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A numerical study on Neumann-Neumann and FETI methods for hp-approximations on geometrically refined boundary layer meshes in two dimensions

by A. Toselli and X. Vasseur

(Report number 2002-20)

Abstract
In this paper, we present extensive numerical tests showing the performance and robustness of certain Balancing Neumann-Neumann and one-level FETI methods for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in two dimensions. The numerical results are in good agreement with the theoretical bounds for the condition numbers of the preconditioned operators derived in [44]. They confirm that the condition numbers are independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients. In addition, they only grow polylogarithmically with the polynomial degree, as in the case of p approximations on shape-regular meshes. Our methods are robust with respect to small parameters of certain singularly-perturbed problems.

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

@Techreport{TV02_306,
  author = {A. Toselli and X. Vasseur},
  title = {A numerical study on Neumann-Neumann and FETI methods for hp-approximations on geometrically refined boundary layer meshes in two dimensions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2002-20},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2002/2002-20.pdf },
  year = {2002}
}

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