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.

p 和 h-p 版本的有限元解的直接和逆近似以及在三维问题、非线性问题和基尔霍夫板问题中的应用。

• 批准号: RGPIN-2014-04642
• 负责人: Guo, Benqi
• 金额: $ 1.31万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635158
• 关键词: Direct; inverse; approximation; finite; element

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.

近几十年来，随着计算机技术的迅速发展，数值模拟在解决工程、科学、金融和卫生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，高阶（p和h-p）有限元法提供了精度高、计算成本低的可靠解。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，数值模拟具有更大的意义，但由于多面体域的几何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困难得多。多面体域问题的p和h-p版本的最佳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版有限元解的逆逼近是三维问题中尚未触及的问题，它可以根据计算数据表明解的规律性，从而提高p和h-p有限元自适应算法的效率。我的研究将对三维有限元逼近理论和实际的科学/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的破坏。腐蚀工程中常用的一种技术是所谓的天主教保护系统，它是用非线性诺伊曼边界条件下的线性椭圆方程来建模的。在该系统中，施加外加电流来控制被保护工程结构表面部分的电势。电势的精确数值解对电磁保护至关重要。由于Neumann条件在由线性型变为非线性极化函数的变动点处是不连续的，因此在该处出现奇点，严重影响了有限元解的全局正则性和收敛性。我的研究将研究可数加权Sobolev空间中变化点附近的奇点，并设计一种与几何网格相关的高精度hp有限元方法。显然，该研究将对腐蚀工程以及一般的非线性分析产生重大影响。基尔霍夫板是板问题的典型模型，其特点是具有多种边界条件的双调和方程。已知解在多边形板的顶点处具有严重的r^s型奇异性，奇异性为s >1。分析了一般准均匀网格上具有C0和C1连续性的p和h-p有限元解的收敛性。有几个关键问题需要处理：(1)在单个元素的Sobolev h ^2范数中测量的奇异函数在p和h上的逼近误差的最优界；(2)构造全局c1连续分段多项式，且不影响全局H^2范数的最优误差界；(3)奇异解的不符合p和h-p有限元。本文的研究进展将为平板问题的数值模拟增添新的知识。