Numerische Mathematik

, Volume 130, Issue 1, pp 125–133 | Cite as

Banach space projections and Petrov–Galerkin estimates

Article

Abstract

We sharpen the classic a priori error estimate of Babuška for Petrov–Galerkin methods on a Banach space. In particular, we do so by (1) introducing a new constant, called the Banach–Mazur constant, to describe the geometry of a normed vector space; (2) showing that, for a nontrivial projection \(P\), it is possible to use the Banach–Mazur constant to improve upon the naïve estimate \( ||I - P ||\le 1 + ||P ||\); and (3) 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.

Mathematics Subject Classification (2010)

65N30 46B20 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsWashington University in St. LouisSt. LouisUSA

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