Super-Closeness Between the Ritz Projection and the Finite Element Solution for Some Elliptic Problems

Abstract

We prove the super-closeness between the finite element solution and the Ritz projection for some second order and fourth order elliptic equations in both the $H^1$ and the $L^2$ norms. For the fourth order problem, a Ciarlet-Raviart type mixed formulation is used in the analysis. The main tool in the proof is a negative norm estimate of the Ritz projection, which requires $H^{q+1}$ regularity for second order elliptic equations. Therefore, the analysis is done on a domain Ω with smooth boundary, and hence we only consider the pure Neumann boundary problems which can be discretized naturally on such domains, if ignoring the effect of numerical integrals. For the fourth order problem, our results amend the gap between the theoretical estimates and the numerical examples in a previous work [22].