Optimal convergence for the finite element method in Campanato spaces

We prove a priori estimates and optimal error estimates for linear finite element approximations of elliptic systems in divergence form with continuous coefficients in Campanato spaces. The proofs rely on discrete analogues of the Campanato inequalities for the solution of the system, which locally measure the decay of the energy. As an application of our results we derive W 1,p–estimates and give a new proof of the well-known W 1,∞–results of Rannacher and Scott.