抛物型方程的若干高效有限元算法

发布时间：2018-02-03 13:27

本文关键词： 有限元方法抛物型方程算子分裂方法 Parareal方法稳定化方法　出处：《新疆大学》2017年硕士论文　论文类型：学位论文

【摘要】：本文主要以有限元方法为基础构造抛物型方程的高效数值算法.首先,针对热传导方程提出有限元算子分裂方法,将高维热传导方程分解为一系列的一维问题进行求解,从而降低求解问题的复杂性.同时给出算法的稳定性分析和误差估计,并通过数值实验表明该算法的高效性.其次,对于Allen-Cahn方程,采用经典算子分裂方法将其分解为线性子问题和非线性子问题,其中线性问题采用标准有限元方法求解,非线性问题则可以解析求解.经典算子分裂格式的求解过程简单,但是时间上的收敛阶比较低.因此,我们考虑将Allen-Cahn方程的算子分裂格式和Parareal算法结合从而构造出高阶的时间离散格式.此外,由于Allen-Cahn方程的算子分裂格式对时间步长有比较严格的限制,所以我们尝试构造稳定化的算子分裂格式.通过增加稳定化项降低数值模拟时对时间步长的要求.
[Abstract]:In this paper, based on the finite element method, the efficient numerical algorithm for the parabolic equation is constructed. Firstly, the finite element operator splitting method is proposed for the heat conduction equation. The high-dimensional heat conduction equation is decomposed into a series of one-dimensional problems to reduce the complexity of the problem, and the stability analysis and error estimation of the algorithm are given. Numerical experiments show that the algorithm is efficient. Secondly, for Allen-Cahn equation, the classical operator splitting method is used to decompose it into linear subproblem and nonlinear subproblem. The linear problem is solved by the standard finite element method, and the nonlinear problem can be solved analytically. The classical operator splitting scheme is simple, but the convergence order in time is low. We consider combining the operator splitting scheme of Allen-Cahn equation with the Parareal algorithm to construct a higher-order time discrete scheme. Because of the operator splitting scheme of Allen-Cahn equation, the time step size is restricted strictly. So we try to construct a stable operator splitting scheme and reduce the requirement of time step in numerical simulation by increasing the stabilizing term.
【学位授予单位】：新疆大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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