The Analysis of Multigrid Algorithms for Nonsymmetric and Indefinite Elliptic Problems

Abstract

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 $\mathcal {V}$-cycle and the $\mathcal {W}$-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 $\mathcal {V}$-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 $\mathcal {V}$-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.