An Efficient EXCMG-Newton Method Combined with Fourth-Order Compact Schemes for Semilinear Poisson Equations

Abstract

A fast solver for nonlinear systems arising from fourth-order compact finite difference schemes for two-dimensional semilinear Poisson equations is constructed. Applying the extrapolation and bi-quartic interpolation to two numerical solutions from the previous two levels of grids, we determine a suitable initial guess for the Newton iterations on the next finer grid. It is fifth-order accurate, which substantially reduces the number of Newton iterations required. Moreover, an extrapolated solution of sixth-order accuracy can be easily constructed on the whole fine grid. Numerical results suggest that the method is much more efficient than the existing multigrid methods for semilinear problems.