A quasi-interpolation operator yielding fully computable error bounds

Abstract: We design a quasi-interpolation operator from the Sobolev space $H^1_0(\Omega)$ to its finite-dimensional finite element subspace formed by piecewise polynomials on a simplicial mesh with a computable approximation constant. The operator 1) is defined on the entire $H^1_0(\Omega)$, no additional regularity is needed; 2) allows for an arbitrary polynomial degree; 3) works in any space dimension; 4) is defined locally, in vertex patches of mesh elements; 5) yields optimal estimates for both the $H^1$ seminorm and the $L^2$ norm error; 6) gives a computable constant for both the $H^1$ seminorm and the $L^2$ norm error; 7) leads to the equivalence of global-best and local-best errors; 8) possesses the projection property. Its construction follows the so-called potential reconstruction from a posteriori error analysis. Numerical experiments illustrate that our quasi-interpolation operator systematically gives the correct convergence rates in both the $H^1$ seminorm and the $L^2$ norm and its certified overestimation factor is rather sharp and stable in all tested situations.