An additive Schwarz method for the h-p version of the boundary element method for hypersingular integral equations in R3

We study a preconditioner for the h-p version of the boundary element method for hypersingular integral equations in three dimensions. The preconditioner is based on a three-level decomposition of the underlying ansatz space, the levels being piecewise bilinear functions on a coarse grid, piecewise bilinear functions on a fine grid, and piecewise polynomials of high degree on the fine grid. We prove that the condition number of the preconditioned linear system is bounded by max(j)(1 + log H(j)p(j)/h(j))(2) where H-j is the diameter of an element Gamma (j) of the coarse grid, h(j) is the size of the elements of the fine grid on Gamma (j), and p(j) is the maximum of the polynomial degrees used in Gamma (i). Numerical results supporting our theory are reported.