Direct and inverse approximation for finite element solutions of the p and h-p versions and applications to three-dimensional propblem, nonlinear problems and Kirchhoff plate problem.

随着近几十年来计算机技术的迅速发展，数值模拟在解决工程、科学、金融和健康科学等各个领域的实际问题方面发挥着极其重要的作用。数值模拟涉及数值方法、误差分析、计算解的评估、算法和应用。有限元法（FEM）是当今最常用的数值模拟方法，而高阶（p和h-p）FEM提供了可靠的解，具有高精度和低计算成本。我的研究将集中在三维p和h-p有限元法及其在非线性问题和板问题中的应用。 我们生活的真实的世界是三维的，三维模型精确地反映了真实的世界中物理问题的本质。由于多面体区域几何形状的复杂性和解的各向异性奇异性，三维模型的数值模拟比简化为一维和二维模型的数值模拟更有意义，也更困难。多面体域上问题的p和h-p形式的最优误差估计是非常困难的，几十年来一直是一个具有挑战性的问题。p型有限元解的逆逼近是三维问题中一个尚未涉及的问题，它可以根据计算数据来指示解的规律性，从而提高p型和h-p型有限元自适应算法的效率。本文的研究将对三维有限元近似理论和实际科学/工程计算产生重要影响。 腐蚀是一种与环境发生的电化学反应，对船舶、桥梁、金属容器等工程结构造成严重破坏。腐蚀工程中常用的技术是所谓的阴极保护系统，其由具有非线性Neumann边界条件的线性椭圆方程建模。在该系统中，施加外加电流以控制工程结构的表面上的待保护部分上的电位。电位的精确数值解是阴极保护的基础。由于Neumann条件在从线性极化函数到非线性极化函数的转换点处是不连续的，奇异性出现在那里，严重影响了有限元解的全局正则性和收敛性.本论文的主要工作是研究可数加权Sobolev空间中变点附近的奇异性，并设计一种与几何网格相结合的高精度h-p有限元方法。显然，这一研究对腐蚀工程以及一般的非线性分析具有重要的意义。 Kirchhoff板是板问题的一种典型模型，其特征在于具有各种边界条件的双调和方程。已知该解在多边形板的顶点处具有r^s-型的严重奇异性，其中s >1。在一般的准均匀网格上，分析了具有C 0和C1连续性的p和h-p有限元解的收敛性。本文主要研究了以下几个关键问题：（1）奇异函数在Sobolev H^2范数下的逼近误差在p和h上的最优界;（2）构造一个全局C1-连续的分段多项式而不牺牲全局H^2范数下的最优误差界;（3）奇异解的非协调p和h-p有限元。本文的研究成果将为薄板问题的数值模拟提供新的知识。

Numerical Simulation plays an extremely important role for solving practical problems arising from various fields of engineering, science, finance, and health science as computer technology has rapidly developed in the recent decades. Numerical simulation involves numerical methods, error analysis, assessment of computed solutions, algorithms, and applications. The finite element method (FEM) is today's most adopted numerical method for numerical simulation, and the high-order (p and h-p) FEM provides reliable solutions with high accuracy and lower computational cost. My proposed research will focus on the p and h-p FEM in three dimensions and their applications to nonlinear problems and plate problems. The real world we are living is three-dimensional, and three dimensional models reflect precisely the nature of physical problems in the real world. Therefore Numerical simulation is of greater significance, and is also much more difficult for three-dimensional models than those simplified to one and two dimensions due to the complexity of geometry of polyhedral domains and anisotropic singularities of solutions. The optimal error estimation of the p and h-p version for problems on polyhedral domains is extremely difficult, and has been a challenging issue for decades. The inverse approximation of FE solution of p-version has been an untouched issue so far for three dimensional problems, which could indicate regularity of the solution based on the computed date to improve the efficiency of adaptive algorithms of p and h-p FEM. My proposed research will have important impact on the approximation theory of FEM and practical scientific/engineering computations in three dimensions. Corrosion is an electrochemical reaction with its environment, which causes severe damage to engineering structures such as ships, bridges, and metal containers. A commonly used technique in corrosion engineering is a so-called catholic protection system, which is modeled by a linear elliptical equation with a nonlinear Neumann boundary condition. In this system, impressed current is applied to control the electrical potential on the portion of surface of the engineering structure to be protected. The accurate numerical solution of the electrical potential is essential for the catholic protection. Since the Neumann condition is discontinuous at the changing point where the Neumann condition changes from a linear type to a nonlinear polarization functions, the singularity occurs there, and severely affects the global regularity and convergence of the FE solutions. My proposed research will investigate singularity near the changing point in a countably weighted Sobolev spaces and design a highly accurate h-p finite element method associated with geometric meshes. Obviously, the research will be of great impact on corrosion engineering as well as to the nonlinear analysis generally. Kirchhoff plate is a typical model of the plate problem, which is characterized by a biharmonic equation with various boundary conditions. It is known that the solutions have severe singularity of r^s-type with s >1 at vertices of the polygonal plate. The convergence of the p and h-p FE solutions with C0 and C1 continuity on a general quasi-uniform mesh will be analyzed. There are several key issues to be dealt with:(1) Optimal bound of approximation error in p and h for the singular function measured in the Sobolev H^2-norm on individual element;(2) Construction of a globally C1-continuous and piecewise polynomial without compromising the optimal error bound in the global H^2-norm;(3) Nonconforming p and h-p FEM with C0-elements for singular solution. The progress of the my proposed research will add new knowledge to numerical simulation for the plate problems.