In this talk, I first give an overview of multigrid methods, which are widely used for the numerical solution of partial differential equations. These methods offer the possibility of solving large and sparse linear systems with linear complexity, yielding substantial improvements in computational efficiency.  With the help of local Fourier analysis, we develop fast multigrid solvers for fluid flow. We present some applications that illustrate the merits of the proposed algorithms. We then move to present the Anderson acceleration scheme for solving nonlinear problems. We provide an overview of our theoretical contributions towards understanding the convergence behavior of the Anderson method.