Iterative Sub-structuring for h-Version Boundary Element Approximation of the Hypersingular Operator in Three Dimensions

Additive Schwarz preconditioners are developed for the h-version of the boundary element method for the hypersingular integral equation on surfaces in three dimensions. The first preconditioner consists of decomposing into local spaces associated with the subdomain interiors, supplemented with a wirebasket space associated with the the subdomain interfaces. The wirebasket correction only involves the inversion of a diagonal matrix, while the interior correction consists of inverting the sub-blocks of the stiffness matrix corresponding to the interior degrees of freedom on each subdomain. It is shown that the condition number of the preconditioned system grows at most as H−1 (1 + log H/h) where H is the size of the subdomains and h is the size of the elements. A second preconditioner is given that incorporates a coarse space associated with the subdomains. This improves the robustness of the method with respect to the number of subdomains: theoretical analysis shows that growth of the condition number of the preconditioned system is now bounded by (1 + log H/h).