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

Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation

Murat Uzunca 1,, and  Ayşe Sarıaydın-Filibelioǧlu 2,

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.

Keywords: Advective Allen-Cahn equation, discontinuous Galerkin method, Rothe's method, adaptivity.
Mathematics Subject Classification: 65M20, 65M22, 65M50, 65M60.
Citation: Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & 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.  Google Scholar

[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.  Google Scholar

[3] I. Babuška and T. Strouboulis, The Finite Element Method and Its Reliability, Numerical Mathematics and Scientific Computation, Clarendon Press, 2001.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[6]

L. Chen, $i$FEM: An Innovative Finite Element Methods Package in MATLAB, Technical Report, Department of Mathematics, University of California, Irvine, 2008. Google Scholar

[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.  Google Scholar

[8]

P. Deuflhard and M. Weiser, Adaptive Numerical Solution of PDEs, De Gruyter Textbook, Walter de Gruyter, 2012. doi: 10.1515/9783110283112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26] P. Solin, K. Segeth and I. Dolezel, Higher-Order Finite Element Methods, Studies in Advanced Mathematics, Chapman & Hall/CRC Press, 2003.   Google Scholar
[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3] I. Babuška and T. Strouboulis, The Finite Element Method and Its Reliability, Numerical Mathematics and Scientific Computation, Clarendon Press, 2001.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[6]

L. Chen, $i$FEM: An Innovative Finite Element Methods Package in MATLAB, Technical Report, Department of Mathematics, University of California, Irvine, 2008. Google Scholar

[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.  Google Scholar

[8]

P. Deuflhard and M. Weiser, Adaptive Numerical Solution of PDEs, De Gruyter Textbook, Walter de Gruyter, 2012. doi: 10.1515/9783110283112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26] P. Solin, K. Segeth and I. Dolezel, Higher-Order Finite Element Methods, Studies in Advanced Mathematics, Chapman & Hall/CRC Press, 2003.   Google Scholar
[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
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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
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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
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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
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