By Jinchao Xu.
Abstract: In this talk, we will present a number of results on multigrid and domain decomposition methods for partial differential equations with strongly discontinuous coefficients. We will first briefly present a related general result which states that the method of subspace correction will converge uniformly for nearly singular problems as long as the near-null space can be fully recovered by its components in all subspaces. We then show that the preconditioned conjugate gradient method will converge uniformly with respect to the discontinuous jumps by using (1) the BPX multilevel preconditioner and (2) a variant of the BPS nonoverlappting domain decomposition method with a simplified coarse space (without using harmonic extension). We will finally report some recent results on robust preconditioners for H(curl) systems and for the Brinkman model for coupling Stokes Stokes equations and Darcy’s law, both with strongly discontinuous coefficients.