ON THE STABILITY OF SCOTT-ZHANG TYPE OPERATORS AND APPLICATION TO MULTILEVEL PRECONDITIONING IN FRACTIONAL DIFFUSION

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,infinity(3/2); for element wise polynomials these are bounded from H-1/2 into B-2,infinity(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.

Automated summary

The article provides an endpoint stability result for Scott-Zhang type operators in Besov spaces, showing that these operators are bounded from H-3/2 into B-2,infinity(3/2) for globally continuous piecewise polynomials, and from H-1/2 into B-2,infinity(1/2) for element wise polynomials. An application of this result is a multilevel decomposition based on Scott-Zhang operators on a hierarchy of meshes generated by newest vertex bisection, with an exclusion of the endpoint case. Additionally, a local multilevel diagonal preconditioner for the fractional Laplacian on locally refined meshes with optimal eigenvalue bounds is presented.