ITERATIVE SUBSTRUCTURING METHODS FOR SPECTRAL ELEMENTS - PROBLEMS IN 3 DIMENSIONS BASED ON NUMERICAL QUADRATURE

Iterative substructuring methods form an important family of domain de composition algorithms for elliptic finite element problems. Two preco nditioners for p-version finite element methods based on continuous, p iecewise Q(p) functions are considered for second order elliptic probl ems in three dimensions; these special methods can also be viewed as s pectral element methods. The first iterative method is designed for th e Galerkin formulation of the problem. The second applies to linear sy stems for a discrete model derived by using Gauss-Lobatto-Legendre qua drature. For both methods, it is established that the condition number of the relevant operator grows only in proportion to (1 + log p)(2). These bounds are independent of the number of elements, into which the given region has been divided, their diameters, as well as the jumps in the coefficients of the elliptic equation between elements. Results of numerical computations are also given, which provide upper bounds on the condition numbers as functions of p and which confirms the corr ectness of our theory.