In recent years several papers have been devoted to stabilityand smoothing properties in maximum-norm offinite element discretizations of parabolic problems.Using the theory of analytic semigroups it has been possibleto rephrase such properties as bounds for the resolventof the associated discrete elliptic operator. In all thesecases the triangulations of the spatial domain has beenassumed to be quasiuniform. In the present paper weshow 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 containsa logarithmic factor.
