Basis function when vertex is in Δl	Basis function when vertex is in Δl+1-Δl

Abstract:

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 we prove that they form strongly stable Riesz bases for the Sobolev spaces H^s(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 our construction is simpler.

Status:
Published

BiBTeX entry:

   @article{ArtMB05b,
      author = "J. Maes and A. Bultheel",
      title = "{$C^1$} hierarchical {R}iesz bases of {L}agrange type on {P}owell-{S}abin triangulations",
      journal = "Journal of Computational and Applied Mathematics",
      volume = "196",
      number = "1",
      pages = "1-19",
      year = "2006",
      url = "http://nalag.cs.kuleuven.be/papers/ade/lagr/index.html",
      DOI = "10.1016/j.cam.2005.08.010",
      LIRIAS = "60293",
      ZBL = "1097.65037",
      MR = "2241571",
   }