An improved accuracy version of the mixed finite-element method for a second-order elliptic equation (Q2367492)

The author considers mixed finite element approximations of the form: \(\text{Find}(\sigma_ h,u_ h)\in V_ h\times W_ h\): \(\alpha_ h(\sigma_ h,\tau_ h)-(\text{div }\tau_ h,u_ h)=-\langle g,\tau_ h\bullet n_{\partial\Omega}\rangle\), \(\forall \tau_ h\in V_ h\), \((\text{div }\sigma_ h,t_ h)+ (cu_ h,t_ h)= (f,t_ h)\), \(\forall t_ h\in W_ h\), to the second-order elliptic Dirichlet boundary value problem \(-\text{div}(a\nabla u)+ cu= f\) in \(\Omega\), \(u=g\) on \(\partial\Omega\), where \(u_ h\) approximates \(u\) and \(\sigma_ h\) approximates \(\sigma=-(a\nabla u)\). The bilinear form \(\alpha_ h(.,.)\) describes the quadrature rule for evaluating \((\sigma,a^{-1}\tau)\). The finite-dimensional spaces \(V_ h\) and \(W_ h\) are constructed on the basis of the lowest-order Raviart-Thomas element. By introducing a particular quadrature rule, the author derives a more accurate mixed finite element scheme. If \(\sigma\) is sufficiently smooth, then the modified scheme has the accuracy \(O(h^ 3)\) instead of \(O(h^ 2)\) (= accuracy of the standard scheme) in some discrete norm. This observation is used in order to develop an a posteriori estimate. The numerical results reported confirm the given theoretical estimates.