General principles of superconvergence in Galerkin finite element methods. (English) Zbl 0902.65046

Wahlbin, Lars B.

Křížek, M. (ed.) et al., Finite element methods. Superconvergence, post-processing, and a posteriori estimates. 1st conference, University of Jyväskylä, Finland. New York, NY: Marcel Dekker, Inc. Lect. Notes Pure Appl. Math. 196, 269-285 (1998).

Summary: In this paper, I highlight four general principles (or, if you like, methods), giving them mnemonic monikers for reference:
COMPARE.\(u_I\): Compare the finite element approximation to an “interpolant” \(u_I\).
LOC$YMM.MESH: A point about which the mesh is locally symmetric is a superconvergent point.
\(\otimes\)PROD.ELEMTS: One-dimensional results translate to tensor product elements.
\(\partial_H\) TRANSL.INV.MESH: Differencing is better than differentiating, in particular on translation invariant meshes.
For the entire collection see [Zbl 0884.00048].

MSC:

65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations

Keywords:

superconvergence; Galerkin finite element methods; second-order elliptic problem; tensor product elements