FOSLL∗ for nonlinear partial differential equations

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 L²(Ω) rather than H¹(Ω) 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 L²(Ω) 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.