【摘要】 在线性有限元中,非常期待其解具有二阶渐进展开.因为,如果二阶渐进展开成立,利用理查德森外推法,可以得到更高收敛阶的解.然而,前人已经构造了一个反例.该例子表明对一维的Poisson方程,其线性有限元解的二阶渐进展开在强意义下不成立.本研究证明了对一维、二维以及三维的Poisson方程,线性有限元解在弱意义下具有期待的渐进展开.更多还原

【Abstract】 A second-order asymptotic expansion of the linear finite element solution is widely expected in the literature. If such an expansion would hold then the Richardson extrapolation can be engaged to achieve higher order of convergence. However,a counterexample is constructed showing that any expansion of this kind for the univariate Poisson equation cannot hold in the strong sense. We prove that the expected asymptotic expansion does hold weakly for 1D,2D and 3D Poisson equations.更多还原