Section: New Results

A posteriori error estimators based on $H\left(\mathrm{div}\right)$-reconstructed fluxes

Participants : Roland Becker, Daniela Capatina, Robert Luce.

Mesh adaptivity is nowadays an essential tool in numerical simulations; in order to achieve it, reliable and efficient, easily computable a posteriori error estimators are needed. Such estimators obtained by reconstructing locally conservative fluxes in the Raviart-Thomas finite element space have been largely employed in the past years.

We have so far considered the convection-diffusion equation and proposed a unified framework for several finite element approximations (conforming, nonconforming and discontinuous Galerkin). The main advantage of our approach is to use, contrarily to the existing references, only the primal mesh for the flux reconstruction, which presents certain facilities from a computational point of view.

For this purpose, the construction of the $H\left(\mathrm{div}\right)$-vector involved in the error estimator is inspired by the hypercircle method cf.  [53] and is achieved on patches, which may overlap. A patch depends on the type of the employed finite elements and is defined as the support of a basis function.

Our first results were presented in [26] . We are working on the extension to higher-order approximations, to quadrilateral meshes and to other model problems.