A quasi-interpolation operator yielding fully computable error bounds

We design a quasi-interpolation operator from the Sobolev space $ H_0^1(\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_0^1(\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 estimates optimal in the mesh size h for both the H1 seminorm and the L2 norm error; (6) yields estimates that bound the error in each computational mesh element by the best-approximation error in the element and in its vertex neighbors; (7) gives a computable constant for both the H1 seminorm and the L2 norm error; (8) leads to the equivalence of global-best and local-best errors; (9) 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 H1 seminorm and the L2 norm and its certified overestimation factor is rather sharp and stable in all tested situations.