[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

A discrete technique of the Schwarz alternating method is presented in this last chapter, to combine the Ritz-Galerkin and finite element methods. This technique is well suited for solving singularity problems in parallel, and requires a little more computation for large overlap of subdomains. The convergence rate of the iterative procedure, which depends upon overlap of subdomains, will be studied. Also a balance strategy will be proposed to couple the iteration number with the element size used in the FEM. For the crack-infinity problem of singularity the total CPU time by the technique in this chapter is much less than that by the nonconforming combination in Chapter 12.

Schwarz Alternating Method

In this paper we present a second order accurate (in time) energy stable numerical scheme for the Cahn-Hilliard (CH) equation, with a mixed finite element approximation in space. Instead of the standard second order Crank-Nicolson methodology, we apply the implicit backward differentiation formula (BDF) concept to derive second order temporal accuracy, but modified so that the concave diffusion term is treated explicitly. This explicit treatment for the concave part of the chemical potential ensures the unique solvability of the scheme without sacrificing energy stability. An additional term Aτ∆(uk+1−uk) is added, which represents a second order Douglas-Dupont-type regularization, and a careful calculation shows that energy stability is guaranteed, provided the mild condition A ≥ 1 16 is enforced. In turn, a uniform in time H1 bound of the numerical solution becomes available. As a result, we are able to establish an l∞(0, T ;L2) convergence analysis for the proposed fully discrete scheme, with full O(τ2+h2) accuracy. This convergence turns out to be unconditional; no scaling law is needed between the time step size τ and the spatial grid size h. A few numerical experiments are presented to conclude the article.

A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation

Classical dynamical density functional theory (DDFT) is one of the cornerstones of modern statistical mechanics. It is an extension of the highly successful method of classical density functional t...

Classical dynamical density functional theory: from fundamentals to applications

An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation

In this article, we discuss the nonlinear stability and convergence of a fully discrete Fourier pseudospectral method coupled with a specially designed second-order time-stepping for the numerical solution of the “good” Boussinesq equation. Our analysis improves the existing results presented in earlier literature in two ways. First, a convergence for the solution and convergence for the time-derivative of the solution are obtained in this article, instead of the convergence for the solution and the convergence for the time-derivative, given in De Frutos, et al., Math Comput 57 (1991), 109–122. In addition, we prove that this method is unconditionally stable and convergent for the time step in terms of the spatial grid size, compared with a severe restriction time step restriction required by the proof in De Frutos, et al., Math Comput 57 (1991), 109–122.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 202–224, 2015

A Fourier pseudospectral method for the “good” Boussinesq equation with second‐order temporal accuracy

Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

In this paper, we study a novel second-order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for the higher oder in time temporal discretization is how to ensure an unconditional energy stability and an efficient numerical implementation. We propose a general framework for designing the higher order in time numerical scheme with unconditional energy stability by using the BDF method with constant coefficient stabilized terms. Based on the unconditional energy stability property, we derive an $L^\infty_h (0,T; H_{h}^2)$ stability for the numerical solution and provide an optimal the convergence analysis. To deal with the 4-Laplacian solver in an $L^{2}$ gradient flow at each time step, we apply an efficient preconditioned steepest descent algorithm and preconditioned nonlinear conjugate gradient algorithm to solve the corresponding nonlinear system. Various numerical simulations are present to demonstrate the stability and efficiency of the proposed schemes and slovers.

A second-order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable and unconditionally energy stable. At each time step, this scheme leads to a system of nonlinear equations that can be efficiently solved by a nonlinear multigrid solver. Owing to the energy stability, we derive an \begin{document}$\ell^2 (0, T; H_h^3)$\end{document} stability of the numerical scheme. To overcome the difficulty associated with the convection term \begin{document}$\nabla · (\phi \mathit{\boldsymbol{u}})$\end{document} , we perform an \begin{document}$\ell^∞ (0, T; H_h^1)$\end{document} error estimate instead of the classical \begin{document}$\ell^∞ (0, T; \ell^2)$\end{document} one to obtain the optimal rate convergence analysis. In addition, various numerical simulations are carried out, which demonstrate the accuracy and efficiency of the proposed numerical scheme.

Wenqiang Feng

Papers

An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation

A Fourier pseudospectral method for the “good” Boussinesq equation with second‐order temporal accuracy

Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

A second-order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

A Second Order Energy Stable Scheme for the Cahn-Hilliard-Hele-Shaw Equations