New class of iterative methods for nonselfadjoint or indefinite problems

A new technique is proposed to solve NSPD (nonsymmetric or indefinite) problems that are 'compact' perturbations of some SPD (symmetric positive definite) problems. In the new algorithm, a direct method is first used to solve the original equation restricted on a coarser space (that has a considerably smaller dimension), then an SPD equation for the residue is solved by using one or a few iterations of a given iterative algorithm. It is shown that for any convergent iterative method for the SPD problem, the new algorithm always converges with essentially the same rate if the coarse space is properly chosen. In applications, for multiplicative domain decomposition methods, the algorithm consists of solving the original NSPD problem on the coarse mesh and solving SPD equations on all subdomains; for multigrid methods, except when the correction on the coarsest mesh is first performed for the original NSPD equation, all smoothings are carried out for SPD equations on all other levels. It is shown that most of these algorithms converge uniformly provided that the coarsest meshsize is reasonably small (but independent of the fine meshsize). Discussions are also devoted to some popular algorithms such as SOR method.