Global estimates for mixed methods for second order elliptic equations

Global error estimates in L 2 ( Ω ) {L^2}(\Omega ) , L ∞ ( Ω ) {L^\infty }(\Omega ) , and H − s ( Ω ) {H^{ - s}}(\Omega ) , Ω \Omega in R 2 {{\mathbf {R}}^2} or R 3 {{\mathbf {R}}^3} , are derived for a mixed finite element method for the Dirichlet problem for the elliptic operator L p = − div ⁡ ( a g r a d p + b p ) + c p Lp = - \operatorname {div}(a\;{\mathbf {grad}}\;p + {\mathbf {b}}p) + cp based on the Raviart-Thomas-Nedelec space V h × W h ⊂ H ( div ; Ω ) × L 2 ( Ω ) {{\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 u = − ( a g r a d p + b p ) {\mathbf {u}} = - (a\;{\mathbf {grad}}\;p + {\mathbf {b}}p) in L 2 ( Ω ) {L^2}(\Omega ) and H − s ( Ω ) {H^{ - s}}(\Omega ) , 0 ⩽ s ⩽ k + 1 0 \leqslant s \leqslant k + 1 , and, if Ω ⊂ R 2 \Omega \subset {{\mathbf {R}}^2} for p in L ∞ ( Ω ) {L^\infty }(\Omega ) .