A numerical study on Neumann-Neumann methods for hp approximations on geometrically refined boundary layer meshes II. Three-dimensional problems

Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland. Xavier.Vasseur@cerfacs.fr

Received: 23 December 2004
Revised: 3 July 2005
Revised: 8 September 2005

Abstract

In this paper, we present extensive numerical tests showing the performance and robustness of a Balancing Neumann-Neumann method for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in three dimensions. The numerical results are in good agreement with the theoretical bound for the condition number of the preconditioned operator derived in [Toselli and Vasseur, IMA J. Numer. Anal. 24 (2004) 123–156]. They confirm that the condition numbers are independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients. Good results are also obtained for certain singularly perturbed problems. The condition numbers only grow polylogarithmically with the polynomial degree, as in the case of p approximations on shape-regular meshes [Pavarino, RAIRO: Modél. Math. Anal. Numér. 31 (1997) 471–493]. This paper follows [Toselli and Vasseur, Comput. Methods Appl. Mech. Engrg. 192 (2003) 4551–4579] on two dimensional problems.