Research article

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

  • 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.

    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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  • 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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    [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. Müller, 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. Filibelioğglu, M. Uzunca and H. Yücel, 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: www.math.utah.edu/~eyre/research/methods/stable.ps.
    [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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