Stokes方程组Hood-Taylor元的分裂外推

考虑拟一致矩形网格上Stokes方程组Hood-Taylor元的多参数渐近误差展开和分裂外推.在每个单元上用Bramble-Hilbert引理确定微分方程精确解与有限元插值之间积分式的主项.由连续性条件相邻两个单元上其主项的某些部分可以相互抵消,经求和后,得到整个求解区域上的主项.对该主项引入辅助问题并利用Stokes问题解的正则性理论给出精确解与有限元插值间的一个误差渐近展开式.有限元解经插值后处理和分裂外推后,与通常的误差估计相比,收敛速度提高了一阶.

The Multi-Parameter Extrapolation of the Hood-Taylor Elements for the Stokes Problem

A multi-parameter asymptotic error expansion and extrapolation of the Hood-Taylor elements for the Stokes problem is considered on the piecewise uniform rectangular meshes. The main term of the error between the exact solution and the finite element interpolating function is determined by Bramble-Hilbert lemma on the individual finite element. A part of the main term of the error on two adjacent finite elements can be cancelled by continuity, and thus the main term on the whole domain is obtained by summation. By introducing an auxiliary problem, the asymptotic error expansion can be achieved by the regularity results of the Stokes problem. Compared with the general error estimate, the multi-parameter extrapolation based on such an expansion increases the rate of convergence by one order.