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

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∂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 ∂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 multilevel projection method (PML [23]) 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 [22] establishes a two-level convergence proof for this new multilevel method.