二维有限元线法自适应分析的若干新进展

NEW PROGRESS IN SELF-ADAPTIVE ANALYSIS OF TWO-DIMENSIONAL FINITE ELEMENT METHOD OF LINES

有限元线法(FEMOL)是一种优良的半解析、半离散方法,将其比拟为广义一维问题,遂可将一维有限元中十分成功的单元能量投影(EEP)超收敛算法以及基于该法的自适应求解方法推广到二维有限元线法分析中,至今已在二维Poisson方程和弹性力学平面问题中取得了令人满意的进展。该文旨在报道这些进展和成果。该文简要介绍了线法的EEP超收敛计算以及相应的自适应求解策略,整套方法思路清晰、高效可靠,可以对任意几何区域上的问题,按最大模度量给出逐点满足事先给定的误差限的位移解答。该文给出充足的数值结果用以展示整套算法的有效性和可靠性。

The finite element method of lines（FEMOL） is a general and powerful semi-discretized method for BVPs.By viewing it as a generalized one-dimensional method,the well-developed Element Energy Projection（EEP） method for super-convergence computation in one-dimensional FEM can readily be extended to the case of two-dimensional FEMOL.In addition,the successful self-adaptive strategy in one-dimensional FEM can also be extended to the two-dimensional FEMOL analysis.By now,a series of satisfactory progress has been made in the two-dimensional problems of the Poisson Equation and the plane elasticity.The present paper intends to give a brief report on the recent progress and some numerical results.The paper briefly describes the idea of the super-convergent EEP method and corresponding self-adaptive strategy for two-dimensional FEMOL analysis,which forms a clean,simple,effective and reliable algorithm that can adaptively produce FEMOL results on arbitrary geometric domains with the displacement accuracy point-wisely satisfying the user specified error tolerance in max-norm.Sufficient and representative numerical examples are given to demonstrate the effectiveness and reliability of the proposed algorithm.