Superconvergent recovery of Raviart--Thomas mixed finite elements on triangular grids

TL;DR

This paper proves superconvergence properties of Raviart--Thomas mixed finite elements on triangular grids, introducing postprocessing operators that enhance solution accuracy in elliptic problems.

Contribution

It establishes supercloseness and super-approximation results for Raviart--Thomas elements, leading to superconvergence of postprocessed solutions on mildly structured meshes.

Findings

Supercloseness of finite element solutions in $H(div)$-norm.

Development of local least squares postprocessing operators.

Superconvergence of postprocessed solutions in $L^2$-norm.

Abstract

For the second lowest-order Raviart--Thomas mixed method, we prove that the canonical interpolant and finite element solution for the vector variable in elliptic problems are superclose in the $H(\text{div})$-norm on mildly structured meshes, where most pairs of adjacent triangles form approximate parallelograms. We then develop a family of postprocessing operators for Raviart--Thomas mixed elements on triangular grids by using the idea of local least squares fittings. Super-approximation property of the postprocessing operators for the lowest and second lowest order Raviart--Thomas elements is proved under mild conditions. Combining the supercloseness and super-approximation results, we prove that the postprocessed solution superconverges to the exact solution in the $L^2$-norm on mildly structured meshes.