A quasi-interpolation operator yielding fully computable error bounds

¹ Inria, Univ. Lille, CNRS, UMR 8524 Laboratoire Paul Painlevé, 40 Av. Halley, 59650 Villeneuve-d'Ascq, France
² Inria, 48 rue Barrault, 75647 Paris, France
³ CERMICS, Ecole nationale des ponts et chaussées, IP Paris, 77455 Marne-la-Vallée, France

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 16 July 2025
Accepted: 17 October 2025

Abstract

We design a quasi-interpolation operator from the Sobolev space H₀¹(Ω) 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 H₀¹(Ω), 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.