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On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion (1912.09160v2)
Published 19 Dec 2019 in math.NA and cs.NA
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.