Two-level method for the discretization of nonlinear boundary value problems

We consider a two-level method for the discretization and solution of nonlinear boundary value problems. The method basically involves (i) solving the nonlinear problem on a very coarse mesh, (ii) linearizing about the coarse mesh solution, and solving the linearized problem on the fine mesh, one time! We analyze the accuracy of this procedure for strongly monotone nonlinear operators ( 2) and general semilinear elliptic boundary value problems (without monotonicity assumptions). in particular, the scaling between the fine and coarse mesh widths required to ensure optimal accuracy of the fine mesh solution is derived as a byproduct of the error estimates herein.