This book is essentially a set of lecture notes from a graduate seminar given at Cornell in Spring 1994. It treats basic mathematical theory for superconvergence in the context of second order elliptic problems. It is aimed at graduate students and researchers. The necessary technical tools are developed in the text although sometimes long proofs are merely referenced.
The book gives a rather complete overview of the field of superconvergence (in time-independent problems). It is the first text with such a scope. It includes a very complete and up-to-date list of references.
Some one-dimensional superconvergence results.- Remarks about some of the tools used in Chapter 1.- Local and global properties of L 2-projections.- to several space dimensions: some results about superconvergence in L 2-projections.- Second order elliptic boundary value problems in any number of space dimensions: preliminary considerations on local and global estimates and presentation of the main technical tools for showing superconvergence.- Superconvergence in tensor-product elements.- Superconvergence by local symmetry.- Superconvergence for difference quotients on translation invariant meshes.- On superconvergence in nonlinear problems.- 10. Superconvergence in isoparametric mappings of translation invariant meshes: an example.- Superconvergence by averaging: mainly, the K-operator.- A computational investigation of superconvergence for first derivatives in the plane.