Super-Closeness Between the Ritz Projection and the Finite Element Solution for Some Elliptic Problems (Communications in Computational Physics, 2020)
We prove the super-closeness between the finite element solution and theRitz 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 estimateof 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 hencewe only consider the pure Neumann boundary problems which can be discretizednaturally on such domains, if ignoring the effect of numerical integrals. For the fourthorder problem, our results amend the gap between the theoretical estimates and thenumerical examples in a previous work [22].