奇异摄动问题DG方法一致超收敛与非线性偏微分方程多解高效算法研究

项目名称： 奇异摄动问题DG方法一致超收敛与非线性偏微分方程多解高效算法研究

项目编号： No.10871066

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 数理科学和化学

项目作者： 谢资清

作者单位： 湖南师范大学

项目金额： 26万元

中文摘要： 随着科学技术的发展，带小或大参数的奇异摄动问题及具有多解的非线性偏微分方程数值方法的研究越来越引起科学和工程界的注意。前者的主要困难是边界层或内部层现象，在拟一致网格下，经典的有限元法产生的解是振荡的，且一般情况下，边界层或内部层的位置无法确定。后者的主要困难是高Morse指标解的不稳定性，其对应物理系统中的不稳定平衡态或瞬时激发态，需要设计高效算法计算分布和结构都很复杂的多解，并发展相应的数学理论。本项目的目的是利用"局部加密等级网格+带权范数估计"的方法分析DG方法求解奇异摄动问题的一致收敛和超收敛性；克服样本点非对称分布带来的困难，发展适合DG方法的后处理重构技术，产生相应的自适应网格，求解一般线性和非线性奇异摄动问题。在"搜索延拓法+新外推多网格法+非线性方程组高效解法"的模式下，研究非线性偏微分方程多解的高效算法，并在"紧性+反证"的框架下，克服问题的非凸性，建立算法的理论基础。

中文关键词： 奇异摄动问题；非线性偏微分方程；间断有限元；一致超收敛；新外推多网格法

英文摘要： With the development of science and technology, more and more attentions have been paid to the investigation on the numerical methods for the singularly perturbed problems with small or large parameters and the nonlinear partial differential equations with multiple solutions in scientific and engineering fields. The main difficulty in the former is the phenomena of boundary layers or interior layers. Under the quasi-uniform meshes, the classical finite element methods often lead to some oscillating solutions. Further, the location of the boundary layers and interior layers cannot be determined in advance. The key difficulty of the latter is the unstability of solutions with high Morse indices, which correpsond to the unstable equilibria or transient excited states in some physical systems. It is necessary to devise some efficient algorithms to compute the complicated solutions in distribution and structure and develop the corresponding mathematical theory. This project is aimed to investigate the uniform convergence and even superconvergence of DG methods for singularly perturbed problems by combining the technique of local refinement of grids and the estimate of norm with weight. Moreover, we want to solve the general linear and nonlinear singularly perturbed problems by overcoming the difficulty due to the nonsymmetry of sample points, developing some postprocessing recovery techniques suitable for DG approaches, and producing the corresponding adpaptive grids. On the other hand, we will study some efficient algorithms for nonlinear partial differential equations by combing the search extension method, new extrapolation multigrid method and the efficient algorithms for nonlinear systems of equations. Furthermore, we will establish the theoretical foundation of it by overcoming the property of non-convexity of the problems under the framework of compactness and contradition arguments.

英文关键词： Singular perturbed problems; Nonlinear partial differential equations; DG; Uniform superconvergence; New extrapolation multigrid method