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

Abstract

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) IntroductionIn recent years hp finite element methods have gained an increasing popularity in both the applied mathematics community and some engineering application fields. The hp finite element method has been first introduced by Gui and Babuška [26] and since then some monographs [30,37,50,54] or part of textbooks ([38], Sect. 8.4), have proposed both theoretical and numerical insights into this topic. These methods are found to be particularly useful when high or extremely high accuracy is needed and when minimal dissipation and dispersion errors in the discrete system are required.Indeed the main reason for the interest in hp finite element methods is that they achieve exponential rates of convergence for both regular and singular solutions [37,50]. In presence of singularities or boundary layers, suitably graded meshes, geometrically refined towards corners, edges and/or faces have to be employed to achieve such an exponential rate of convergence [37,50]. Thus highly stretched meshes with huge aspect ratios are obtained in practice. Consequently, the condition number of the stiffness matrix severely deteriorates: [32] (see also the references therein). Unfortunately, up to now, no iterative substructuring method has been proven to be efficient when very thin elements and/or subdomains (involving meshes with high aspect ratio) or general non quasiuniform meshes are employed. Thus our goal is to fill this gap by proposing a domain decomposition preconditioner that is robust with the mesh aspect ratio and possible large jumps in the coefficients. Its theoretical derivation has been presented in [58]. In this paper the main emphasis is devoted to an extensive numerical study of its performances. These robustness aspects will be carefully summarized and analyzed hereafter for some three-dimensional elliptic model problems of diffusion or reaction-diffusion type. These model problems defined on simple geometries have been chosen here in order to be easily reproducible. Nevertheless some of them...