PROJECTION MULTILEVEL METHODS FOR QUASILINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS: THEORETICAL RESULTS

Abstract. In a companion paper [8], we propose a new multilevel solver for two-dimensional elliptic systems of partial di®erential equations (PDEs) with nonlinearity of type u@v. The approach is based on a multilevel projection method (PML [9]) applied to a ¯rst-order system least-squares (FOSLS) functional that allows us to treat the nonlinearity directly. While [8] focuses on compu-tation, here we concentrate on developing a theoretical framework that con¯rms optimal two-level convergence. To do so, we choose a ¯rst-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