The Analysis of Multigrid Algorithms for Nonsymmetric and Indefinite Elliptic Problems*

We prove some new estimates for the convergence of multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems. We provide results for the so-called 'symmetric' multigrid schemes. We show that for the variable 2^-cycle and the ^-cycle schemes, multigrid algorithms with any amount of smoothing on the finest grid converge at a rate that is independent of the number of levels or unknowns, provided that the initial grid is sufficiently fine. We show that the 2^-cycle algorithm also converges (under appropriate assumptions on the coarsest grid) but at a rate which may deteriorate as the number of levels increases. This deterioration for the ^-cycle may occur even in the case of full elliptic regularity. Finally, the results of numerical experiments are given which illustrate the convergence behavior suggested by the theory.