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Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations. (English) Zbl 1203.76153
Summary: We present an unconditionally energy stable and solvable finite difference scheme for the Cahn-Hilliard-Hele-Shaw (CHHS) equations, which arise in models for spinodal decomposition of a binary fluid in a Hele-Shaw cell, tumor growth and cell sorting, and two phase flows in porous media. We show that the CHHS system is a specialized conserved gradient-flow with respect to the usual Cahn-Hilliard (CH) energy, and thus techniques for bistable gradient equations are applicable. In particular, the scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step-size. Owing to energy stability, we show that the scheme is stable in the $$L_{s}^{\infty}(0,T;H_{h}^{1})$$ norm, and, assuming two spatial dimensions, we show in an appendix that the scheme is also stable in the $$L_{s}^{2}(0,T;H_{h}^{2})$$ norm. We demonstrate an efficient, practical nonlinear multigrid method for solving the equations. In particular, we provide evidence that the solver has nearly optimal complexity. We also include a convergence test that suggests that the global error is of first order in time and of second order in space.

##### MSC:
 76S05 Flows in porous media; filtration; seepage 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76M20 Finite difference methods applied to problems in fluid mechanics 65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs 76D27 Other free boundary flows; Hele-Shaw flows 80A20 Heat and mass transfer, heat flow (MSC2010)
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##### References:
 [1] Bertozzi, A., Esedoglu, S., Gillette, A.: Inpainting of binary images using the Cahn-Hilliard equation. IEEE Trans. Image Process. 16, 285–291 (2007) · Zbl 1279.94008 · doi:10.1109/TIP.2006.887728 [2] Bramble, J.: A second order finite difference analog of the first biharmonic boundary value problem. Numer. Math. 9, 236–249 (1966) · Zbl 0154.41105 · doi:10.1007/BF02162087 [3] Brinkman, H.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. AI, 27–34 (1949) · Zbl 0041.54204 [4] Cahn, J.: On spinodal decomposition. Acta Metall. 9, 795 (1961) · doi:10.1016/0001-6160(61)90182-1 [5] Cahn, J., Hilliard, J.: Free energy of a nonuniform system. i. interfacial free energy. J. Chem. Phys. 28, 258 (1958) · doi:10.1063/1.1744102 [6] Elliot, C., Stuart, A.: The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30, 1622–1663 (1993) · Zbl 0792.65066 · doi:10.1137/0730084 [7] Eyre, D.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. In: Bullard, J.W., Kalia, R., Stoneham, M., Chen, L. (eds.) Computational and Mathematical Models of Microstructural Evolution, vol. 53, pp. 1686–1712. Materials Research Society, Warrendale (1998) [8] Feng, X.: Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44, 1049–1072 (2006) · Zbl 1344.76052 · doi:10.1137/050638333 [9] Feng, X., Wise, S.: Analysis of a Fully Discrete Finite Element Approximation of a Darcy-Cahn-Hilliard Diffuse Interface Model for the Hele-Shaw Flow (in preparation) · Zbl 1426.76258 [10] Furihata, D.: A stable and conservative finite difference scheme for the Cahn-Hilliard equation. Numer. Math. 87, 675–699 (2001) · Zbl 0974.65086 · doi:10.1007/PL00005429 [11] Hu, Z., Wise, S., Wang, C., Lowengrub, J.: Stable and efficient finite-difference nonlinear-multigrid schemes for the phase-field crystal equation. J. Comput. Phys. 228, 5323–5339 (2009) · Zbl 1171.82015 · doi:10.1016/j.jcp.2009.04.020 [12] Kay, D., Welford, R.: A multigrid finite element solver for the Cahn-Hilliard equation. J. Comput. Phys. 212, 288–304 (2006) · Zbl 1081.65091 · doi:10.1016/j.jcp.2005.07.004 [13] Kay, D., Welford, R.: Efficient numerical solution of Cahn-Hilliard-Navier Stokes fluids in 2d. SIAM J. Sci. Comput. 29, 2241–2257 (2007) · Zbl 1154.76033 · doi:10.1137/050648110 [14] Kim, J., Kang, K., Lowengrub, J.: Conservative multigrid methods for Cahn-Hilliard fluids. J. Comput. Phys. 193, 511–543 (2003) · Zbl 1109.76348 · doi:10.1016/j.jcp.2003.07.035 [15] Lee, H., Lowengrub, J., Goodman, J.: Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration. Phys. Fluids 14, 492–513 (2002) · Zbl 1184.76316 · doi:10.1063/1.1425843 [16] Lee, H., Lowengrub, J., Goodman, J.: Modeling pinchoff and reconnection in a Hele-Shaw cell. II. Analysis and simulation in the nonlinear regime. Phys. Fluids 14, 514–545 (2002) · Zbl 1184.76317 · doi:10.1063/1.1425844 [17] Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D 179, 211–228 (2003) · Zbl 1092.76069 · doi:10.1016/S0167-2789(03)00030-7 [18] Lowengrub, J., Truskinovsky, L.: Cahn-Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A 454, 2617–2654 (1998) · Zbl 0927.76007 · doi:10.1098/rspa.1998.0273 [19] Shinozaki, A., Oono, Y.: Spinodal decomposition in a Hele-Shaw cell. Phys. Rev. A 45, R2161–R2164 (1992) · doi:10.1103/PhysRevA.45.R2161 [20] Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic Press, New York (2005) [21] Vollmayr-Lee, B., Rutenberg, A.: Fast and accurate coarsening simulation with an unconditionally stable time step. Phys. Rev. E 68, 066,703 (2003) · doi:10.1103/PhysRevE.68.066703 [22] Wang, C., Wang, X., Wise, S.: Unconditionally stable schemes for equations of thin film epitaxy. Discrete Cont. Dyn. Sys. A 28, 405–423 (2010) · Zbl 1201.65166 · doi:10.3934/dcds.2010.28.405 [23] Wang, C., Wise, S.: An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. (in review) · Zbl 1230.82005 [24] Wise, S., Lowengrub, J., Cristini, V.: An adaptive algorithm for simulating solid tumor growth using mixture models. Math. Comput. Model. (in review) · Zbl 1211.65123 [25] Wise, S., Lowengrub, J., Frieboes, H., Cristini, V.: Three-dimensional multispecies nonlinear tumor growth–I model and numerical method. J. Theor. Biol. 253, 524–543 (2008) · Zbl 1398.92135 · doi:10.1016/j.jtbi.2008.03.027 [26] Wise, S., Wang, C., Lowengrub, J.: An energy stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal. 47, 2269–2288 (2009) · Zbl 1201.35027 · doi:10.1137/080738143 [27] Zheng, X., Wise, S., Cristini, V.: Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method. Bull. Math. Biol. 67, 211–259 (2005) · Zbl 1334.92214 · doi:10.1016/j.bulm.2004.08.001
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