Some Schwarz Algorithms for the P-version Finite Element Method

Domain decomposition methods based on the Schwarz framework were originally proposed for the h-version nite element method for elliptic problems. In this paper, we consider instead the p-version, in which increased accuracy is achieved by increasing the degree of the elements while the mesh is xed. We consider linear, scalar, self adjoint, second order elliptic problems and quadrilateral elements in the nite element discretization. For a class of overlapping additive Schwarz methods, we prove a constant bound, independent of the degree p and the number of elements N, for the condition number of the iteration operator. This optimal result holds in two and three dimensions for additive and multiplicative schemes, as well as variants on the interface. We then study local reenement for the same class of overlapping methods in two dimensions. A constant bound holds under certain hypotheses on the reenement region, while in general an almost optimal bound with logarithmic growth in p is obtained.