Quasi-Optimal Schwarz Methods for the
Conforming Spectral Element Discretization
Mario Casarin
Courant Institute of Mathematical Sciences
Abstract
Fast methods are proposed for solving the system K_{N}x = b generated by the
discretization 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 finite element space on the GLL mesh consisting of piecewise Q_{1}
elements produces a stiffness matrix K_h that is spectrally equivalent to the
spectral element stiffness matrix K_N. The action of the inverse of K_h is
expensive for large problems, and so is replaced by a Schwarz preconditioner
B_h of this finite element stiffness matrix. The preconditioned operator is
B_h^{-1} K_N.
The technical difficulties 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 and
applied in a few examples. This technique also provides a new analysis for an
iterative substructuring method proposed by Pavarino and Widlund for the
spectral element discretization.
Keywords: domain decomposition, Schwarz methods, spectral element method,
preconditioned conjugate gradients, iterative substructuring
AMS(MOS) subject classifications: 41A10, 65N30, 65N35, 65N55