Volume 55, Number 2, March-April 2021
|Page(s)||595 - 625|
|Published online||01 April 2021|
On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion
Institute of Analysis and Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8-10, A–1040 Wien, Austria
* Corresponding author: firstname.lastname@example.org
Accepted: 13 November 2020
We provide an endpoint stability result for Scott-Zhang type operators in Besov spaces. For globally continuous piecewise polynomials these are bounded from H3/2 into B3/22,∞; for element wise polynomials these are bounded from H1/2 into B1/22,∞. 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.
Mathematics Subject Classification: 65D05 / 65F08 / 65N30 / 35R11
Key words: Scott-Zhang operator / Besov space / multilevel decomposition / fractional Laplacian / preconditioning
© EDP Sciences, SMAI 2021
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