PROJECTION MULTILEVEL METHODS FOR QUASILINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS: THEORETICAL RESULTS THOMAS A. MANTEUFFEL ‡ , STEPHEN F. MCCORMICK ‡ , AND OLIVER R ¨ OHRLE † Abstract. In a companion paper [8], we propose a new multilevel solver for two-dimensional elliptic systems of partial differential equations (PDEs) with nonlinearity of type u∂v. The approach is based on a multilevel projection method (PML [9]) applied to a first-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 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. Key words. Projection Method, Multigrid, Least Squares, Finite Elements, Quasilinear PDEs, Navier-Stokes AMS subject classifications. 35J60, 65N12, 65N30, 65N55 1. Introduction. Our companion paper [8] introduces a new multilevel solver for two-dimensional elliptic systems of partial differential equations (PDEs) with non- linearity of type u∂v. The approach is based on a multilevel projection method (PML [9]) applied to a first-order system least-squares (FOSLS) functional, where the non- linearity is treated directly, with no need for linearization anywhere in the algorithm. While [8] focuses on computation, the key objective of the present paper is to establish local well-posedness of our functional minimization problem. This result leads to the immediate conclusion that our two-level solver converges linearly with grid indepen- dent factors, as observed numerically in [8]. This two-grid result can be extended to W -cycles in the usual way. However, an important alternative would be to establish a V-cycle result based on the general theory developed in [11] and [12]. This alter- native would naturally yield grid-dependent convergence bounds because of the weak smoothness assumptions on the problem formulation (i.e., only Lipschitz continuity on the domain boundary). We base our theory for a two-level PML method on the first-order formulation of the Navier-Stokes formulation given in (2.1) below. Although we choose this formu- lation as a foundation for our theoretical framework, it is not limited to it: similar results can be established for other PDEs of this class. This paper is organized in the following way. Section 2 provides the first-order system formulation, with definitions, notation, and description of one two-level PML † Bioengineering Institute, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand. Email: o.rohrle@acukland.ac.nz. This work was sponsored by the Department of Energy under grant numbers DE-FC02-01ER25479 and DE-FG02-03ER25574, Lawrence Livermore National Laboratory under contract number B533502, Sandia National Laboratory under contract number 15268, and the National Science Foundation under VIGRE grant number DMS-9810751. ‡ Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO, 80309–0526. Email: {tmanteuf, stevem}@colorado.edu. This work was sponsored by the Department of Energy under grant numbers DE-FC02-01ER25479 and DE-FG02-03ER25574, Lawrence Livermore National Laboratory under contract number B533502, Sandia National Lab- oratory under contract number 15268, and the National Science Foundation under VIGRE grant number DMS-9810751. 1