Fully computable a posteriori error estimates for the primal hybrid variational formulation of Poisson’s equation

Resumo

We present a new fully computable a posteriori error estimates for the primal hybrid formulation applied to Poisson’s problem. The estimates are based on the reconstruction of a continuous potential field and an equilibrated flux, which are computed using the potential and Lagrange multipliers solutions. The potential reconstruction is the result of orthogonally projecting the potential solution onto a function over the mesh skeleton, smoothing this function into a continuous trace, and solving local pure Dirchlet problems. This procedure for reconstructing the potential were used to develop error estimates for the mixed formulation in [1, 2]. The equilibrated flux is obtained from solving local mixed problems using Lagrange multipliers at a pure Neumann boundary condition. This technique is similar to the flux recovery strategy based on the Arnold–Boffi–Falk spaces described in [3], but for the divergent compatible pair of spaces described in [4]. An adaptive refinement strategy is developed, and numerical results illustrate the efficiency of the error estimates.