Abstract
In this work, we propose a bi-grid scheme framework for the Allen–Cahn equation in finite element method. The new methods are based on the use of two FEM spaces, a coarse one and a fine one, and on a decomposition of the solution into mean and fluctuant parts. This separation of the scales, in both space and frequency, allows to build a stabilization on the high-mode components: the main computational effort is concentrated on the coarse space on which an implicit scheme is used while the fluctuant components of the fine space are updated with a simple semi-implicit scheme; they are smoothed without damaging the consistency. The numerical examples we give show the good stability and the robustness of the new methods. An important reduction of the computation time is also obtained when comparing our methods with fully implicit ones.
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References
Abboud H, Girault V, Sayah T (2009) A second order accuracy for a full discretized time-dependent Navier–Stokes equations by a two-grid scheme. Numer Math. https://doi.org/10.1007/s00211-009-0251-5
Abounouh M, Al Moatassime H, Chehab J-P, Dumont S, Goubet O (2008) Discrete Schrodinger equations and dissipative dynamical systems. Commun Pure Appl Anal 7(2):211–227
Allen SM, Cahn JW (1979) A microscopic theory for antiphase boundary mo- tion and its application to antiphase domain coarsening. Acta Metall Mater 27:10851095
Allen SM, Cahn JW (1972) Ground state structures in ordered binary alloys with second neighbor interactions. Acta Metall 20:423
Allen SM, Cahn JW (1973) A correction to the ground state of FCC binary ordered alloys with first and second neighbor pairwise interactions. Scr Metall 7:1261
Bank RE (1996) Hierarchical bases and the finite element. Acta Numer 5:1–43
Bank RE, Xu J (1996) An algorithm for coarsening unstructured meshes. Numer Math 73:1–36
Bartels S (2015) Numerical methods for nonlinear partial differential equations, vol 47. Springer Series in Computational Mathematics, New York
Brezinski C, Chehab J-P (1998) Nonlinear hybrid procedures and fixed point iterations. Num Funct Anal Optim 19:465–487
Brachet M, Chehab J-P (2016) Stabilized times schemes for high accurate finite differences solutions of nonlinear parabolic equations. J Sci Comput 69(3):946–982
Chehab J-P, Costa B (2003) Multiparameter schemes for evolutive PDEs. Numer Algorithms 34:245–257
Chehab J-P, Costa B (2004) Time explicit schemes and spatial finite differences splittings. J Sci Comput 20(2):159–189
Costa B, Dettori L, Gottlieb D, Temam R (2001) Time marching techniques for the nonlinear Galerkin method. SIAM J Sci. Comput. 23(1):46–65
Dubois T, Jauberteau F, Temam R (1998) Dynamic multilevel methods and the numerical simulation of turbulence. Cambridge University Press, Cambridge
Dubois T, Jauberteau F, Temam R (1998) Incremental unknowns, multilevel methods and the numerical simulation of turbulence. Comp. Meth. in Appl. Mech. and Engrg. (CMAME). Elsevier Science Publishers, Amsterdam
Elliott CM (1989) The Cahn–Hilliard Model for the kinetics of phase separation. Mathematical models for phase change problems, international series of numerical mathematics, vol 88. Birkhäuser, Boston
Emmerich H (2003) The diffuse interface approach in materials science thermodynamic,concepts and applications of phase-field models. Lecture notes in physics monographs. Springer, Heidelberg
Ern A, Guermond J-L (2004) Theory and practice of finite elements, applied mathematical science, vol 159. Springer, New York
Eyre DJ (1998) Unconditionallly stable one-step scheme for gradient systems. http://www.math.utah.edu/HrBeyre/research/methods/stable.psHrB
FreeFem++. Available from: http://www.freefem.org
He Y, Liu K-M, Sun W (2005) Multi-level spectral galerkin method for the navier-stokes problem I: spatial discretization. Numer Math 101:501–522
Jiang J, Shi J (2009) Bistability dynamics in structured ecological models. In: Cantrellz S, Cosner C, Ruan S (eds) Spatial ecology. CRC Press, Boca Raton
Lemaréchal C (1971) Une méthode de résolution de certains systèmes non linéaires bien posés. C R Acad Sci Paris Sér A 272:605–607
Li Y, Jeong D, Choi J, Lee S, Kim J (2015) Fast local image inpainting based on the Allen–Cahn model. Digital Signal Process 37:65–74
Marion M, Xu J (1995) Error estimates on a new nonlinear Galerkin method based on two-grid finite elements. SIAM J Numer Anal 32(4):1170–1184
Merlet B, Pierre M (2010) Convergence to equilibrium for the backward Euler scheme and applications. Commun Pure Appl Anal 9(3):685–702
Shen J, Yang X (2010) Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. DCDS Ser A 28:1669–1691
Xu J (1994) Some two-grids finite element methods. In: Proceedings of the 6th international conference on domain decomposition method, vol 157, pp 79–87
Acknowledgements
The authors would like to thank the anonymous referees for their valuable and constructive remarks that helped to improve the paper. This project has been partly founded with support from the National Council for Scientific Research in Lebanon and the Lebanese University; it was also supported by LAMFA, UMR CNRS 7352, at Université de Picardie Jules Verne, Amiens, France and by the Fédération de Recherche ARC, CNRS FR 3399.
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Abboud, H., Kosseifi, C.A. & Chehab, JP. A stabilized bi-grid method for Allen–Cahn equation in finite elements. Comp. Appl. Math. 38, 35 (2019). https://doi.org/10.1007/s40314-019-0781-0
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DOI: https://doi.org/10.1007/s40314-019-0781-0