Multilevel First-Order System Least Squares for Nonlinear Elliptic Partial Differential Equations

A fully variational approach is developed for solving nonlinear elliptic equations that enables accurate discretization and fast solution methods. The equations are converted to a first-order system that is then linearized via Newton’s method. First-order system least squares (FOSLS) is used to formulate and discretize the Newton step, and the resulting matrix equation is solved using algebraic multigrid (AMG). The approach is coupled with nested iteration to provide an accurate initial guess for finer levels using coarse-level computation. A general theory is developed that confirms the usual full multigrid efficiency: accuracy comparable to the finest-level discretization is achieved at a cost proportional to the number of finest-level degrees of freedom. In a companion paper, the theory is applied to elliptic grid generation (EGG) and supported by numerical results.