Newton-LL∗ method for the second-order semi-linear elliptic partial differential equations

Newton's method with first-order system least squares (FOSLS) finite element method has been widely used to approximately solve a system of nonlinear partial differential equations (Adler et al., 2010  [9], Codd et al., 2003  [10], Manteuffel et al., 2006  [11]). In this paper, we propose to use the first order system LL∗ method to find a correction in each Newton's iteration which provides an L2-approximation of the second-order semi-linear elliptic partial differential equations while the typical Newton-FOSLS method providesH1-approximations. The numerical tests have been conducted to validate the theory.