Banach space projections and Petrov-Galerkin estimates

TL;DR

This paper improves error estimates for Petrov-Galerkin methods in Banach spaces by introducing the Banach-Mazur constant, which refines classical bounds and extends previous Hilbert space results.

Contribution

It introduces the Banach-Mazur constant to better describe Banach space geometry and uses it to refine projection error estimates in Petrov-Galerkin methods.

Findings

Improved a priori error estimate for Petrov-Galerkin methods.

Introduction of the Banach-Mazur constant for Banach space geometry.

Extension of Hilbert space results to general Banach spaces.

Abstract

We sharpen the classic a priori error estimate of Babuska for Petrov-Galerkin methods on a Banach space. In particular, we do so by (i) introducing a new constant, called the Banach-Mazur constant, to describe the geometry of a normed vector space; (ii) showing that, for a nontrivial projection $P$, it is possible to use the Banach-Mazur constant to improve upon the naive estimate $ \| I - P \| \leq 1 + \| P \| $; and (iii) applying that improved estimate to the Petrov-Galerkin projection operator. This generalizes and extends a 2003 result of Xu and Zikatanov for the special case of Hilbert spaces.