二次有限元解的渐近展开与外推

在Poisson方程的求解域Ω存在一致的三角剖分,并且相邻两初始单元构成平行四边形的假设下,证明了若Poisson方程的解u属于H6(Ω),那么二次有限元的误差有h4的渐近展开.基于误差的渐近展开,可以利用h4-Richardson外推进一步提高数值解的精度阶,并且能够得到一个后验误差估计.最后,一个数值算例验证了理论分析.

Asymptotic Expansion and Extrapolation for Quadratic Finite Element

Suppose that there exists a uniform triangular partition on the domain ${\it \Omega}$, and two adjacent elements form a parallelogram, then it is proven that approximation error of quadratic finite element has asymptotic expansion with power $h^{4}$, if solution $u$ of Poisson equation belongs to $H^{6}({\it \Omega})$. By virtue of $h^{4}$-Richardson extrapolation based on the asymptotic expansion, the authors not only the accuracy order of the approximation is improved, but also a posteriori error estimation is given. Anumerical example demonstrates these theoretical results.