The importance of reconstructing conservative local fluxes from a primal discrete solution is widely recognized in the literature. One major application is in a posteriori error analysis: the difference between the numerical flux and a recovered equilibrated flux provides an a posteriori error indicator, with a reliability constant equal to 1, which is further used in adaptive mesh refinement.
In this talk, we consider an elliptic diffusion problem with discontinuous coefficients, approximated by conforming and nonconforming finite elements of arbitrary polynomial degree. We recover a conservative flux in the Raviart-Thomas space following the approach proposed by Becker, Capatina and Luce in 2016 for the Poisson equation. The idea is to introduce an equivalent hybrid mixed formulation and to use the Lagrange multiplier, which is defined on the sides of the mesh, as correction of the degrees of freedom of the flux. The main difficulty lies in the choice of the multiplier's space, which should allow to establish a uniform inf-sup condition and to compute the multiplier locally. Thus, we do not solve the mixed formulation but only use it for the error analysis. Contrarily to other approaches, no local nor global mixed problem needs to be solved. In addition, the previous approach provides a unified framework for several standard finite element methods and differential operators.
This study generalizes the methodology described above to diffusion problems, the main focus being on the robustness of the reconstruction and of the a posteriori error bounds with respect to the diffusion coefficients.
Firstly, we consider a conforming finite element approximation on triangular meshes. We express the inf-sup constant of the equivalent mixed formulation in terms of the coefficients and we also establish a local bound for the multiplier, with a constant whose dependence on the coefficients is given explicitly. This allows us to deduce, besides the usual sharp reliability of the a posteriori error indicator, its local efficiency with an explicit constant. This result is new, at the best of our knowledge; for quasi-monotone coefficients, we retrieve the complete robustness, which already exists in the literature in the quasi-monotone case.
Secondly, we consider a nonconforming finite element approximation of arbitrary polynomial degree, based on the space introduced by Matthies and Tobiska in 2005. The standard nonconforming space of odd degree rises no difficulty and the reconstruction of conservative fluxes in this case is well-known. Meanwhile, this is no longer true for even degree, due to the loss of unisolvence. Our contribution is the extension of the previous approach to the spaces of Matthies and L. Tobiska, which are well-defined for any degree and also inf-sup stable for the Stokes problem, in a robust way with respect to the diffusion coefficients.
Finally, we will present several numerical experiments illustrating the theoretical results.
