Efficient nonlinear solvers for Laplace-Beltrami smoothing of three-dimensional unstructured grids

The Laplace-Beltrami system of nonlinear, elliptic, partial differential equations has utility in the generation of computational grids on complex and highly curved geometry. Discretization of this system using the finite element method accommodates unstructured grids, but generates a large, sparse, ill-conditioned system of nonlinear discrete equations. The use of the Laplace-Beltrami approach, particularly in large-scale applications, has been limited by the scalability and efficiency of solvers. This paper addresses this limitation by developing two nonlinear solvers based on the Jacobian-Free Newton-Krylov (JFNK) methodology. A key feature of these methods is that the Jacobian is not formed explicitly for use by the underlying linear solver. Iterative linear solvers such as the Generalized Minimal RESidual (GMRES) method do not technically require the stand-alone Jacobian;