Chad Westphal
FOSLL$^*$ For Nonlinear Partial Differential Equations

Dept of Math & CS
Wabash College
Crawfordsville
IN 47933
westphac@wabash.edu
Eunjung Lee
Thomas Manteuffel

Least-squares finite element methods are designed around the idea of minimizing discretization error in an appropriate norm. For sufficiently regular, elliptic-like problems, a first-order system least squares (FOSLS) approach may be continuous and coercive in the $H^1$ norm, yielding solutions accurate in $H^1$ . For problems posed in high aspect ratio domains (e.g., flow through a long channel), approximations on coarse resolutions may have error whose $L^2$ norm is large relative to the $H^1$ seminorm. The first order system $LL^*$ (FOSLL$^*$ ) approach seeks to minimize the residual of the equations in a dual norm induced by the differential operator, yielding a better $L^2$ approximation. Mass conservation in fluid flow, for example, is greatly enhanced by such an approach. In this talk, we extend this general framework to nonlinear problems.

Newton's method is a typical outer iteration for an efficient finite element approximation of nonlinear partial differential equations. We present the framework for an inexact Newton iteration based on a FOSLL$^*$ approximation to each linearization step and establish theory for convergence. Numerical results are presented for a velocity-vorticity-pressure formulation of the steady incompressible Navier-Stokes equations, and we discuss extensions and comparisons to the more typical Newton-FOSLS approach.