2025/07/16 by Théophile Chaumont-Frelet, Chaumont-Frelet, T., Martin Vohralı́k +1
Computer Science · Engineering · #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical Methods and Algorithms #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2507.11819
openalex publication_date 2025/07/16 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28
We design a quasi-interpolation operator from the Sobolev space H10(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 H10(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 optimal estimates for both the H1 seminorm and the L2 norm error; 6) gives a computable constant for both the H1 seminorm and the L2 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 H1 seminorm and the L2 norm and its certified overestimation factor is rather sharp and stable in all tested situations.