Stability, analyticity, and almost best approximation in maximum norm for parabolic finite element equations

We consider semidiscrete solutions in quasi-uniform finite element spaces of order O(hr) of the initial boundary value problem with Neumann boundary conditions for a second-order parabolic differential equation with time-independent coefficients in a bounded domain in R^N. We show that the semigroup on L∞, defined by the semidiscrete solution of the homogeneous equation, is bounded and analytic uniformly in h. We also show that the semidiscrete solution of the inhomogeneous equation is bounded in the space-time L∞-norm, modulo a logarithmic factor for r = 2, and we give a corresponding almost best approximation property.