Robust localization of the best error with finite elements in the reaction-diffusion norm (Q745858)

Finite element approach to the homogeneous Dirichlet problem for Poisson's equation involves approximation of a function \(u\in H^1_0(\Omega) \) in the \(H^1\)-seminorm in the space of continuous piecewise polynomials of degree \(\ell \) on simplicial mesh. In this context \textit{A. Veeser} [``Approximating gradients with continuous piecewise polynomial functions'', Found. Comput. Math. (2015). \url{doi:10.1007/s10208-015-9262-z}] proved that the global best error is equivalent to the \(\ell_2\)-norm of the local best errors on elements. Observing that a modification of this for the reaction-diffusion norm should invoke local best errors which take the continuity constraint across faces into account, the authors provide here a robust alternative by replacing the \(H^1\)-seminorm by the reaction-diffusion norm.