A quadratic finite element wavelet Riesz basis

TL;DR

This paper constructs stable quadratic finite element wavelets on polygons in 2D, with two vanishing moments, suitable for multilevel methods, and provides numerical condition numbers for practical applications.

Contribution

It introduces a new construction of quadratic finite element wavelets on general polygons with proven stability and vanishing moments, expanding their applicability.

Findings

Wavelets are stable in $H^s$ for $|s|<1.5$

Each wavelet combines 11 or 13 basis functions

Numerical condition numbers are computed for the unit square

Abstract

In this paper, continuous piecewise quadratic finite element wavelets are constructed on general polygons in $\mathbb{R}^2$. The wavelets are stable in $H^s$ for $|s|<\frac{3}{2}$ and have two vanishing moments. Each wavelet is a linear combination of 11 or 13 nodal basis functions. Numerically computed condition numbers for $s \in \{-1,0,1\}$ are provided for the unit square.