非线性抛物方程的半隐式全离散有限元法

By the traditional approach, optimal error estimates of semi-implicit finite element methods for nonlinear parabolic equations often require certain stability conditions (time-step size restrictions). These stability conditions suggest people choose very small time-step size in practical computations, which severely increase computational cost and accumulate round-off errors. In this project, we study stability and convergence of semi-implicit fully discrete finite element methods for nonlinear parabolic equations via a new approach, by introducing a system of time-discrete parabolic equations. By this new approach, we split the error into two parts: the error due to the time discretisation of the parabolic equations and the error due to the spatial discretisation of certain elliptic equations. The first part of the error depends only on the time-step size and the second part of the error depends only on the spatial mesh size. By analysing the two parts separately, we prove that the fully discrete finite element methods are stable and possess optimal convergence rate without any restriction on the time-step size. Besides, we shall study semi-implicit fully discrete finite element methods for some nonlinear parabolic equations from mathematical physics, such as the thermistor problem, heat and moisture transport in porous media and some other equations. Optimal error estimates will be provided, which show that the proposed numerical schemes are unconditionally stable with optimal convergence rate. Numerical examples will be provided to support our theoretical results.

非线性抛物型偏微分方程的半隐式全离散有限元法的最优误差估计常常建立在一定的稳定性条件（时间步长条件）下。这些理论分析中产生的时间步长条件常常影响到实际计算，导致计算规模的严重增加和舍入误差的大量积累。在本项目中，我们将以新的误差分析方法深入系统地研究非线性抛物型偏微分方程的半隐式全离散有限元法的稳定性和收敛性。新的误差分析方法使我们能够清楚地区分出有限元解的误差中仅依赖于时间步长的部分和仅依赖于空间网格的部分。对两部分误差分别加以分析，最终我们将彻底地去除这些稳定性条件。此外，我们将把新的误差分析方法应用于电热耦合方程组、纺织多孔介质中的水汽热传导方程组等具体的数学物理方程，提出无条件稳定的半隐式全离散有限元法并给出最优误差估计。这些研究将使人们对非线性抛物方程的全离散有限元法和广泛使用的线性化半隐格式有一个新的认识，并为大规模科学计算提供理论支持。

本项目研究了多类非线性抛物型数学物理偏微分方程的线性化全离散有限元解法的稳定性和收敛性，提出并发展了一套误差分裂的分析框架，使人们对非线性抛物方程的全离散有限元解法和广泛使用的线性化半隐格式有了一个新的认识，并为大规模科学计算提供了理论支持。