Functional-Type A Posteriori Error Estimates for Mixed Finite Element Methods

The work concerns the a posteriori error estimation for the primal and the dual mixed finite element methods applied to the diffusion problem. The problem is considered in a general setting, with inhomogeneous mixed Dirichlet/Neumann boundary conditions. The new, functional-type a posteriori error estimators are proposed that exhibit the ability both to indicate the local error distribution and to provide guaranteed upper bounds for the discretization errors in the primal and the dual (flux) variables. The latter property is a direct consequence of the absence in the estimators of any mesh-dependent constants; the only constants present in the estimates stem from the Friedrichs and the trace inequalities and, thus, are global and dependent solely on the domain geometry and the bounds of the diffusion matrix. The estimators are computationally cheap and require only the projections of piecewise constant functions onto the spaces of the lowest-order Raviart-Thomas or of the continuous piecewise linear elements. It is shown how these projections can be easily realized with a simple local averaging.