Iterative Methods for Nonlinear Elliptic Equations

‖f(u)− f(v)‖−1 ≤ L‖u− v‖−1. We drop the dependence of f on x to emphasis the nonlinearity in u. Let Vh ⊂ H 0 (Ω) be the linear finite element space associated to a quasi-uniform mesh Th. The nonlinear Garlerkin method is: find uh ∈ Vh such that (2) Auh = f(uh) in Vh, where A : H 0 (Ω)→ H−1(Ω) is the elliptic operator defined by −∆. We shall assume the existence and locally uniqueness of the solution to the nonlinear equation (1). Then it can be shown that (c.f. [11] and the references cited therein) if h is sufficiently small, the equation (2) for the nonlinear Garlerkin method has a (locally unique) solution uh satisfying the error estimate (3) ‖u− uh‖0,p + h‖u− uh‖1,p ≤ C(‖u‖2,p)h for all 2 ≤ p <∞. We are interested in iterative methods to compute uh or a good approximation of uh. We shall drop the subscript h when the size h does not play a role in the method.