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

We consider a generalized conjugate gradient method for solving sparse, 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