自适应人工边界方法及其在非线性问题中的应用

无界区域上非线性偏微分方程的数值求解是一个既有广泛的工程应用背景，又具有挑战性的困难课题。传统的人工边界方法求解此类问题，不论非线性人工边界条件的构造还是有限元边界元耦合法数值求解非线性问题，都面临很多困难。本项目将结合自适应算法处理非线性问题和含奇性问题的优势，发展基于自适应有限元和边界元耦合法的自适应人工边界方法，并将其应用到典型的非线性外问题的数值求解中。具体研究内容如下：1. 发展非线性弹性问题的自适应有限元方法；2. 构造非线性传输问题的精确非线性人工边界条件，研究超奇异积分的自适应算法及其超收敛性；3. 研究非线性传输问题的自适应人工边界法；4. 将自适应人工边界方法用于计算简单电磁场问题。本项目将推广边界元方法和人工边界方法，研究成果不仅为一般非线性外问题的数值求解提供理论基础和参考, 而且将丰富弹性力学、电磁学、超奇异积分数值计算。

Numerical method of nonlinear partial differential equations has been very popular in science and engineering computation. It is a difficult and challenging task. The traditional artificial boundary method which based on the coupling of finite element method (FEM) and boundary element method (BEM) will be inefficient if we use the method for general nonlinear partial differential equation, not only on the construction of nonlinear artificial boundary conditions but also on the numerical methods of the nonlinear problems in bounded domain. Our research group mainly focus on the adaptive artificial boundary element methods study of nonlinear problems, which based on the coupling of adaptive finite element method and boundary element method. Our research topics are given in the following: 1.Developing the adaptive finite element method of nonlinear degenerate convex minimization problem; 2.Constructing the nonlinear artificial boundary conditions, and studying the adaptive method and superconvergence phenomenon of the supersingular integral; 3.Developing adaptive artificial boundary method based on adaptive finite element method and boundary element method for nonlinear transmission problem; 4.The adaptive artificial boundary method will be used to numerically simulate a class of electromagnetic field problem. This research will generalize the boundary element method and artificial boundary method. This proposal will not only provide benchmark references to the numerical method of nonlinear problems, but also enrich the contents of elasticity mechanics, electromagnetic mechanics, and the computation methods of supersingular integral.