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


  • 1. R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, arXiv:0805.3823, 2008.
  • 2. D. A. Benson, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advectiondispersion equation, Water Resour. Res., 36 (2000), 1403–1412.
  • 3. S. Capuani, M. Palombo, A. Orlandi, et al. Spatio-temporal anomalous diffusion imaging: results in controlled phantoms and in excised human meningiomas, Magn. Reson. Imaging, 31 (2013), 359–365.
  • 4. G. R. Hernández-Labrodo, R. E. Constreas-Donayre, J. E. Collazos-Castro, et al. Subdiffision behaviour in poly(3,4-ethylenedioxythiophene): polystyrene sulfonate (PEDOT:PSS/) evidenced by electrochemical impedance spectroscopy, J. Electroanal. Chem., 659 (2011), 201–204.
  • 5. E. Scales, R. Gorenflo and F. Mainardi, Fractional calculus and continuous time-finance, Phys. A, 284 (2000), 376–384.
  • 6. R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1–77.
  • 7. J. Huang, Y. T. L. Vázquez, Converge analysis of a block-by-block method for fractional differential equations, Numer. Math. Theory Methods Appl., 5 (2012), 229–241.
  • 8. F. Liu, P. Zuang, V. Anh, et al. Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput., 191 (2007), 12–20.
  • 9. C. Tadjeran, M. M. Meerschaert, A second-order accurate numerical method for the twodimensional fractional diffusion equation, J. Comput. Phys., 220 (2007), 813–823.
  • 10.T. Breiten, V. Simoncini and M. Stoll, Low-rank solvers for fractional differential equations, ETNA, 45 (2016), 107–132.
  • 11.X. Zhao, Z. Z. Sun, Compact Crank-Nicolson schemes for a class of fractional Cattaneo equation inhomegeneous medium, J. Sci. Comput., 62 (2015), 747–771.
  • 12.N. Nie, J. Huang, W. Wang, et al. Solving spatial-fractional partial differential diffusion equations by spectral method, J. Stat. Comput. Simul., 84 (2014), 1173–1189.
  • 13.Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552.
  • 14.W. H. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47 (2008), 204–226.
  • 15.V. J. Ervin, J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods. Partial Differ. Eqs., 22 (2006), 558–576.
  • 16.Z. Zhao, C. Li, Fractional difference/finite element approximations for the time-space fractional telegraph equation, Appl. Math. Comput., 219 (2012), 2975–2988.
  • 17.W. Bu, Y. Tang and J. Yang, Galerkin finite element method for two dimensional Riesz space fractional diffusion equations, J. Comput. Phys., 276 (2014), 26–38.
  • 18.W. H. Dong, J. S. Hesthaven, Local discontinuous Galerkin methods for fractional diffusion equations, ESAIM: Math. Model. Numer. Anal., 47 (2013), 1845–1864.
  • 19.L. Qiu,W. Deng and J. S. Hesthaven, Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangular meshes, J. Comput. Phys., 298 (2015), 678–694.
  • 20.Q. Yang, I. Turner, F. Liu, et al. Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM J. Sci. Comput., 33 (2011), 1159–1180.
  • 21.K. Burrage, N. Hale and D. Kay, An efficient implicit fem scheme for fractional-in-space reactiondiffusion equations, SIAM J. Sci. Comput., 34 (2012), A2145–A2172.
  • 22.S. Bartels, R. M¨uller, Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential, Numer. Math., 119 (2011), 409–435.
  • 23.M. I. M. Copetti, C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39–65.
  • 24.H. Gomez, V. M. Calo, Y. Bazilevs, et al. Isogeometric analysis of the Cahn-Hilliard phase field model, Comput. Methods Appl. Mech. Engrg., 197 (2008), 4333–4352.
  • 25.H. Gomez, T. J. R. Hughes, Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models, J. Comput. Phys., 230 (2011), 5310–5327.
  • 26.C. M. Elliott, D. A. French, A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation, SIAM J. Numer. Anal., 26 (1989), 884–903.
  • 27.X. Feng, A. Prohl, Error analysis of a mixed finite element method for the Cahn-Hilliard equation, Numer. Math., 99 (2004), 47–84.
  • 28.M. Ilić, F. Liu, I. Turner, et al. Numerical approximation of a fractional-in-space diffusion equation (II) with nonhomogeneous boundary conditions, Frac. Calc. and App. Anal., 9 (2006), 333–349.
