Abstract. Let Ω 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 Ω 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(θ)-stable rational functions, with θ small.

Abstract. This paper is concerned with the stability of rational one-step approximations of C 0 semigroups. Particular emphasis is laid on long-term stability bounds. The analysis is based on a general Banach space framework and allows variable stepsize sequences. Under reasonable assumptions on the stepsize sequence, asymptotic stability bounds for general C 0 semigroups are derived. The bounds are typical in the sense that they contain, in general, a factor that grows with the number of steps. Under additional hypotheses on the approximation, more favorable stability bounds are obtained for the subclass of holomorphic semigroups.

Abstract. In recent years several papers have been devoted to stability and smoothing properties in maximum-norm of finite element discretizations of parabolic problems. Using the theory of analytic semigroups it has been possible to rephrase such properties as bounds for the resolvent of the associated discrete elliptic operator. In all these cases the triangulations of the spatial domain has been assumed to be quasiuniform. In the present paper we show a resolvent estimate, in one and two space dimensions, under weaker conditions on the triangulations than quasiuniformity. In the two-dimensional case, the bound for the resolvent contains a logarithmic factor.Mathematics Subject Classification. 65M06, 65M12, 65M60.

We consider fully discrete approximations to a parabolic initialboundary value problem with rough or distribution-valued initial data in two space dimensions. For discretization in time and space, we apply single step methods and the standard Galerkin method with piecewise linear test functions, respectively. For spatial discretization of the initial condition, we are however forced to use more involved constructions. Our main result is stability and error estimates of the discrete solutions.

No abstract