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.