Superconvergent Recovery of Raviart–Thomas Mixed Finite Elements on Triangular Grids

• Published in: Journal of scientific computing Vol. 81; no. 3; pp. 1882 - 1905
• Main Authors: Bank, Randolph E., Li, Yuwen
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2019; Springer Nature B.V
• Subjects: Algorithms; Approximation; Computational Mathematics and Numerical Analysis; Exact solutions; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Norms; Operators; Parallelograms; Theoretical; Mildly structured grids; Second order elliptic equations; Mixed methods; 65N50; Superconvergence; 65N30; Raviart–Thomas elements
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-019-01068-0

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 ( 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.