An additive Schwarz method for the h-p version of the boundary element method for hypersingular integral equations in R~3

摘要

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_jp_j)/(h_j))~2 where H_j is the diameter of an element Γ_j of the coarse grid, h_j is the size of the elements of the fine grid on Γ_j, and p_j is the maximum of the polynomial degrees used in Γ_j. Numerical results supporting our theory are reported.