非线性抛物方程的高效两层网格三步格式

发布时间：2018-11-01 12:51

【摘要】：一直以来,流体力学总是很多科学家研究的重要课题.大多数人都比较关注多孔介质中的流体运动,它是一种很复杂的物理运动,与它相应的数学模型也很复杂.因为反应扩散方程中含有两个变量:速度和压力,因此很多研究者通常运用混合有限元方法来研究该方程,我们知道该方程组离散以后得到的方程组一般是非线性的.对于求解非线性问题,两层网格算法是很多研究者热爱的课题.非线性抛物方程是非常典型的非线性偏微分方程,应用很广泛.已经有很多研究者应用各种两层网格算法对求解压缩系数为线性情况的反应扩散方程的数值解进行了研究,而对于这种压缩系数为非线性项的研究比较少,本课题就是基于扩张混合有限元方法和两层网格算法,用两层网格三步格式求解压缩系数为非线性项的非线性抛物方程.在本文中,我们在陈艳萍教授和陈罗平所研究的求解非线性抛物方程的两层网格两步格式的基础上,用两层网格三步格式来求解非线性抛物方程的数值解.我们所说的两步格式,通常是指先对非线性方程组在粗网格上求解,再对将其线性化以后的线性方程组在细网格上求解.而我们得到的两种三步格式,第一种格式是在原来的两步格式之后在细网格上进行校正,与第一种格式不同的是第二种格式要求我们要在粗网格上再做一次校正.在分析证明过程中,我们主要利用的是混合有限元的超收敛性,最终我们发现,对于这两种格式,只要我们选取的粗网格的步长H分别满足H=O(h1/4)或H=O(h1/3)时,我们所构造的算法就可以对混合有限元方法的解进行最优逼近.
[Abstract]:Fluid mechanics has always been an important subject for many scientists. Most people pay more attention to the fluid movement in porous media, which is a very complex physical movement, and the corresponding mathematical model is also very complex. Because there are two variables in the reaction-diffusion equation: velocity and pressure, many researchers usually use the hybrid finite element method to study the equation. We know that the equations obtained after the discretization of the equations are generally nonlinear. For solving nonlinear problems, two-layer mesh algorithm is a topic that many researchers love. Nonlinear parabolic equation is a typical nonlinear partial differential equation, which is widely used. Many researchers have used various two-layer mesh algorithms to study the numerical solutions of the reaction-diffusion equations whose compressibility coefficients are linear, but less research has been done on the nonlinear terms of the compressibility coefficients. Based on the extended mixed finite element method and the two-layer mesh algorithm, the nonlinear parabolic equation with nonlinear compressibility coefficient is solved by using the two-layer mesh three-step scheme. In this paper, on the basis of the two-layer grid two-step scheme for solving nonlinear parabolic equation studied by Professor Chen Yanping and Chen Luoping, we use the two-layer grid three-step scheme to solve the numerical solution of nonlinear parabolic equation. The two-step scheme is usually used to solve the nonlinear equations on rough grids and then to linearize the linear equations on fine meshes. The first one is corrected on fine mesh after the original two-step scheme. The second format, unlike the first one, requires us to make a correction on the coarse grid again. In the process of analysis and proof, we mainly use the superconvergence of mixed finite element. Finally, we find that for these two schemes, as long as the step size H of the coarse mesh is satisfied with H0 (H1 / 4) or H0 (H1 / 3), respectively, The algorithm we constructed can approach the solution of the hybrid finite element method optimally.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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