Quasi-optimal Schwarz Methods for the Conforming Spectral Element Discretization

Fast methods are proposed for solving the system KNx = b resulting from the dis-cretization of elliptic self-adjoint equations in three dimensional domains by the spectral element method. The domain is decomposed into hexahedral elements, and in each of these elements the discretization space is formed by polynomials of degree N in each variable. Gauss-Lobatto-Legendre (GLL) quadrature rules replace the integrals in the Galerkin formulation. This system is solved by the preconditioned conjugate gradients method. The conforming nite element space on the GLL mesh consisting of piecewise Q1 elements produces a stiiness matrix Kh that is spectrally equivalent to the spectral element stiiness matrix KN. The action of the inverse of Kh is expensive for large problems, and so is replaced by a Schwarz preconditioner Bh of this nite element stiiness matrix. The preconditioned operator used is B ?1 h KN. The technical diiculties stem from the non-regularity of the mesh. The tools to estimate the convergence of a large class of new iterative substructuring and overlapping Schwarz preconditioners are developed. This technique also provides a new analysis for an iterative substructuring method proposed by Pavarino and Widlund for the spectral element discretization. 1. Introduction. In the past decade, many preconditioners have been developed for the large systems of linear equations arising from the nite element discretization of elliptic self-adjoint partial diierential equations; see e.g. 6], 14], 27]. A specially challenging problem is the design of preconditioners for three dimensional problems. More recently, spectral element discretizations of such equations have been proposed, and their eeciency has been demonstrated; see 15], 16], and references therein. In large scale problems, long range interactions of the basis elements produce quite dense and expensive factorizations of the stiiness matrix, and the use of direct methods is not economical due to the large memory requirements 12]. Early work on preconditioners for these equations was done by Pavarino 20], 21], 19]. His algorithms are numerically scalable (i.e., the number of iterations is independent of the number of substructures) and quasi-optimal (the number of iterations grows slowly with the degree of the polynomials.) However, each application of the preconditioner can be very expensive. Several iterative substructuring methods which preserve scalability and quasi-optimality was introduced by Pavarino and Widlund 22], 24]. These preconditioners can be viewed as block-Jacobi methods after transforming the matrix to a particular basis. The subspaces used are the analogues of those proposed by Smith …