Journal of Computational and Applied Mathematics vol:196 issue:1 pages:1-19
In this paper we construct C^1 continuous piecewise quadratic hierarchical bases on Powell-Sabin triangulations of arbitrary polygonal domains in R^2. 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 H^s(Omega) with s is an element of (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 H-s(Omega), 1 < s < 5/2. Constructive Approximation, to appear) our construction is simpler.