改进的预条件最速下降法求解P-Laplacian方程.pdfVIP

Improved Preconditioned Descent Algorithms for P-Laplacian Candidate Lijun Chen Supervisor Professor Yunqing Huang College Institute of Mathematics and Computer Science Program Computational Mathematics Specialization Numerical Methods for PDE:Theory and Application Degree Master of Science University Xiangtan University Date April 15th, 2015 III 摘 要 P-Laplacian方程是退化非线性系统中的典型问题，被应用于物理学等的研 究，故研究它的数值解具有重要的意义。 本论文主要包含以下几个方面的内容： 第一章介绍了p-Laplacian方程的研究背景及其现状和一些准备知识。 第二章通过算例比较了预条件最速下降法和FR-PRP混合共轭梯度算法的数 值效果，发现当p 较大时，FR-PRP混合共轭梯度算法的效率更高；而p 取1.1附近 的值时，预条件最速下降法的表现更好。研究了参数ϵ 的选择对数值效果的影响， 从很多数值实验发现ϵ取10−5 ∼ 10−3 能使算法效果达到最佳。在预条件最速下降 法的基础上，提出了自适应选择固定步长的预条件最速下降法，并通过数值实验 验证了算法的可行性。 关键词：p-Laplacian方程，预条件最速下降法，共轭梯度算法，自适应选择固定步长 IV Abstract The p-Laplacian equation is one of the typical examples of degenerate non-linear systems, which widely occurs in the study of physics. Then, it is very important to research its numerical solution. This paper mainly includes the following contents. The first chapter introduces the research backgroud and current status of the p- Laplacian equation, and some preliminary knowledge. The second chapter compares the efficiency of precondioned steepest descen- t method and FR-PRP conjugate gradient methods through numerical example. Finding that FR-PRP conjugate gradient methods has higher efficiency when the value of p is big. However, precondioned steepest descent method performs better when the value of p is near 1.1. Also, the choice of parameter ϵ is studied in its influence on numer- ical results, through many numerical experiments finding t