Abstract
On arbitrary polygonal domains $\Omega \subset \RR^2$, we construct $C^1$ hierarchical Riesz bases for Sobolev spaces $H^s(\Omega)$. In contrast to an earlier construction by Dahmen, Oswald, and Shi (1994), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from $s \in (2,\frac{5}{2})$ to $s \in (1,\frac{5}{2})$. Since the latter range includes $s=2$, with respect to the present basis, the stiffness matrices of fourth-order elliptic problems are uniformly well-conditioned.
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Davydov, O., Stevenson, R. Hierarchical Riesz Bases for Hs(Ω), 1 < s < 5/2. Constr Approx 22, 365–394 (2005). https://doi.org/10.1007/s00365-004-0593-2
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DOI: https://doi.org/10.1007/s00365-004-0593-2