发展型方程的高性能各向异性非协调有限元方法研究

项目名称： 发展型方程的高性能各向异性非协调有限元方法研究

项目编号： No.11271340

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 石东洋

作者单位： 郑州大学

项目金额： 60万元

中文摘要： 研究发展型方程的各向异性非协调有限元的构造、理论分析及数值计算的一般框架；重点解决好自由度少、精度高的低阶非协调元新模式对非线性方程诸如热传导对流方程、反应扩散方程、Navier-Stokes方程、积分微分方程及结构特殊的Maxwell方程等有难度问题的应用；研究各向异性任意三角形及四边形剖分下可以导出相应的最优的误差估计、超逼近、超收敛及后验估计等结果的条件；探索并提出各向异性元的稳定化方法、加罚方法、二重网格法及辛算法等新方法，提升各向异性有限元研究各个方面的数学品位。由于我们较早在国内开展这一独具特色且有挑战性的工作，国际上在这方面的相关报道也很少，其创新性和突破性进展对丰富和发展非协调有限元的内容有重要的理论意义和应用价值。

中文关键词： 高性能；各向异性；非协调元；发展型方程；数值算法

英文摘要： The project aims to give the general framework of anisotropic nonconforming finite element's construction, theoretical analysis and numerical calculation to time-dependent equations; we focus on the applications of new low order nonconforming finite element schemes with less degrees of freedom and high accuracy to nonlinear equations such as heat conduction convection equations, reaction diffusion equations, Navier-Stokes equations, itegro-differential equations and Maxwell's equations which have special structures; investigating the conditions which can result in the corresponding optimal error estimates, superclose, superconvergence and posterior estimation results for anisotropic arbitrary triangular and quadrilateral meshes; exploring and proposing some new methods of the anisotropic finite elements, such as stabilization methods, penalty methods, two-grid methods，symplectic methods and so on for enhancing the mathematical taste of anisotropic finite element analysis on various aspects. As we have been carrying out this unique and challenging work in the country earlier and there are few international reports on this direction, its innovative and breakthrough results may enrich and develop the nonconforming finite element contents, which have important theoretical significance and application value.

英文关键词： high-performance；anisotropy；nonconforming elements；time-dependent equations；numerical algorithms