一维EEP自适应技术新进展:从线性到非线性

NEW PROGRESS IN 1D SELF-ADAPTIVE ANALYSIS BASED ON EEP TECHNOLOGY:FROM LINEARITY TO NONLINEARITY

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 （Element Energy Projection） method for super-convergence calculation in the post-processing stage of FEM and the self-adaptive strategy based on the EEP technology are fully successful for linear ODEs and hence pave the way for extending to the self-adaptive analysis of nonlinear ODEs. With recent intensive studying, the technology transfer from linearity to nonlinearity in the self-adaptive analysis of lD problems has successfully been achieved and the present paper gives a brief overview and report about this progress. A new self-adaptive finite element （FE） strategy for nonlinear ODE problems was proposed. In this method, by means of linearization, the existing linear self-adaptive strategy based on the EEP method is incorporated directly into the solution of nonlinear ODEs to avoid constructing super-convergent formulae and the self-adaptive algorism for each specific and individual nonlinear problem. As a result, a general and unified self-adaptive algorism was proposed. The numerical examples show that the proposed method is highly efficient, stable, general and reliable with the results satisfying the user-preset error tolerance by maximum norm, and hence can serve as the core theory and the algorithm of an advanced and efficient FE solver for nonlinear ODEs.