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.