A fourth‐order difference method for elliptic equations with nonlinear first derivative terms

We present a nine‐point fourth‐order finite difference method for the nonlinear second‐order elliptic differential equation Au xx + Bu yy = f ( x, y, u, u x , u y ) on a rectangular region R subject to Dirichlet boundary conditions u ( x, y ) = g ( x, y ) on ∂ R . We establish, under appropriate conditions O( h 4 )‐convergence of the finite difference scheme. Numerical examples are given to illustrate the method and its fourth‐order convergence.