Two-Level Local Refinement Preconditioners for Nonsymmetric and Indefinite Elliptic Problems

Preconditioners of optimal order for nonselfadjoint and indefinite elliptic boundary value problems discretized on grids with local refinement are constructed. The proposed technique utilizes solution of a discrete problem on a uniform coarse grid; then, the reduced problem is handled by a generalized conjugate gradient (GCG) method. The reduced problem is coercive if the initial coarse mesh is sufficiently fine and is local, solving only for the unknowns on the subdomains where local refinement has been introduced. The reduced problem can be preconditioned by a preconditioner for the symmetric positive definite matrix arising from the symmetric and coercive principal part of the original bilinear form restricted to the subdomains containing local refinement. This problem also utilizes a uniform grid. In the numerical tests, the recent algebraic multilevel (AMLI) preconditioners [Axelsson and Vassilevski, SIAM j. Numer. Anal., 27 (1990), pp. 1569–1590; Saad and Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869], which are of optimal order for selfadjoint and coercive elliptic problems, were used.