C1 hierarchical Riesz bases of Lagrange type on Powell–Sabin triangulations

摘要

In this paper we construct C1 continuous piecewise quadratic hierarchical bases on Powell–Sabin triangulations of arbitrary polygonal domains in R2. Our bases are of Lagrange type instead of the usual Hermite type and under some weak regularity assumptions on the underlying triangulations we prove that they form strongly stable Riesz bases for the Sobolev spaces Hs(Ω) with s∈(1,5/2). Especially the case s=2 is of interest, because we can use the corresponding hierarchical basis for preconditioning fourth-order elliptic equations leading to uniformly well-conditioned stiffness matrices. Compared to the hierarchical Riesz bases by Davydov and Stevenson (Hierarchical Riesz bases for Hs(Ω), 1