A Black Box Generalized Conjugate Gradient Solver with Inner Iterations and Variable-Step Preconditioning

The generalized conjugate gradient method proposed by Axelsson is studied in the case when a variable-step preconditioning is used. This can be the case when the preconditioned system is solved approximately by an auxiliary (inner) conjugate gradient method, for instance, and the thus-obtained quasi residuals are used to construct the next search vector in the outer generalized cg-iteration method. A monotone convergence of the method is proved and a rough convergence rate estimate is derived, provided the variable-step preconditioner (generally, a nonlinear mapping) satisfies a continuity and a coercivity assumption. These assumptions are verified for application of the method for two-level grids and indefinite problems. This variable-step preconditioning involves, for the two-level case, the solution of the coarse grid problem and problems for the nodes on the rest of the grid—both by auxiliary (inner) iterative methods. For the indefinite problems that are considered, the special block structure of the matrix is utilized—also in an outer-inner iterative method. For both the outer and inner iterations, parameter-free preconditioned generalized conjugate gradient methods are advocated. For indefinite problems the method used offers an alternative to the well-known Uzawa algorithm.