Iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems. Two preconditioners for p-version finite element methods based on continuous, piecewise Qp functions are considered for second order elliptic problems in three dimensions; these special methods can also be viewed as spectral element methods. The first iterative method is designed for the Galerkin formulation of the problem. The second applies to linear systems for a discrete model derived by using Gauss-Lobatto-Legendre quadrature. 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 correctness of our theory.
Pavarino, L. F., & Widlund, O. B. (1997). Iterative substructuring methods for spectral elements: Problems in three dimensions based on numerical quadrature. Computers and Mathematics with Applications, 33(1–2), 193–209. https://doi.org/10.1016/s0898-1221(96)00230-1