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.