Superconvergence for triangular order $\text{k=1}$ Raviart–Thomas mixed finite elements and for triangular standard quadratic finite element methods

Abstract

We will prove, for a model problem, that on regular families of uniform triangulations, the vector variable of the order $\text{k=1}$ Raviart–Thomas type mixed finite element method, is superconvergent with respect to Fortin interpolation. For lowest order $\text{k=0}$ this was already proved in (Brandts, 1994). As a side product of the present analysis, we obtain similar results for the gradient of the standard quadratic finite element method, also with respect to Fortin interpolation.

Although the use of Fortin interpolation instead of Lagrange interpolation in the setting of standard finite elements is somewhat unusual, it turns out that the superconvergence for standard quadratic elements with respect to Lagrange interpolation, proved in (Goodsell and Whiteman, 1991), is a direct corollary of it. As a result, the post-processing scheme that was developed in (Goodsell and Whiteman, 1991) to raise the approximation order of the gradient of the standard finite element approximation, can be adapted to improve the approximation quality of the mixed finite element vector variable in a similar fashion.

The Fortin interpolation approach results moreover in $\text{L}{}^2\text{(Ω)}$-superconvergence for the scalar variable.