Finite Element Wavelets on Manifolds

Preprint No. 1199, Department of Mathematics, University of Utrecht, June 2001. Submitted to IMA J. Numer. Anal. We construct locally supported, continuous wavelets on manifolds Γ that are given as the closure of a disjoint union of general smooth parametric images of an n-simplex. The wavelets are proven to generate Riesz bases for Sobolev spaces Hs(Γ) when s ∈ (−1, 3 2 ), if not limited by the global smoothness of Γ. These results generalize the findings from [DSt99], where it was assumed that each parametrization has a constant Jacobian determinant. The wavelets can be arranged to satisfy the cancellation property of in principal any order, except for wavelets with supports that extend to different patches, which generally satisfy the cancellation property of only order 1.