Global Estimates for Mixed Methods for Second Order Elliptic Equations

Abstract

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