$C^1$ spline wavelets on triangulations

Abstract

Abstract. In this paper we investigate spline wavelets on general triangulations. In particular, we are interested in C 1 wavelets generated from piecewise quadratic polynomials. By using the Powell-Sabin elements, we set up a nested family of spaces of C 1 quadratic splines, which are suitable for multiresolution analysis of Besov spaces. Consequently, we construct C 1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three-direction mesh. A general theory is developed so as to verify the global stability of these wavelets in Besov spaces. The wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations. IntroductionIn this paper we investigate spline wavelets on general triangulations. In particular, we are interested in C 1 wavelets generated from piecewise quadratic polynomials.In µ with 1 < µ < 5/2. In this paper we shall employ the Hermite interpolation property of the PowellSabin elements to construct C 1 spline wavelets on general triangulations. These wavelet bases will be shown to be H 2 -stable. In the process we shall establish a general theory for wavelet bases in Besov spaces, which will be useful for future research.Here is an outline of the paper. In Section 2 we review some basic properties of Besov spaces as well as Bernstein type inequalities for spline functions on polygonal domains. In Section 3, using the Powell-Sabin elements, we formulate bases for the concerned spaces of spline functions and establish Jackson type inequalities. In Section 4 we discuss multiresolution analysis of Besov spaces and related norm equivalence. In Section 5 we develop a general theory for wavelet bases in Besov spaces. Finally, in Section 6, we construct C 1 wavelet bases on general triangulations and show that the wavelet bases are H 2 -stable. In particular, for the three-direction mesh, the wavelets are explicitly given.It is expected that the C 1 spline wavelets constructed in this paper will have applications to numerical solutions of partial differential equations. These wavelet bases are particularly suitable for the biharmonic equation (see the discussion in [27] and [10]). The condition numbers of the corresponding discretization matrices will be uniformly bounded. Our wavelet bases could have applications to a wide range of problems in numerical analysis, such as numerical solutions of integral equations and operator equations, and singular perturbation problems. See the related work of Chen, Micchelli and Xu [2], Liu and Xu [25], and Shen and Lin [32]. The refinable spline functions discussed in this paper also could have applications to computer graphics and multi-level data representation. See the related work of Chui and Jiang [4,5] on surface subdivision schemes.

TheC1wavelet method for numerical solutions of the Neumann problem

“…For a three-direction mesh, the wavelets are explicitly given. For further details, we refer to the articles [16,19].…”

“…In fact, it has been a challenging problem to construct spline wavelets on general triangulations [16]. Recently, some researchers have done some good works [16,17] for that.…”

“…In fact, it has been a challenging problem to construct spline wavelets on general triangulations [16]. Recently, some researchers have done some good works [16,17] for that. In [16], Jia and Liu developed a method of constructing the C 1 spline wavelets on Powell-Sabin triangulations.…”

“…Recently, some researchers have done some good works [16,17] for that. In [16], Jia and Liu developed a method of constructing the C 1 spline wavelets on Powell-Sabin triangulations. The Powell-Sabin element method constructs piecewise polynomial functions with continuous derivative [18,19].…”

Riesz bases of wavelets and applications to numerical solutions of elliptic equations

“…For polygonal domains Riesz bases of spline wavelets were constructed by Davydov and Stevenson [10] on quadrangulation, and by Jia and Liu [21] on arbitrary triangulations. However, numerical schemes based on these wavelet bases have yet to be implemented.…”