C¹ piecewise quadratic hierarchical bases

Abstract

We present a construction of $C^1$ piecewise quadratic hierarchical bases of Lagrange type on arbitrary polygonal domains $\mathrm{\Omega }\subset {\mathbb{R}}^2$. Properly normalized, these bases are Riesz bases for Sobolev spaces $H^s(\mathrm{\Omega })$, with $s\in (1,\frac{5}{2})$. The method is applicable to arbitrary initial triangulations of polygonal domains, and does not require a checkerboard quadrangulation needed for earlier $C^1$ cubic hierarchical Lagrange bases. Homogeneous boundary conditions can be taken into account in a natural way, and lead to Riesz bases for Sobolev spaces $H_0^s(\mathrm{\Omega })$, $s\in (1,5/2)\setminus \{3/2\}$, and $H_{00}^{3/2}(\mathrm{\Omega })$.

Introduction

Hierarchical bases were first constructed by Yserentant in [21], based on continuous piecewise linear functions on general triangulations. For domains in ${\mathbb{R}}^2$ they give rise to Riesz bases for Sobolev spaces $H^s(\mathrm{\Omega })$, $s\in (1,3/2)$, and are used in the finite element method as multilevel preconditioners for elliptic partial differential equations of second order. Motivated by applications to the fourth order elliptic equations, Oswald [19] and Dahmen, Oswald and Shi [3] constructed $C^1$ hierarchical bases of Hermite type by applying piecewise quadratic, cubic or quintic polynomials. Properly scaled, they produce Riesz bases for $H^s(\mathrm{\Omega })$, $s\in (2,5/2)$. In [12] Hong and Schumaker demonstrated applications of $C^1$ piecewise cubic Hermite hierarchical bases to the problem of surface compression introduced by DeVore, Jawerth and Lucier in [10].

Stable $C^1$ hierarchical bases of Lagrange type generate Riesz bases for $H^s(\mathrm{\Omega })$ with an extended range $s\in (1,5/2)$. They are however difficult to obtain because stable and local Lagrange bases that are already rare for splines on triangulations [17] have to be found for nested spline spaces and the interpolation sets have to be nested themselves. The first construction is due to Davydov and Stevenson [7]. It is based on piecewise cubic polynomials with respect to triangulated checkerboard quadrangulations that are rather difficult to produce for general polygonal domains, and the treatment of homogeneous boundary conditions is cumbersome. Maes and Bultheel [16] suggested a $C^1$ piecewise quadratic hierarchical basis of Lagrange type based on the Powell-Sabin-6 split of triangulated checkerboard quadrangulations. This basis is however only stable under the assumption that the resulting nested sequence of triangulations is quasi-uniform, which seems difficult to enforce unless the quadrangulation is uniform. In [13] Jia and Liu constructed $C^1$ piecewise quadratic wavelet type bases for arbitrary polygonal domains that are Riesz bases for $H^s(\mathrm{\Omega })$ with a smaller range $s\in (1.618,5/2)$. Classical bivariate smooth wavelet constructions are difficult to adapt to general bounded domains if the Riesz basis property in $H^s(\mathrm{\Omega })$ for $s\ge 3/2$ is required. In particular, the only general method of this type due to Dahmen and Schneider [4] requires implementation of the Hestenes extension operator.

In this paper we construct Lagrange hierarchical bases for the spaces of $C^1$ piecewise quadratic polynomials on a combination of Powell-Sabin-6 and Powell-Sabin-12 splits of uniform refinements of an arbitrary initial triangulation of any polygonal domain. We prove the Riesz basis property for $H^s(\mathrm{\Omega })$, $s\in (1,5/2)$. Moreover, the construction is easily adapted to the spaces of functions satisfying homogeneous boundary conditions and leads to Riesz bases for $H_0^s(\mathrm{\Omega })$, $s\in (1,5/2)\setminus \{3/2\}$, and $H_{00}^{3/2}(\mathrm{\Omega })$. The results are based in part on the thesis of the second named author [22].

The paper is organized as follows. In Section 2 we briefly summarize the general theory of hierarchical Riesz bases of Lagrange type following [9], and in Section 3 describe a nested sequence of spaces of $C^1$ piecewise quadratics introduced in [13]. Section 4 presents our construction of nested interpolation sets for these spaces and the proof of the key stability and locality properties of the corresponding hierarchical Lagrange bases that imply the main result of the paper (Theorem 4.9). Section 5 includes a numerical experiment that illustrates the Riesz stability of Lagrange and Hermite hierarchical expansions of particular functions for appropriate ranges of s.