Convergence of the highest derivatives in projection methods

Assume that for the approximate solution of an elliptic differential equation in a bounded domain Ω, under a natural boundary condition, one applies the Galerkin method with polynomial coordinate functions. One gives sufficient conditions, imposed on the exact solutionu*, which ensure the convergence of the derivatives of order k of the approximate solutions, uniformly or in the mean in Ω or in any interior subdomain. For example, ifu*∈Wk2, then the derivatives of order k converge in L2(Ω′), where Ω′ is an interior subdomain of Ω. Somewhat weaker statements are obtained in the case of the Dirchlet problem.