On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

Abstract

We provide an endpoint stability result for Scott-Zhang type operators in Besov spaces. For globally continuous piecewise polynomials these are bounded from $H^{3/2}$ into $B_{2,\infty }^{3/2}$; for elementwise polynomials these are bounded from $H^{1/2}$ into $B_{2,\infty }^{1/2}$. As an application, we obtain a multilevel decomposition based on Scott-Zhang operators on a hierarchy of meshes generated by newest vertex bisection with equivalent norms up to (but excluding) the endpoint case. A local multilevel diagonal preconditioner for the fractional Laplacian on locally refined meshes with optimal eigenvalue bounds is presented.

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Cited by in corpus (1)

• Optimal operator preconditioning for pseudodifferential boundary problems