Krylov subspace methods for large linear systems of equations

When solving PDE's by means of numerical methods one often has to deal with large systems of linear equations, specifically if the PDE is time-independent or if the time-integrator is implicit. For real life problems, these large systems can often only be solved by means of some iterative method. Even if the systems are preconditioned, the basic iterative method often converges slowly or even diverges. We discuss and classify algebraic techniques to accelerate the basic iterative method. Our discussion includes methods like CG, GCR, ORTHODIR, GMRES, CGNR, Bi-CG and their modifications like GMRESR, CG-S, Bi- CGSTAB.We place them in a frame, discuss their convergence behavior and their advantages and drawbacks