Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems

Igor Mozolevski (1) , Serge Prudhomme (2)

1 MTM - Departamento de Matemática = Mathematics Department [Florianópolis]
2 Département de mathématiques et de génie industriel

We propose an approach for goal-oriented error estimation in finite element approximations of second-order elliptic problems that combines the dual-weighted residual method and equilibrated-flux reconstruction methods for the primal and dual problems. The objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution. It is only assumed here that the discretization methods come with a priori error estimates and an equilibrated-flux reconstruction algorithm. A high-order discontinuous Galerkin (dG) method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes. One contribution of the paper is to show how the order of the dG method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method, a dG method, or a mixed Raviart- Thomas method are used for the solution of the primal problem. Numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size.

Mots clés: Finite element method; goal-oriented estimates; asymptotically-exact error estimates

Domaines: Analyse numérique [math.NA]

Igor Mozolevski, Serge Prudhomme. Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems. 2014. ⟨hal-00985971⟩