A Generalized Conjugate Gradient Method for the Numerical Solution of Elliptic Partial Differential Equations

We consider a generalized conjugate gradient method for solvingsparse, symmetric, positive-definite systems of linear equations,
principally those arising from the discretization of boundary value
problems for elliptic partial differential equations. The method
is based on splitting off from the original coefficient matrix a
symmetric, positive-definiteonethat corresponds to a more easily
solvable system of equations, and then accelerating the associated
iteration using conjugate gradients. Optimality and convergence
properties are presented, and the relation to other methods is
discussed.
Several splittings for which the method seems particularly
effective are also discussed, and for some, numerical examples are given