Problems and Techniques

Abstract

This installment of Problems and Techniques features articles on extremal properties of the Schur complement of a matrix and the computation of superconvergent functionals from approximate solutions of partial differential equations. In the first article, Chi-Kwong Li and Roy Mathias describe extremal properties of the generalized Schur complement of a positive semidefinite matrix. The generalization refers to the use of a pseudoinverse when the matrix is not invertible. Using the extremal properties, the authors derive several operator and eigenvalue inequalities involving generalized Schur complements. While some of these results were known, the extremal principles simplify their analysis and also provide a technology for generating several new inequalities. The authors also provide an interesting historical perspective of the development and use of Schur complements. The use of the adjoint solutions has been suggested by several investigators as a means of constructing a posteriori error bounds for finite element solutions and their integral functionals. Niles Pierce and Michael Giles use this approach more generally, without reliance on the finite element method, to correct the leading order error term in integral functional estimates. This is a very powerful technique that can be used to construct superconvergent approximations of the integral functionals or to control an adaptive enrichment process. The authors indicate several problems where accuracy of an integral functional is of primary concern. In optimal design, for example, the adjoint solution is often needed as part of the optimization. The importance of this work was recognized when Niles Pierce received the Fox Prize in Numerical Analysis last June. I hope that you enjoy these articles as much as I did. Please join me in thanking our authors for their very interesting contributions.