特征值问题协调/非协调有限元后验误差分析

本文研究对称椭圆特征值问题的有限元后验误差估计,包括协调元和非协调元,具有下列特色:(1)对协调/非协调元建立了有限元特征函数uh的误差与相应的边值问题有限元解的误差在局部能量模意义下的恒等关系式,该边值问题的右端为有限元特征值λh与uh的乘积,有限元解恰好为uh.从而边值问题有限元解在能量模意义下的局部后验误差指示子,包括残差型和重构型后验误差指示子,成为有限元特征函数在能量模意义下的局部后验误差指示子.(2)讨论了协调有限元特征函数的基于插值后处理的梯度重构型后验误差估计,对有限元特征函数的导数得到了最大模意义下的渐近准确局部后验误差指示子.

A posteriori error analysis of conforming/nonconforming finite elements

This paper discusses a posteriori error estimates of finite element method for symmetric elliptic eigenvalue problems, including conforming and non-conforming finite element. Special work of this paper are as follows： （1） Both for conforming and non-conforming finite element method, we establishes the identity relation between the error of finite element eigenfunction Uh and the error of finite element solution of the corresponding boundary value problem in the sense of local energy norm. The term in the right-hand side of the boundary value problem is the product of finite element eigenvalue ）kh and uh. And the finite element solution is just Uh. Then the local a posteriori error estimators, including recovery type and residual type, of finite element solution of the boundary value problem in the sense of energy norm become the local a posteriori error estimators of finite element eigenfunction in the sense of energy norm. （2） We studies the a posteriori error estimation of gradient reconstruction type of conforming finite element eigenfunction based on post-processing interpolation, and obtains the asymptotically exact local a posteriori error estimators of the derivative of finite element eigenfunction in the sense of maximum norm.