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 boundaryelement method for hypersingular integral equations in threedimensions. The preconditioner is based on a three-level decompositionof the underlying ansatz space, the levels being piecewise bilinearfunctions on a coarse grid, piecewise bilinear functions ona fine grid, and piecewise polynomials of high degree on thefine grid. We prove that the condition number of the preconditionedlinear system is bounded by max_j (1 + log H_jp_j/h_j)² where H_jis the diameter of an element _j of the coarse grid, h_j is thesize of the elements of the fine grid on _j, and p_j is the maximumof the polynomial degrees used in _j. Numerical results supportingour theory are reported. Received 9 March 1999. Accepted 19 July 1999.