Finite element quasi-interpolation and best approximation

TL;DR

This paper develops a quasi-interpolation operator for finite element spaces that achieves optimal approximation estimates in various norms, applicable to scalar and vector fields on affine, shape-regular meshes.

Contribution

It introduces a new quasi-interpolation operator with optimal approximation properties and stability in $L^1$, applicable to multiple finite element spaces with minimal regularity assumptions.

Findings

Provides optimal error estimates in $L^p$-norms for the operator.

Ensures stability in $L^1$ and invariance in finite element spaces.

Applicable to $H^1$, $ extbf{H}( ext{curl})$, and $ extbf{H}( ext{div})$ spaces.

Abstract

This paper introduces a quasi-interpolation operator for scalar- and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces.This operator gives optimal estimates of the best approximation error in any $L^p$-norm assuming regularity in the fractional Sobolev spaces $W^{r,p}$, where $p\in [1,\infty]$ and the smoothness index $r$ can be arbitrarily close to zero. The operator is stable in $L^1$, leaves the corresponding finite element space point-wise invariant whether homogeneous boundary conditions are imposed or not. The theory is illustrated on $H^1$-, $\mathbf{H}(\text{curl})$- and $\mathbf{H}(\text{div})$-conforming spaces.