  • 29.M. Ilić, I. Turner, F. Liu, et al. Analytical and numerical solutions of a one-dimensional fractionalin-space diffusion equation in a composite medium, Appl. Math. Comput., 216 (2010), 2248–2262.
  • 30.N. Hale, N. J. Higham and L. N. Trefethen, Computing Aα, log(A), and related matrix functions by contour integrals, SIAM J. Numer. Anal., 46 (2008), 2505–2323.
  • 31.J. W. Cahn, J. E. Hilliard, Free energy of a nonuniform system. i. interfacialfree energy, J. Chem. Phys., 28 (1958), 258–267.
  • 32.S. M. Allen, J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1159–1180.
  • 33.G. Barles, H. M. Soner and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim., 31 (1993), 439–469.
  • 34.G. B. Mcfadden, Phase field models of solidification, Contemp. Math., 295 (2007), 107–145.
  • 35.A. Christlieb, J. Jones, B. Wetton, et al. High accuracy solutions to energy gradient flows from material science models, J. Comput. Phys., 257 (2014), 193–215.
  • 36.J. Zhu, L.-Q. Chen, J. Shen, et al. Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: Application of a semi-implicit Fourier spectral method, Phys. Rev. E, 60 (1999), 3564–3572.
  • 37.H. Gomez, A. Reali and G. Sangalli, Accurate, efficient, and (iso)geometrically flexible collocation methods for phase-field models, J. Comput. Phys., 262 (2014), 153–171.
  • 38.J.W. Barrett, J. F. Blowey and H. Garcke, Finite element approximation of the Cahn-Hilliard equation with degenerate mobility, SIAM J. Numer. Anal., 37 (2000), 286–318.
  • 39.G. Wells, E. Kuhl and K. Garikipati, A discontinuous Galerkin method for the Cahn-Hilliard equation, J. Comput. Phys., 218 (2006), 860–877.
  • 40.X. B. Feng, O. A. Karakashian, Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard euation of phase transition, Math. Comp., 76 (2007), 1093–1117.
  • 41.R. Guo, Y. Xu, Efficient solvers of discontinuous Galerkin discretization for the Cahn-Hilliard equations, J. Sci. Comput., 58 (2014), 380–408.
  • 42.Y. Xia, Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for the Cahn-Hilliard type equations, J. Comput. Phys., 227 (2007), 472–491.
  • 43.F. Liu, J. Shen, Stabilized semi-implicit spectral deferred correction methods for Allen-Cahn and Cahn-Hilliard equations, Math. Methods Appl. Sci., 38 (2013), 4564–4575.
  • 44.X. Feng, Y. Li, Analysis of symmetric interior penalty discontinuous Galerkin methods for the Allen-Cahn equation and the mean curvature flow, IMA J. Numer. Anal., (2014), 193–215.
  • 45.B. Karasözen, A. S. Filibelio˘glu, M. Uzunca and H. Y¨ucel, Energy stable discontinuous Galerkin finite element method for the Allen-Cahn equation, Int. J. Comput. Methods, In Press, 2018.
  • 46.J. Hua, P. Lin, C. Liu, et al. Energy law preserving C0 finite element schemes for phase field models in two-phase flow computations, J. Comput. Phys., 230 (2011), 7115–7131.
  • 47.J. Shen, X. Yang, Numerical approximations of Allen-Cahn and CahnHilliard equations, Discret. Contin. Dyn-A, 28 (2010), 1669–1691.
  • 48.X. Feng, H. Song, T. Tang, et al. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation, Inverse Probl. Imag., 7 (2013), 679–695.
  • 49.X. Feng, Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model two-phase fluid flows, SIAM J. Numer. Anal., 44 (2006), 1049–1072.
  • 50.E. Celledoni, V. Grimm, R. I. Mclachlan, et al. Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method, J. Comput. Phys., 231 (2012), 6770–6789.
  • 51.C. M. Elliott, A. M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30 (1993), 1622–1663.
  • 52.J. D. Eyre, An unconditionally stable one-step scheme for gradient systems, Available from:˜eyre/research/methods/
  • 53.E. V. L. Mello, O. T. S. Filho, Numerical study of the Cahn-Hilliard equation in one, two, three dimensions, Physica A, 347 (2005), 429–443.
  • 54.J. Shen, C. Wang, X. Wang, et al. Second-order convex splitting schemes for gradient flows with Enhrich-Schwoebel type energy: application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), 105–125.
  • 55.S. M. Wise, Unconditionally stable finite difference, nonlinear multigrid solutions of the Cahn-Hilliard-Hele-Shaw system of equations, J. Sci. Comput., 44 (2010), 38–68.
