ABSTRACT

In this paper, we present a two-grid scheme of expanded mixed finite element method combined with two second-order time discretization schemes for semilinear parabolic integro-differential equations and give a detailed convergence analysis. On the coarse grid space, we first use the Crank–Nicolson scheme to solve the original nonlinear problem at the first time step, then we utilize the Leap-Frog scheme at the rest time levels. Next, we make use of the known coarse mesh solution and Taylor expansion to infer the solution on the fine mesh space. Thus, we only need to solve the nonlinear problem once on the coarse grid space of the two-grid scheme. Finally, a numerical example is presented to verify the effectiveness of the proposed two-grid scheme.

摘要

在本文中，我们提出了一种扩展混合有限元法的双网格方案，结合了半线性抛物型积分微分方程的两个二阶时间离散化方案，并给出了详细的收敛性分析。在粗网格空间上，我们首先在第一个时间步使用 Crank-Nicolson 方案来解决原始非线性问题，然后在休息时间级别使用 Leap-Frog 方案。接下来，我们利用已知的粗网格解和泰勒展开来推断细网格空间上的解。因此，我们只需要在两网格方案的粗网格空间上求解一次非线性问题。最后，给出了一个数值例子来验证所提出的两网格方案的有效性。