In this paper, we propose and analyze a second order accurate in time, masslumped mixed finite element scheme for the Cahn-Hilliard equation with a logarithmic Flory-Huggins energy potential. The standard backward dif...In this paper, we propose and analyze a second order accurate in time, masslumped mixed finite element scheme for the Cahn-Hilliard equation with a logarithmic Flory-Huggins energy potential. The standard backward differentiation formula (BDF)stencil is applied in the temporal discretization. In the chemical potential approximation,both the logarithmic singular terms and the surface diffusion term are treatedimplicitly, while the expansive term is explicitly updated via a second-order Adams-Bashforth extrapolation formula, following the idea of the convex-concave decompositionof the energy functional. In addition, an artificial Douglas-Dupont regularizationterm is added to ensure the energy dissipativity. In the spatial discretization, the masslumped finite element method is adopted. We provide a theoretical justification of theunique solvability of the mass lumped finite element scheme, using a piecewise linearelement. In particular, the positivity is always preserved for the logarithmic argumentsin the sense that the phase variable is always located between -1 and 1. In fact, thesingular nature of the implicit terms and the mass lumped approach play an essentialrole in the positivity preservation in the discrete setting. Subsequently, an unconditionalenergy stability is proven for the proposed numerical scheme. In addition, theconvergence analysis and error estimate of the numerical scheme are also presented.Two numerical experiments are carried out to verify the theoretical properties.展开更多
In this paper we propose and analyze a second order accurate numericalscheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the l...In this paper we propose and analyze a second order accurate numericalscheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order AdamsBashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme,which ensures the positivity-preserving property, i.e., the numerical value of the phasevariable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special formof the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearizedstability analysis. A few numerical results, including both the constant-mobility andsolution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.展开更多
基金NSFC(No.12071090)the National Key R&D Program of China(No.2019YFA0709502)+2 种基金Z.R.Zhang is partially supported by NSFC No.11871105 and Science Challenge Project No.TZ2018002C.Wang is partially supported by the NSF DMS-2012269S.M.Wise is partially supported by the NSF DMS-1719854,DMS-2012634.
文摘In this paper, we propose and analyze a second order accurate in time, masslumped mixed finite element scheme for the Cahn-Hilliard equation with a logarithmic Flory-Huggins energy potential. The standard backward differentiation formula (BDF)stencil is applied in the temporal discretization. In the chemical potential approximation,both the logarithmic singular terms and the surface diffusion term are treatedimplicitly, while the expansive term is explicitly updated via a second-order Adams-Bashforth extrapolation formula, following the idea of the convex-concave decompositionof the energy functional. In addition, an artificial Douglas-Dupont regularizationterm is added to ensure the energy dissipativity. In the spatial discretization, the masslumped finite element method is adopted. We provide a theoretical justification of theunique solvability of the mass lumped finite element scheme, using a piecewise linearelement. In particular, the positivity is always preserved for the logarithmic argumentsin the sense that the phase variable is always located between -1 and 1. In fact, thesingular nature of the implicit terms and the mass lumped approach play an essentialrole in the positivity preservation in the discrete setting. Subsequently, an unconditionalenergy stability is proven for the proposed numerical scheme. In addition, theconvergence analysis and error estimate of the numerical scheme are also presented.Two numerical experiments are carried out to verify the theoretical properties.
基金This work is supported in part by the grants NSFC 12071090(W.Chen)NSF DMS-2012669(C.Wang)+2 种基金NSFC 11871159Guangdong Provincial Key Laboratory for Computational Science and Material Design 2019B030301001(X.Wang)NSF DMS-1719854,DMS-2012634(S.Wise).C.Wang also thanks the Key Laboratory of Mathematics for Nonlinear Sciences,Fudan University,for the support.
文摘In this paper we propose and analyze a second order accurate numericalscheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order AdamsBashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme,which ensures the positivity-preserving property, i.e., the numerical value of the phasevariable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special formof the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearizedstability analysis. A few numerical results, including both the constant-mobility andsolution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.
