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有限元后处理中超收敛计算的EEP(单元能量投影)法以及基于该法的自适应分析方法对线性ODE(常微分方程)问题的求解已经获得了全面成功,也推动了非线性ODE问题自适应求解的研究。经过研究,已经实现了一维有限元自适应分析技术从线性到非线性的跨越,该文意在对这方面的进展作一简要综述与报道。该文提出一种基于EEP法的一维非线性有限元自适应求解方法,其基本思想是通过线性化,将现有的线性问题自适应求解方法直接引入非线性问题求解,而无需单独建立非线性问题的超收敛计算公式和自适应算法,从而构成一个统一的、通用的非线性问题自适应求解算法。该文给出的数值算例表明所提出的算法高效、稳定、通用、可靠,解答可逐点按最大模度量满足用户给定的误差限,可作为先进高效的非线性ODE求解器的核心理论和算法。
The EEP (unit energy projection) method and the adaptive analysis method based on this method have been fully successful in solving the problem of linear ODE (Ordinary Differential Equations), and also promote the nonlinear ODE problem adaptive Solve the research. After research, the one-dimensional finite element adaptive analysis technology has been achieved from linear to nonlinear span, the article is intended to make a brief overview of the progress in this area and coverage. This paper proposes a one-dimensional nonlinear finite element method based on the EEP method. The basic idea of this method is to linearize the existing linear problem adaptive solution method directly into the nonlinear problem solving without the need to establish a separate non-linear The linear convergence problem is solved by the super convergent formula and the adaptive algorithm. Thus, a unified and universal adaptive algorithm for solving nonlinear problems is formulated. The numerical examples given in this paper show that the proposed algorithm is efficient, stable, universal and reliable. The proposed algorithm can meet user-specified error limits point-by-point by maximum modulus, which can be used as the core theory of advanced and efficient nonlinear ODE solvers And algorithm.