Abstract
In previous work, the first-order system LL* (FOSLL*) method was developed for linear partial differential equations. This approach seeks to minimize the residual of the equations in a dual norm induced by the differential operator, yielding approximations accurate in $L^2(\Omega)$ rather than $H^1(\Omega)$ or $H(Div)$. In this paper, the general framework of FOSLL* is extended to a wide range of nonlinear problems. Four approaches to propagating an inexact Newton iteration based on a FOSLL* approximation are presented, and theory for robust convergence in $L^2(\Omega)$ is established. Numerical results are presented for two formulations of the steady incompressible Navier--Stokes equations and for a diffusion equation with reduced regularity due to a discontinuous diffusion coefficient.
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