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An efficient fully-discrete local discontinuous Galerkin method for the Cahn-Hilliard-Hele-Shaw system. (English) Zbl 1349.76211
Summary: In this paper, we develop an efficient and energy stable fully-discrete local discontinuous Galerkin (LDG) method for the Cahn-Hilliard-Hele-Shaw (CHHS) system. The semi-discrete energy stability of the LDG method is proved firstly. Due to the strict time step restriction $$(\Delta t=O(\Delta x^4))$$ of explicit time discretization methods for stability, we introduce a semi-implicit time integration scheme which is based on a convex splitting of the discrete Cahn-Hilliard energy. The unconditional energy stability has been proved for this fully-discrete LDG scheme. The fully-discrete equations at the implicit time level are nonlinear. Thus, the nonlinear Full Approximation Scheme (FAS) multigrid method has been applied to solve this system of algebraic equations, which has been shown the nearly optimal complexity numerically. Numerical results are also given to illustrate the accuracy and capability of the LDG method coupled with the multigrid solver.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 76D27 Other free boundary flows; Hele-Shaw flows
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##### References:
 [1] Barrett, J. W.; Blowey, J. F., Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy, Numer. Math., 77, 1-34, (1997) · Zbl 0882.65129 [2] Barrett, J. W.; Blowey, J. F.; Garcke, H., Finite element approximation of the Cahn-Hilliard equation with degenerate mobility, SIAM J. Numer. Anal., 37, 286-318, (1999) · Zbl 0947.65109 [3] Barrett, J. W.; Blowey, J. F.; Garcke, H., On fully practical finite element approximations of degenerate Cahn-Hilliard systems, Math. Model. Numer. Anal., 35, 713-748, (2001) · Zbl 0987.35071 [4] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131, 267-279, (1997) · Zbl 0871.76040 [5] Cahn, J. W., On spinodal decomposition, Acta Metall., 9, 795-801, (1961) [6] Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput., 52, 411-435, (1989) · Zbl 0662.65083 [7] Cockburn, B.; Lin, S.-Y.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems, J. Comput. Phys., 84, 90-113, (1989) · Zbl 0677.65093 [8] Cockburn, B.; Hou, S.; Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comput., 54, 545-581, (1990) · Zbl 0695.65066 [9] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 199-224, (1998) · Zbl 0920.65059 [10] Cockburn, B.; Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35, 2440-2463, (1998) · Zbl 0927.65118 [11] Collins, C.; Shen, J.; Wise, S. M., An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system, Commun. Comput. Phys., 13, 929-957, (2013) · Zbl 1373.76161 [12] Eyre, D. J., Systems of Cahn-Hilliard equations, SIAM J. Appl. Math., 53, 1686-1712, (1993) · Zbl 0853.73060 [13] Feng, X. B.; Wise, S. M., Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation, SIAM J. Numer. Anal., 50, 1320-1343, (2012) · Zbl 1426.76258 [14] Furihata, D., A stable and conservative finite difference scheme for the Cahn-Hilliard equation, Numer. Math., 87, 675-699, (2001) · Zbl 0974.65086 [15] Guo, R.; Xu, Y., Efficient solvers of discontinuous Galerkin discretization for the Cahn-Hilliard equations, J. Sci. Comput., 58, 380-408, (2014) · Zbl 1296.65134 [16] Kay, D.; Welford, R., A multigrid finite element solver for the Cahn-Hilliard equation, J. Comput. Phys., 212, 288-304, (2006) · Zbl 1081.65091 [17] Kim, J.; Kang, K.; Lowengrub, J., Conservative multigrid methods for Cahn-Hilliard fluids, J. Comput. Phys., 193, 511-543, (2004) · Zbl 1109.76348 [18] Kim, J.; Kang, K.; Lowengrub, J., Conservative multigrid methods for ternary Cahn-Hilliard systems, Commun. Math. Sci., 2, 53-77, (2004) · Zbl 1085.65093 [19] Reed, W. H.; Hill, T. R., Triangular mesh method for the neutron transport equation, (1973), Los Alamos Scientific Laboratory Los Alamos, NM, Technical report LA-UR-73-479 [20] Sun, Z. Z., A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation, Math. Comput., 64, 1463-1471, (1995) · Zbl 0847.65056 [21] Trottenberg, U.; Oosterlee, C.; Schüller, A., Multigrid, (2005), Academic Press New York [22] Wise, S. M., Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, J. Sci. Comput., 44, 38-68, (2010) · Zbl 1203.76153 [23] Xia, Y.; Xu, Y.; Shu, C.-W., Local discontinuous Galerkin methods for the Cahn-Hilliard type equations, J. Comput. Phys., 227, 472-491, (2007) · Zbl 1131.65088 [24] Xia, Y.; Xu, Y.; Shu, C.-W., Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system, Commun. Comput. Phys., 5, 821-835, (2009) · Zbl 1364.65203 [25] Xu, Y.; Shu, C.-W., Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys., 7, 1-46, (2010) · Zbl 1364.65205 [26] Yan, J.; Shu, C.-W., A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal., 40, 769-791, (2002) · Zbl 1021.65050 [27] Yan, J.; Shu, C.-W., Local discontinuous Galerkin methods for partial differential equations with higher order derivatives, J. Sci. Comput., 17, 27-47, (2002) · Zbl 1003.65115
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