3, Issue 1: 66-95. doi: 10.3934/Math.2018.1.66

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Research article

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

Martin Stoll ,
Hamdullah Yücel ^,

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

Received: 29 January 2018 Accepted: 05 February 2018 Published: 01 March 2018

Fractional differential equations are becoming increasingly popular as a modelling tool to describe a wide range of non-classical phenomena with spatial heterogeneities throughout the applied sciences and engineering. However, the non-local nature of the fractional operators causes essential difficulties and challenges for numerical approximations. We here investigate the numerical solution of fractional-in-space phase-field models such as Allen-Cahn and Cahn-Hilliard equations via the contour integral method (CIM) for computing the fractional power of a matrix times a vector. Time discretization is performed by the first-and second-order implicit-explicit schemes with an adaptive time-step size 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 fractional power.
- Allen-Cahn/Cahn-Hilliard equations,
- fractional diffusion,
- contour integral method,
- implicit-explicit methods,
- discontinuous Galerkin methods
Citation: Martin Stoll, Hamdullah Yücel. Symmetric interior penalty Galerkin method for fractional-in-space phase-field equations[J]. AIMS Mathematics, 2018, 3(1): 66-95. doi: 10.3934/Math.2018.1.66

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Abstract

Fractional differential equations are becoming increasingly popular as a modelling tool to describe a wide range of non-classical phenomena with spatial heterogeneities throughout the applied sciences and engineering. However, the non-local nature of the fractional operators causes essential difficulties and challenges for numerical approximations. We here investigate the numerical solution of fractional-in-space phase-field models such as Allen-Cahn and Cahn-Hilliard equations via the contour integral method (CIM) for computing the fractional power of a matrix times a vector. Time discretization is performed by the first-and second-order implicit-explicit schemes with an adaptive time-step size 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 fractional power.

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