Efficient Exponential Integrator Finite Element Method for Semilinear Parabolic Equations

Jianguo Huang, L. Ju, Y. Xu
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引用次数: 1

Abstract

In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model equation using the finite element approximation with continuous multilinear rectangular basis functions, and then takes the explicit exponential Runge-Kutta approach for time integration of the resulting semi-discrete system to produce fully-discrete numerical solution. Under certain regularity assumptions, error estimates measured in $H^1$-norm are successfully derived for the proposed schemes with one and two RK stages. More remarkably, the mass and coefficient matrices of the proposed method can be simultaneously diagonalized with an orthogonal matrix, which provides a fast solution process based on tensor product spectral decomposition and fast Fourier transform. Various numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the excellent performance of the proposed method.
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半线性抛物型方程的有效指数积分有限元法
在本文中,我们提出了一种有效的指数积分有限元法来求解矩形区域上的一类半线性抛物型方程。该方法首先采用具有连续多线性矩形基函数的有限元近似对模型方程进行空间离散化,然后采用显式指数龙格-库塔方法对得到的半离散系统进行时间积分,得到全离散的数值解。在一定的正则性假设下,成功地导出了具有一个和两个RK阶段的方案的H^1 -范数的误差估计。更值得注意的是,该方法的质量矩阵和系数矩阵可以同时与正交矩阵对角化,提供了基于张量积谱分解和快速傅里叶变换的快速求解过程。通过二维和三维的数值实验验证了理论结果,证明了所提方法的优良性能。
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