Superconvergence for finite element approximation of a convection-diffusion equation using graded meshes

DURAN, RICARDO GUILLERMO; LOMBARDI, ARIEL LUIS; PRIETO, MARIANA INES

IMA JOURNAL OF NUMERICAL ANALYSIS

OXFORD UNIV PRESS

Lugar: Oxford; Año: 2012 vol. 32 p. 511 - 533

0272-4979

In this paper we analyse the approximation of a model convection?diffusion equation by standard bilinear finite elements using the graded meshes introduced in Durán & Lombardi (2006, Finite element approximation of convection?diffusion problems using graded meshes. Appl. Numer. Math., 56, 1314:1325). Our main goal is to prove superconvergence results of the type known for standard elliptic problems, namely, that the difference between the finite element solution and the Lagrange interpolation of the exact solution, in the ε-weighted H1-norm, is of higher order than the error itself. The constant in our estimate depends only weakly on the singular perturbation parameter. As a consequence of the superconvergence result we obtain optimal order error estimates in the L2-norm. Also we show how to obtain a higher order approximation by a local postprocessing of the computed solution.