Global estimates for mixed methods for second order elliptic equations

Global error estimates in ${L^2}(\Omega )$, ${L^\infty }(\Omega )$, and ${H^{ - s}}(\Omega )$, $\Omega$ in ${{\mathbf{R}}^2}$ or ${{\mathbf{R}}^3}$, are derived for a mixed finite element method for the Dirichlet problem for the elliptic operator $Lp = - \operatorname{div}(a\;{\mathbf{grad}}\;p + {\mathbf{b}}p) + cp$ based on the Raviart-Thomas-Nedelec space ${{\mathbf{V}}_h} \times {W_h} \subset {\mathbf{H}}(\operatorname{div};\Omega ) \times {L^2}(\Omega )$ . Optimal order estimates are obtained for the approximation of p and the associated velocity field ${\mathbf{u}} = - (a\;{\mathbf{grad}}\;p + {\mathbf{b}}p)$ in ${L^2}(\Omega )$ and ${H^{ - s}}(\Omega )$, $0 \leqslant s \leqslant k + 1$, and, if $\Omega \subset {{\mathbf{R}}^2}$ for p in ${L^\infty }(\Omega )$.