Abstract:In this study, we use a residual-norm-based algorithm (RNBA) to solve nonlinear elliptic boundary value problems (BVPs) on an arbitrary planar domain. For complex geometries, BVPs are very difficult and time-consuming to solve using conventional finite difference methods (FDMs). To overcome these problems, we apply a novel finite difference method (NFDM). By adding a fictitious rectangular domain and a bilinear function, we can easily treat geometrically complex boundary conditions. Then, through the use of the internal residual and the boundary residual, we can easily obtain the solution without the necessity of computing a matrix inverse. The RNBA avoids the oscillations that can occur in the manifold-based exponentially convergent algorithm (MBECA) by maintaining the manifold properties while guaranteeing convergence order greater than one. The accuracy and the convergence behaviour of this new method are demonstrated with several examples.