An additive Schwarz preconditioner for p -version boundary element approximation of the hypersingular operator in three dimensions

Abstract

Additive Schwarz preconditioners are developed for the p-version of the boundary element method for the hypersingular integral equation on surfaces in three dimensions. The principal preconditioner consists of decomposing the subspace into local spaces associated with the element interiors supplemented with a wirebasket space associated with the the element interfaces. The wirebasket correction involves inverting a diagonal matrix. If exact solvers are used on the element interiors then theoretical analysis shows that growth of the condition number of the preconditioned system is bounded by \((1+\log p)^2\) for an open surface and \((1+\log p)\) for a closed surface. A modified form of the preconditioner only requires the inversion of a diagonal matrix but results in a further degradation of the condition number by a factor \(p(1+\log p)\).