Traditional iterative methods are stalling numerical processes, in which the error has relatively small changes from one iteration to the next. Multigrid methods overcome the limitations of iterative methods and are computationally efficient. Convergence of iterative methods for elliptic partial differential equations is extremely slow. In particular, the convergence of the non-linear elliptic Poisson grid generation equations used for elliptic grid generation is very slow. Multigrid methods are fast converging methods when applied to elliptic partial differential equations. In this dissertation, a non-linear multigrid algorithm is used to accelerate the convergence of the non-linear elliptic Poisson grid generation method. The non-linear multigrid algorithm alters the performance characteristics of the non-linear elliptic Poisson grid generation method making it robust and fast in convergence. The elliptic grid generation method is based on the use of a composite mapping. It consists of a nonlinear transfinite algebraic transformation and an elliptic transformation. The composite mapping is a differentiable one-to-one mapping from the computational space onto the domains. Compact finite difference schemes are used for the discretization of the grid generation equations. Compared to traditional schemes, compact finite difference schemes provide better representation of shorter length scales and this feature brings them closer to spectral methods
