Projection Multilevel Methods for Quasilinear Elliptic Partial Differential Equations: Numerical Results

Abstract

The goal of this paper is to introduce a new multilevel solver for two-dimensional elliptic systems of nonlinear partial differential equations (PDEs), where the nonlinearity is of the type \(u \partial v\). The incompressible Navier--Stokes equations are an important representative of this class and are the target of this study. Using a first-order system least-squares (FOSLS) approach and introducing a new variable for $\partial v$, for this class of PDEs we obtain a formulation in which the nonlinearity appears as a product of two different dependent variables. The result is a system that is linear within each variable but nonlinear in the cross terms. In this paper, we introduce a new multilevel method that treats the nonlinearities directly. This approach is based on a projection multilevel (PML) method [S. F. McCormick, Multilevel Projection Methods for Partial Differential Equations, SIAM, Philadelphia, 1992] applied to the FOSLS functional. The implementation of the discretization process, relaxation, coarse-grid correction, and cycling strategies is discussed, and optimal performance is established numerically. A companion paper [T. A. Manteuffel, S. F. McCormick, and O. Rohrle, SIAM J. Numer. Anal., 44 (2006), pp. 139-152] establishes a two-level convergence proof for this new multilevel method.