Superconvergence analysis of a two-grid method for an energy-stable Ciarlet-Raviart type scheme of Cahn-Hilliard equatio
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Superconvergence analysis of a two-grid method for an energy-stable Ciarlet-Raviart type scheme of Cahn-Hilliard equation Qian Liu1 · Dongyang Shi1 Received: 20 March 2019 / Accepted: 4 October 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper, superconvergence analysis of a mixed finite element method (MFEM) combined with the two-grid method (TGM) is presented for the Cahn-Hilliard (CH) equation for the first time. In particular, the discrete energy-stable Ciarlet-Raviart scheme is constructed with the bilinear element. By use of the high accuracy character of the element, the superclose estimates are deduced for both of the traditional MFEM and of the TGM. Crucially, the main difficulty brought by the coupling of the unknowns is dealt with by some technical methods. Furthermore, the global superconvergent results are achieved by interpolation postprocessing skill. Numerical results illustrate that the proposed TGM is very effective and its computing cost is almost one-third of that of the traditional FEM without loss of accuracy. Keywords CH equation · MFEM · TGM · Superconvergent estimates
1 Introduction In this paper, we consider the Cahn-Hilliard (CH) problem in 2D: ⎧ −1 ⎨ ut + (γ u − γ f (u)) = 0, (X, t) ∈ × (0, T ], ∂(γ u−γ −1 f (u)) ∂u = 0, = 0, (X, t) ∈ ∂ × (0, T ], ∂n ⎩ ∂n u(X, 0) = u0 (X), with the following form of energy [1]: γ 1 2 2 2 E(u) = (u − 1) + |∇u| dX, 2 4γ
Dongyang Shi
shi [email protected]; [email protected] 1
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China
(1.1)
(1.2)
Numerical Algorithms
where ⊂ R 2 is a rectangular domain with boundary ∂ parallel to the axes. The phase equilibria are represented by u ≈ ±1. T > 0 is a fixed constant, and f (u) = F (u), F (u) = 14 (u2 − 1)2 . γ is a positive constant, which represents the interaction length. n is the unit outward vector over ∂. The problem (1.1) was originally introduced by Cahn and Hilliard [2, 3] to describe the phenomena of complicated phase separation in a solid, where only two different phases can exist stably. It is a challenging fourth-order, nonlinear parabolic type partial differential equation (PDE). For more physical background of the CH problems and related equations, we refer to [4–7] and the references therein. Recently, some studies have been devoted to theoretical analysis of the CH problems (see [8–11]). Due to the nonlinearity term of f (u), developing efficient numerical algorithms for problem (1.1) has become a significant manner for understanding the phase model, such as the spectral Galerkin method [12, 13], the finite difference method [14, 15], the finite volume method [16], and the FEMs [17–25]. Particularly, [17] set up a nonconforming FEM with Morley element and derived optimal error estimate for (1.1). Liu et al. [18] proposed a two-level method for the traditional FEM for (1.1) with P5 element, where the corresponding finite element space involved a large number of degrees of freedom from the computatio
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