# American Institute of Mathematical Sciences

June  2021, 11(2): 269-281. doi: 10.3934/naco.2020025

 1 Department of Mathematics, Sinop University, Sinop, 57000, Turkey 2 Turkish Scientific and Technological Research Council (TÜBİTAK), Ankara, 06100, Turkey

* Corresponding author: Murat Uzunca

Received  January 2020 Revised  January 2020 Published  June 2021 Early access  March 2020

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.

Citation: Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
##### References:
 [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. Strouboulis, The Finite Element Method and Its Reliability, Numerical Mathematics and Scientific Computation, Clarendon Press, 2001. [4] J. W. Barrett, J. 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. Cangiani, E. 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. Ern, A. 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. Hoppe, G. 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. Houston, C. 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. Liu, A. 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. Shen, T. 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. Solin, K. Segeth and I. Dolezel, Higher-Order Finite Element Methods, Studies in Advanced Mathematics, Chapman & Hall/CRC Press, 2003. [27] M. Uzunca, B. 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. Vasconcelos, A. 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.

show all references

##### References:
 [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. Strouboulis, The Finite Element Method and Its Reliability, Numerical Mathematics and Scientific Computation, Clarendon Press, 2001. [4] J. W. Barrett, J. 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. Cangiani, E. 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. Ern, A. 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. Hoppe, G. 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. Houston, C. 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. Liu, A. 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. Shen, T. 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. Solin, K. Segeth and I. Dolezel, Higher-Order Finite Element Methods, Studies in Advanced Mathematics, Chapman & Hall/CRC Press, 2003. [27] M. Uzunca, B. 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. Vasconcelos, A. 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.
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
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
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
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] Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154 [2] Xiaofeng Yang. Error analysis of stabilized semi-implicit method of Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 1057-1070. doi: 10.3934/dcdsb.2009.11.1057 [3] Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489 [4] Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703 [5] Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic and Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139 [6] Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127 [7] Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015 [8] Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani. Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4907-4925. doi: 10.3934/dcds.2020205 [9] Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems and Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679 [10] Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure and Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577 [11] Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319 [12] Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009 [13] Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024 [14] Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407 [15] Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077 [16] Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817 [17] Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185 [18] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319 [19] Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations and Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012 [20] Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations and Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517

Impact Factor: