A preconditioner based on domain decomposition for h-p finite-element approximation on quasi-uniform meshes

The problem of solving the algebraic systems arising from the discretization of a symmetric, elliptic boundary value problem using h-p finite-element methods in two dimensions is addressed. A preconditioning technique similar to those of Bramble, Pasciak, and Schatz [Math. Comp., 47 (1986), pp. 103-135] based on domain decomposition (also known as substructuring) is developed. The method is applicable to problems on general domains involving differential operators with quite general coefficients. The algorithm reduces to that of Bramble, Pasciak, and Schatz if linear elements are used. It is shown that the condition number of the preconditioned system (which determines the rate of convergence of the iterative method) grows at most as (1 + log(2) p) (1 + log(2) (Hp/h)). where p is the polynomial degree, h is the size of the elements, and H is the size of the subdomains.