三维椭圆问题 P 和 H-P Version 有限元法理论及其在工程中的应用研究

p-version 和 h-p version 有限元法是传统h-version 有限元法的进一步推广, 它们具有收敛速度快和计算精度高的优点。一维和二维的 p-version 和 h-p version 有限元法理论已经建立并且在工程和科学计算等多学科领域得到了广泛应用，但三维p-version 和h-p version 有限元法的研究近年来也才刚刚开始,取得的成果也还不多。本项目将在申请者在国外和国内的研究基础上,建立三维空间多面体域中拟一致网格上具有光滑和奇性解的椭圆问题p-version 和 h-p version 有限元法的收敛性理论，并对相关算法编制计算机程序或利用已有的有限元商业软件进行数值验证,同时研究高阶有限元法在工程中特别是在固体力学和流体力学中的应用,推动相关领域的进一步发展。

P-version and h-p version finite element method are the development of Classical h-version finite element method, they have the advantageous of faster convergence rate and higher accuracy. The theory of one and two dimensional p-version and h-p version finite element method had been established and widely used in many different fields shch as engineering and scientific computation, but the research of three dimensional p-version and h-p version finite element method just begins in recent years, and researchers only obtained a few results. This project will establish the convergence of the p-version and h-p version of the finite element method with quasi uniform meshes for elliptic problems with smooth solutions and singular solutions on polyhedral domains in three dimensions, and program relative computer software or use finite element commercial software to verify numerical results. At the same time, we will study the application of high-order finite element method in engineering practice, espicially the application of high-order finite element method in solid mechnics and fluid mechnics, and promote the development of relative fields.