Additive Schwarz methods for the \(p\)-version finite element method (Q1326487)

We study some additive Schwarz methods (ASM) for the \(p\)-version finite element method. We consider linear, scalar, self adjoint, second order elliptic problems and quadrilateral elements in the finite element discretization. We prove a constant bound, independent of the degree \(p\) and the number of subdomains \(N\), for the condition number of the ASM iteration operator. This optimal result is obtained first in dimension two. It is then generalized to dimension \(n\) and to a variant of the method on the interface. Numerical experiments confirming these results are reported. As is the case for other additive Schwarz methods, our algorithms are highly parallel and scalable.