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arXiv · 1912.09160

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

Markus Faustmann·Jens Markus Melenk·Maryam Parvizi
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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.

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Markus Faustmann, Jens Markus Melenk, Maryam Parvizi. 2019-12-19. On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion. https://doi.org/10.1051/m2an/2020079

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