  • 56.J. Kim, K. Kang, J. Lowengrub, Conservative multigrid methods for Cahn-Hilliard fluids, J. Comput. Phys., 193 (2004), 511–543.
  • 57.S. Badia, F. Guill´en-Gonzales, J. V. Gutiérrez-Santacreu, Finite element approximation of nematic liquid crystal flows using a saddle-point structure, J. Comput. Phys., 230 (2011), 1686–1706.
  • 58.F. Guillén-Gonzales, G. Tierra, On linear schemes for a Cahn-Hilllard diffuse interface model, J. Comput. Phys., 234 (2013), 140–171.
  • 59.M. Ainsworth, Z. Mao, Analysis and approximation of a fractional Cahn-Hilliard equation, SIAM J. Numer. Anal., 55 (2017), 1689–1718.
  • 60.G. Akagi, G. Schimperna and A. Segatti, Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations, J. Differ. Equations, 261 (2016), 2935–2985.
  • 61.P. W. Bates, J. Han, The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equations, J. Math. Anal. Appl., 311 (2005), 289–312.
  • 62.P. Colli, S. Frigeri, M. Graselli, Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Math. Anal. Appl., 386 (2012), 428–444.
  • 63.S. Zhai, Z. Weng, X. Feng, Fast explicit operator splitting method and time-step adaptivity for fractional non-local Allen-Cahn model, Appl. Math. Model., 40 (2016), 1315–1324.
  • 64.W. Feller, On a generalization of Marcel Riesz' potentials and the semi-groups generated by them, meddelanden Lunds Universitets Matmatiska Seminarium, 1952.
  • 65.M. D. Ruiz-Medina, V. V. Anh, J. M. Angula, Fractional generalized random fields of variable order, Stoch. Anal. Appl., 22 (2004), 775–779.
  • 66.M. Ilić, F. Liu, I. Turner, et al. Numerical approximation of a fractional-in-space diffusion equation, I, Frac. Calc. Appl. Anal., 8 (2005), 323–341.
  • 67.D. N. Arnold, F. Brezzi, B. Cockburn, et al. Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749–1779.
  • 68.N. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, 2008.
  • 69.T. A. Driscoll, Improvement to the Schwarz-Christoffel toolbox for MATLAB, ACM Trans. Math. Software, 31 (2005), 239–251.
  • 70.U. M. Ascher, S. J. Ruuth, R. J. Spiteri, Implicit-explicit Runge-Kutta method for time dependent partial differential equations, Appl. Numer. Math., 25 (1997), 151–167.
  • 71.U. M. Ascher, S. J. Ruuth, T. R. Wetton, Implicit-explicit method for time dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797–823.
  • 72.H. K. Pang, H. W. Sun, Fast numerical contour integral method for fractional diffusion equations, J. Sci. Comput., 66 (2016), 41–66.
  • 73.G. Benderskaya, M. Clemens, H. De Gersem, et al. Embedded Runge-Kutta methods for field-circuit coupled problems with switching elements, IEEE Trans. Magn., 41 (2005), 1612–1615.
  • 74.P. J. van der Houwen, B. P. Sommeijer, W. Couzy, Embedded diagonally implicit Runge-Kutta algorithms on parallel computers, Math. Comput., 58 (1992), 135–159.
  • 75.J. Lang, Two-dimensional fully adaptive solutions of reaction-diffusion equations, Appl. Numer. Math., 18 (1995), 223–240.
  • 76.E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential Algebraic Problems, Springer Series in Computational Mathematics, Vol. 14, Springer Verlag, Berlin, Heidelberg, New York, 1991.
  • 77.H. G. Lee, J. Y. Lee, A semi-analytical Fourier spectral method for the Allen-Cahn equation, Comput. Math. Appl., 68 (2014), 174–184.
  • 78.X. Feng, Y. Li, A posteriori error estimates and an adaptive finite element method for the Allen-Cahn equation and the mean curvature flow, J. Sci. Comput., 24 (2005), 121–146.
  • 79.S. C. Hardy, P.W. Voorhess, Ostwald ripening in a system with a high volume fraction of coarsening phase, Metall. Mater. Trans. A, 19 (1988), 2713–2721.
  • 80.R. V. Kohn, F. Otto, Upper bounds for coarsening rates, Comm. Math. Phys., 229 (2002), 375–395.
  • 81.T. Tang, J. Yang, Implicit-explicit scheme for the Allen-Cahn equation preserves the maximum principle, J. Comput. Math., 34 (2016), 451–461.


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