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Domain decomposition preconditioners of Neumann-Neumann type for hp-approximations on boundary layer meshes in three dimensions

by A. Toselli and X. Vasseur

(Report number 2003-01)

Abstract
We develop and analyze Neumann-Neumann methods for hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in three dimensions. These are meshes that are highly anisotropic where the aspect ratio typically grows exponentially with the polynomial degree. The condition number of our preconditioners is shown to be independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients. In addition, it only grows polylogarithmically with the polynomial degree, as in the case of p approximations on shape-regular meshes. This work generalizes our previous one on two dimensional problems in [40,42] and the estimates derived here can be employed to prove condition number bounds for certain types of FETI methods.

Keywords: domain decomposition, preconditioning, hp finite elements, spectral elements, anisotropic meshes

@Techreport{TV03_313,
  author = {A. Toselli and X. Vasseur},
  title = {Domain decomposition preconditioners of Neumann-Neumann type for hp-approximations on boundary layer meshes in three dimensions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2003-01},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2003/2003-01.pdf },
  year = {2003}
}

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