Convergence of the highest derivatives in projection methods

Assume that for the approximate solution of an elliptic differential equation in a bounded domain $\Omega$, under a natural boundary condition, one applies the Galerkin method with polynomial coordinate functions. One gives sufficient conditions, imposed on the exact solution $u^*$, which ensure the convergence of the derivatives of order $k$ of the approximate solutions, uniformly or in the mean in $\Omega$ or in any interior subdomain. For example, if $u^*\in W_2^{(k)}$, then the derivatives of order k converge in $L_2(\Omega')$, where $\Omega'$ is an interior subdomain of $\Omega$. Somewhat weaker statements are obtained in the case of the Dirchlet problem.