Maximum-norm Resolvent Estimates for Elliptic Finite Element Operators on Nonquasiuniform Triangulations

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. Received: May 2, 2006. Introduction Consider the initial-value problem ut −∆u = 0 in Ω and u = 0 on ∂Ω, for t > 0, with u(·, 0) = v in Ω, (0.1) where Ω is a domain in R, and denote by E(t) the solution operator related to this problem and defined by u(t) = E(t)v. Then it is a special case of a result of Stewart [11] that if ∂Ω is smooth, then E(t) is an analytic semigroup on C0(Ω̄) = {v ∈ C(Ω̄) : v = 0 on ∂Ω} generated by ∆. This follows from the resolvent estimate ‖(λI +∆)v‖C ≤ C 1+|λ|‖v‖C, for v ∈ C0(Ω̄) and λ / ∈ Σδ = {λ : | argλ| ≤ δ}, (0.2)