An engineering approach to generalized conjugate gradient methods and beyond

Abstract

This is an elementary introductory paper (rather a tutorial) for generalized conjugate gradient (CG) methods. The emphasis is on “understanding” the approach of the now rather sophisticated CG-type methods and beyond. Practical aspects like normalization and smoothing are discussed. Then the development of different types of methods with their characteristic properties is presented. For a model of the Navier-Stokes equations that varies between strong and weak ellipticity, the methods are compared and their typical properties are demonstrated.