===affil2:
Yonsei University
===firstname:
Chad
===firstname4:

===firstname3:
Thomas
===lastname2:
Lee
===lastname:
Westphal
===firstname5:

===affil6:

===lastname3:
Manteuffel
===email:
westphac@wabash.edu
===lastname6:

===affil5:

===otherauths:

===lastname4:

===affil4:

===lastname7:

===affil7:

===firstname7:

===postal:
Dept. of Math \& CS
Wabash College
Crawfordsville, IN 47933
===firstname6:

===ABSTRACT:
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.
===affil3:
University of Colorado
===lastname5:

===affilother:

===title:
FOSLL$^*$ For Nonlinear Partial Differential Equations
===firstname2:
Eunjung