Banach space projections and Petrov–Galerkin estimates
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
- 6.John, F.: Extremum problems with inequalities as subsidiary conditions. Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, pp. 187–204. Interscience Publishers Inc, New York (1948)Google Scholar
- 9.Kato, M., Takahashi, Y.: Some recent results on geometric constants of Banach spaces. In: Simos, T.E., Psihoyios, G., Tsitouras, C. (eds.) Numerical Analysis and Applied Mathematics (Rhodes, 2010), AIP Conference Proceedings, vol. 1281, pp. 494–497. American Institute of Physics (2010)Google Scholar