Maximum-norm estimates for resolvents of elliptic finite element operators

Let be a convex domain with smooth boundary in . It has been shown recently that the semigroup generated by the discrete Laplacian for quasi-uniform families of piecewise linear finite element spaces on 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 -stable rational functions, with small.