A quasi-interpolation operator yielding fully computable error bounds

TL;DR

This paper introduces a new quasi-interpolation operator for finite element methods that provides fully computable, optimal error bounds in multiple norms, applicable in any dimension and polynomial degree, with demonstrated numerical effectiveness.

Contribution

The paper presents a novel quasi-interpolation operator with fully computable constants, applicable to any polynomial degree and dimension, improving error estimation in finite element analysis.

Findings

Provides optimal error estimates in $H^1$ and $L^2$ norms.

Numerical experiments confirm correct convergence rates.

Operator's overestimation factor is sharp and stable.

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.