A Galerkin method for a nonlinear Dirichlet problem

Abstract

A Galerkin method due to Nitsche for treating the Dirichlet problem for a linear second-order elliptic equation is extended to cover the nonlinear equation ∇ ⋅ ( a ( x , u ) ∇ u ) = f \nabla \cdot (a(x,u)\nabla u) = f . The asymptotic error estimates are of the same form as in the linear case. Newton’s method can be used to solve the nonlinear algebraic equations.