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

TL;DR

This paper establishes endpoint stability of Scott-Zhang operators in Besov spaces and applies these results to develop an efficient multilevel preconditioner for fractional Laplacian problems on adaptively refined meshes.

Contribution

It provides new endpoint stability bounds for Scott-Zhang operators in Besov spaces and introduces a multilevel preconditioner with optimal eigenvalue bounds for fractional diffusion.

Findings

Endpoint stability results for Scott-Zhang operators in Besov spaces.

A multilevel decomposition framework for adaptively refined meshes.

A local multilevel diagonal preconditioner with optimal eigenvalue bounds.

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 $B^{3/2}_{2,\infty}$; for elementwise polynomials these are bounded from $H^{1/2}$ into $B^{1/2}_{2,\infty}$. 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.