This is a joint work with Alexandre Ern and Iain Smears.
We consider in this talk two fundamental model partial differential equations: the elliptic linear reaction-diffusion equation and the parabolic linear heat equation. When a numerical approximation of these equations is performed by the finite element method, a question arises about the size of the error between the unknown exact solution and the available numerical approximation. We derive quantities fully computable from the numerical approximations (guaranteed a posteriori error estimates) that are upper bounds on the energy errors. Moreover, we prove that a converse relation also holds, in that the estimators are also lower bounds of the errors, up to a generic constant. Crucially, this generic constant only depends on the mesh shape regularity and the space dimension (and additionally on the polynomial degree of the approximation in the reaction-diffusion case); it is in particular independent of the exact solution and its regularity, the computational domain, the singular perturbation parameter, or the final time. This is called robustness. Moreover, the local contributions to the estimators are also lower bounds of the local contributions to the errors, theoretically underpinning the fact that the estimates allow to predict the distribution of the errors in space (and in time). The analysis for the heat equation hinges on an equivalence between global and local norms and allows for arbitrary space mesh refinement and coarsening between time steps. In the reaction-diffusion case, explicit bounds for some inverse inequality constants on a simplex need to be derived. Numerical experiments illustrate the theoretical developments.