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.

近几十年来，随着计算机技术的飞速发展，数值模拟在解决工程、科学、金融、卫生等各个领域的实际问题中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。有限元方法是当今数值模拟中最常用的数值方法，而高阶(p和h-p)有限元方法以其高精度和低计算量提供了可靠的解。我的研究将集中在三维p和h-p有限元及其在非线性问题和板问题中的应用。我们生活的现实世界是三维的，三维模型准确地反映了现实世界中物理问题的本质。因此，数值模拟具有更重要的意义，而且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也比简化为一维和二维的模型要困难得多。对于多面体区域上的问题，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。对于三维问题，p-型有限元解的逆逼近一直是一个未被触及的问题，它可以根据计算数据指示解的规律性，从而提高p-p和h-p有限元自适应算法的效率。本文的研究将对有限元逼近理论和实际的三维科学/工程计算产生重要的影响。腐蚀是一种与环境发生的电化学反应，对船舶、桥梁、金属集装箱等工程结构造成严重破坏。腐蚀工程中常用的一种技术是所谓的阴极保护系统，它由一个带有非线性诺伊曼边界条件的线性椭圆型方程来模拟。在该系统中，施加外加电流来控制工程结构表面被保护部分的电位。准确的电势数值解对于阴极保护是至关重要的。由于Neumann条件在Neumann条件从线型转变为非线性极化函数的变化点处是不连续的，因此会出现奇异性，严重影响有限元解的整体正则性和收敛。我的研究将研究可数加权Sobolev空间中变点附近的奇异性，并设计一种结合几何网格的高精度h-p有限元方法。显然，该研究将对腐蚀工程和一般的非线性分析产生重大影响。Kirchhoff板是板问题的一个典型模型，其特征是具有不同边界条件的双调和方程。已知解在多边形板的顶点具有与S和gt；1严重的r^S型奇性。在一般的拟均匀网格上，分析了具有C0和C1连续性的p和h-p有限元解的收敛问题。需要解决的几个关键问题是：(1)奇异函数在单个单元上按Sobolev H^2范数的逼近误差在p和h上的最优界；(2)在不影响整体H^2范数最优误差界的情况下构造整体连续的分段多项式；(3)奇异解的C0元非协调p和h-p有限元。本文研究的进展将为板问题的数值模拟增添新的知识。

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.