A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative
A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative
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DOI：
10.1016/j.camwa.2015.09.012
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发表时间：
2015-11
期刊：
影响因子：
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通讯作者：
Yang Liu;Yanwei Du;Hong Li;Jichun Li;Siriguleng He
中科院分区：
文献类型：
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作者：
Yang Liu;Yanwei Du;Hong Li;Jichun Li;Siriguleng He
本文提出了一种基于混合有限元法的双网格算法，用于求解具有caputo型时间分数阶导数的非线性四阶反应扩散方程。我们将该问题表述为一个非线性全离散MFE系统，其中时间整数导数和分数阶导数用有限差分方法逼近，空间导数用MFE方法逼近。为了更有效地求解非线性MFE系统，提出了一种两网格算法，该算法由两步组成：首先通过非线性迭代在粗网格上求解非线性MFE系统，然后通过牛顿迭代在细网格上求解线性化MFE系统。证明了两网格格式在l2范数下的数值稳定性和最优误差估计O (k Δ 2−α+ h r+ 1+ h 2r + 2)，其中k Δ、h和h分别为时间步长、粗网格尺寸和细网格尺寸。我们实现了两网格算法，并给出了数值结果来验证我们的理论误差估计。数值试验也表明，双网格法比直接求解非线性MFE系统要有效得多。
In this article, we develop a two-grid algorithm based on the mixed finite element (MFE) method for a nonlinear fourth-order reaction–diffusion equation with the time-fractional derivative of Caputo-type. We formulate the problem as a nonlinear fully discrete MFE system, where the time integer and fractional derivatives are approximated by finite difference methods and the spatial derivatives are approximated by the MFE method. To solve the nonlinear MFE system more efficiently, we propose a two-grid algorithm, which is composed of two steps: we first solve a nonlinear MFE system on a coarse grid by nonlinear iterations, then solve the linearized MFE system on the fine grid by Newton iteration. Numerical stability and optimal error estimate O (k Δ 2− α+ h r+ 1+ H 2 r+ 2) in L 2-norm are proved for our two-grid scheme, where k Δ, h and H are the time step size, coarse grid mesh size, and fine grid mesh size, respectively. We implement the two-grid algorithm, and present the numerical results justifying our theoretical error estimate. The numerical tests also show that the two-grid method is much more efficient than solving the nonlinear MFE system directly.