Maximum-norm estimates for resolvents of elliptic finite element operators

Bakaev, Nikolai; Thomée, Vidar; Wahlbin, Lars

doi:10.1090/S0025-5718-02-01488-6

Maximum-norm estimates for resolvents of elliptic finite element operators
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by Nikolai Yu. Bakaev, Vidar Thomée and Lars B. Wahlbin;

Math. Comp. 72 (2003), 1597-1610

DOI: https://doi.org/10.1090/S0025-5718-02-01488-6

Published electronically: December 3, 2002

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Abstract:

Let $\Omega$ be a convex domain with smooth boundary in $R^d$. It has been shown recently that the semigroup generated by the discrete Laplacian for quasi-uniform families of piecewise linear finite element spaces on $\Omega$ is analytic with respect to the maximum-norm, uniformly in the mesh-width. This implies a resolvent estimate of standard form in the maximum-norm outside some sector in the right halfplane, and conversely. Here we show directly that such a resolvent estimate holds outside any sector around the positive real axis, with arbitrarily small angle. This is useful in the study of fully discrete approximations based on $A(\theta )$-stable rational functions, with $\theta$ small.

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