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Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation

  • * Corresponding author: Murat Uzunca

    * Corresponding author: Murat Uzunca 
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  • We apply a space adaptive interior penalty discontinuous Galerkin method for solving advective Allen-Cahn equation with expanding and contracting velocity fields. The advective Allen-Cahn equation is first discretized in time and the resulting semi-linear elliptic PDE is solved by an adaptive algorithm using a residual-based a posteriori error estimator. The a posteriori error estimator contains additional terms due to the non-divergence-free velocity field. Numerical examples demonstrate the effectiveness and accuracy of the adaptive approach by resolving the sharp layers accurately.

    Mathematics Subject Classification: 65M20, 65M22, 65M50, 65M60.


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  • Figure 1.  Expanding flow: (Top) Uniform solutions, (middle) adaptive solutions and (bottom) adaptive meshes at time instances $ t = 0 $, $ t = 0.03 $ and $ t = 0.06 $ from left to right

    Figure 2.  Expanding flow: (Left) Maximum element error propagation over time, and (right) evaluation of DoFs over time; dashed line indicates the DoFs used for uniform solutions

    Figure 3.  Sheering flow: (Top) Uniform solutions, (middle) adaptive solutions and (bottom) adaptive meshes at time instances $ t = 0 $, $ t = 0.01 $ and $ t = 0.06 $ from left to right

    Figure 4.  Sheering flow: (Left) Maximum element error propagation over time, and (right) evaluation of DoFs over time; dashed line indicates the DoFs used for uniform solutions

  • [1] M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Pure and Applied Mathematics, John Wiley & Sons, Inc., 2000. doi: 10.1002/9781118032824.
    [2] D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 724-760.  doi: 10.1137/0719052.
    [3] I. Babuška and  T. StrouboulisThe Finite Element Method and Its Reliability, Numerical Mathematics and Scientific Computation, Clarendon Press, 2001. 
    [4] J. W. BarrettJ. F. Blowey and H. Garcke, Finite element approximation of a fourth order nonlinear degenerate parabolic equation, Numerische Mathematik, 80 (1998), 525-556.  doi: 10.1007/s002110050377.
    [5] A. CangianiE. H. Georgoulis and S. Metcalfe, Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems, IMA Journal of Numerical Analysis, 34 (2013), 1578-1597.  doi: 10.1093/imanum/drt052.
    [6] L. Chen, $i$FEM: An Innovative Finite Element Methods Package in MATLAB, Technical Report, Department of Mathematics, University of California, Irvine, 2008.
    [7] L. Q. Chen, Phase-field models for microstructure evolution, Annual Review of Materials Research, 32 (2002), 113-140.  doi: 10.1146/annurev.matsci.32.112001.132041.
    [8] P. Deuflhard and M. Weiser, Adaptive Numerical Solution of PDEs, De Gruyter Textbook, Walter de Gruyter, 2012. doi: 10.1515/9783110283112.
    [9] A. ErnA. F. Stephansen and M. Vohralík, Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems, Journal of Computational and Applied Mathematics, 234 (2010), 114-130.  doi: 10.1016/j.cam.2009.12.009.
    [10] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Reviews of Modern Physics, 49 (1977), 435-479.  doi: 10.1103/RevModPhys.49.435.
    [11] R. H. W. HoppeG. Kanschat and T. Warburton, Convergence analysis of an adaptive interior penalty discontinuous Galerkin method, SIAM Journal on Numerical Analysis, 47 (2008), 534-550.  doi: 10.1137/070704599.
    [12] P. HoustonC. Schwab and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 39 (2002), 2133-2163.  doi: 10.1137/S0036142900374111.
    [13] O. A. Karakashian and F. Pascal, Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems, SIAM Journal on Numerical Analysis, 45 (2007), 641-665.  doi: 10.1137/05063979X.
    [14] B. Karasözen and M. Uzunca, Time-space adaptive discontinuous Galerkin method for advection-diffusion equations with non-linear reaction mechanism, GEM - International Journal on Geomathematics, 5 (2014), 255-288.  doi: 10.1007/s13137-014-0067-z.
    [15] M. S. Khan, Phase Field Methods for Multi-Phase Flow Simulations, PhD Thesis, Technische Universitat Dortmund, Institut für Angewandte Mathematik (LS III), Dortmund, 2009.
    [16] P. Lesaint and P. A. Raviert, On a finite element for solving the neutron transport equation, mathematical aspects of finite elements in partial differential equations, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, (1974), 89–123.
    [17] W. Liu, Two Dynamical System Models on Real–World Scenarios: A Swarming Control Model and a Surface Tension Model, PhD thesis, University of California, Los Angeles, 2011.
    [18] W. LiuA. Bertozzi and T. Kolokolnikov, Diffuse interface surface tension models in an expanding flow, Comm. Math. Sci., 10 (2012), 387-418.  doi: 10.4310/CMS.2012.v10.n1.a16.
    [19] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2.
    [20] W. H. Reed and T. R. Hill, Triangular Mesh Methods for the Neutron Transport Equation, Technical Report LA-UR-73-479, Los Alomos Scientific Laboratory, Los Alomos, NM, 1973.
    [21] B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, Theory and Implementation, Frontiers in Applied Mathematics, SIAM, 2008. doi: 10.1137/1.9780898717440.
    [22] B. Rivière and M. F. Wheeler, A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems, Computers Math. with Applications, 46 (2003), 141-163.  doi: 10.1016/S0898-1221(03)90086-1.
    [23] D. Schötzau and L. Zhu, A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations, Applied Numerical Mathematics, 59 (2009), 2236-2255.  doi: 10.1016/j.apnum.2008.12.014.
    [24] J. ShenT. Tang and J. Yang, On the maximum principle preserving schemes for the generalized Allen-Cahn equation, Commun. Math. Sci., 14 (2016), 1517-1534.  doi: 10.4310/CMS.2016.v14.n6.a3.
    [25] J. Shen and X. Yang, A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities, SIAM Journal on Scientific Computing, 32 (2010), 1159-1179.  doi: 10.1137/09075860X.
    [26] P. SolinK. Segeth and  I. DolezelHigher-Order Finite Element Methods, Studies in Advanced Mathematics, Chapman & Hall/CRC Press, 2003. 
    [27] M. UzuncaB. Karasözen and M. Manguoğlu, Adaptive discontinuous Galerkin methods for non-linear diffusion-convection-reaction equations, Computers and Chemical Engineering, 68 (2014), 24-37. 
    [28] D. F. M. VasconcelosA. L. Rossa and A. L. G. A. Coutinho, A residual-based Allen-Cahn phase field model for the mixture of incompressible fluid flows, International Journal for Numerical Methods in Fluids, 75 (2014), 645-667. 
    [29] R. Verfürth, Robust a posteriori error estimates for stationary convection-diffusion equations, SIAM Journal on Numerical Analysis, 43 (2005), 1766-1782.  doi: 10.1137/040604261.
    [30] Z. Zhang and H. Tang, An adaptive phase field method for the mixture of two incompressible fluids, Computers & Fluids, 36 (2007), 1307-1318.  doi: 10.1016/j.compfluid.2006.12.003.
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