论文标题
半线性抛物线方程的有效指数积分元素方法
Efficient Exponential Integrator Finite Element Method for Semilinear Parabolic Equations
论文作者
论文摘要
在本文中，我们提出了一种有效的指数积分元素方法，用于求解矩形域中的半连接抛物线方程。所提出的方法首先使用具有连续多线性矩形基函数的有限元近似进行模型方程的空间离散化，然后采用明确的指数runge-kutta方法，以便将产生的半污物系统的时间整合以产生全偏差的数值解决方案。在某些规律性假设下，以$ h^1 $ norm测量的错误估计成功得出了一个和两个RK阶段的提议方案。更明显的是，可以用正交矩阵同时对所提出的方法的质量和系数矩阵进行对角线，该基质提供了基于张量产品光谱分解和快速傅立叶变换的快速溶液过程。还进行了两个和三个维度的各种数值实验，以验证理论结果并证明该方法的出色性能。
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.
