Projection multilevel methods for quasilinear elliptic partial differential equations:: Theoretical results

In a companion paper [T. A. Manteuffel et al., SIAM J. Numer. Anal., 44 (2006), pp. 120 - 138], we propose a new multilevel solver for two-dimensional elliptic systems of partial differential equations with nonlinearity of type u partial derivative v. The approach is based on a multilevel projection method (PML) [S. F. McCormick, Multilevel Projection Methods for Partial Differential Equations, SIAM, Philadelphia, 1992] applied to a first-order system least-squares functional that allows us to treat the nonlinearity directly. While the companion paper focuses on computation, here we concentrate on developing a theoretical framework that confirms optimal two-level convergence. To do so, we choose a first-order formulation of the Navier-Stokes equations as a basis of our theory. We establish continuity and coercivity bounds for the linearized Navier-Stokes equations and the full nonquadratic least-squares functional, as well as existence and uniqueness of a functional minimizer. This leads to the immediate result that one cycle of the two-level PML method reduces the functional norm by a factor that is uniformly less than 1.