Keywords: inexact solver, aggregation algorithm, BOSS method, preconditioning.

The largest systems of algebraic equations treatable on single processor computers are solved by a preconditioned iterative method. Direct methods cannot be used as a result of their large memory demands and iterative methods without preconditioning may converge very slowly. Analyses of plates or shells are typical examples of problems where the convergence is poor.

This paper describes application of the black-box overlapping Schwarz method with smoothed coarse space (BOSS) as preconditioner of the conjugate gradient method. The BOSS method collects unknowns into aggregates with no overlap. Piecewise constant indicator function is defined for each aggregate. If the coarse problem is assembled, the indicator functions are smoothed and the aggregates become overlapping. The smoothed indicator functions serve as basis functions of the coarse problem. The size of overlap depends on the degree of smoothing. The higher degree of smoothing leads to the larger overlap.

Local to global mapping between an aggregate and the whole problem is represented by localization matrix which is based on the smoothed indicator function. Matrices of enlarged aggregates are assembled from the system matrix with the help of the localization matrices. Correction operators on aggregates are defined with respect to the localization matrices and aggregate matrices.

The BOSS method contains two loops over all aggregates. Local vectors of particular aggregates are selected from a vector of the whole problem and the correction operator is applied. After correction, a new global vector is assembled from the local vectors. This new vector is projected to the coarse problem and coarse level correction is done. The second loop over the aggregates goes in the reverse order with respect to the first loop.

Any correction contains solution of local system of equations. The local system is usually solved exactly by the direct method. In the case of very large problems with large aggregates, solution of the local systems may be very expensive or it can even restrict the maximum number of unknowns which can be treated on a given computer. The exact solution can be exchanged by an inexact solution. It means, the factorization of local matrices is not performed and the local systems are solved by an iterative method. Requirements on the residual may be relaxed because the BOSS method serves as a preconditioner of the conjugate gradient method.

Numerical experiments show that significant amount of memory is saved if the exact local solvers are replaced by an inexact solver. Our implementation should be improved because we have not obtained any speedup of computation.

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