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

Faustmann, Markus; Melenk, Jens Markus; Parvizi, Maryam

doi:10.1051/m2an/2020079

Faustmann, Markus ; Melenk, Jens Markus ; Parvizi, Maryam

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 2, pp. 595-625

Résumé

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 $B_{2, \infty}^{3 / 2}$ ; for element wise polynomials these are bounded from H^1/2 into $B_{2, \infty}^{1 / 2}$ . 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.

MR

DOI : 10.1051/m2an/2020079

Classification : 65D05, 65F08, 65N30, 35R11
Keywords: Scott-Zhang operator, Besov space, multilevel decomposition, fractional Laplacian, preconditioning

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     author = {Faustmann, Markus and Melenk, Jens Markus and Parvizi, Maryam},
     title = {On the stability of {Scott-Zhang} type operators and application to multilevel preconditioning in fractional diffusion},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {595--625},
     year = {2021},
     publisher = {EDP-Sciences},
     volume = {55},
     number = {2},
     doi = {10.1051/m2an/2020079},
     mrnumber = {4238777},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2020079/}
}

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Faustmann, Markus; Melenk, Jens Markus; Parvizi, Maryam. On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 2, pp. 595-625. doi: 10.1051/m2an/2020079

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