A parallel Schwarz preconditioner for the Chebyshev Gauss–Lobatto collocation (d2dx2−h2) operator

A Schwarz method is proposed to solve a Chebyshev one-dimensional Helmholtz problem of polynomial order k⁎nk⁎n by iteratively solving problems of polynomial order up to n, which allows the use of a parallel algorithm. Its efficiency is commented. A multigrid global approach is implemented to accelerate the convergence, leading to an additive system of size 2k or 3k to be solved, depending on the kind of used boundary conditions. A preliminary application to a 2D problem is proposed.