Numerical grid generation: foundations and applications
Numerical grid generation: foundations and applications
An adaptive grid with directional control
Journal of Computational Physics
Elliptic grid generation based on Laplace equations and algebraic transformations
Journal of Computational Physics
Computational conformal mapping for surface grid generation
Journal of Computational Physics
NITSOL: A Newton Iterative Solver for Nonlinear Systems
SIAM Journal on Scientific Computing
An object-oriented framework for block preconditioning
ACM Transactions on Mathematical Software (TOMS)
An r-adaptive finite element method based upon moving mesh PDEs
Journal of Computational Physics
A multgrid Newton-Krylov method for multimaterial equilibrium radiation diffusion
Journal of Computational Physics
A Multigrid Preconditioned Newton--Krylov Method
SIAM Journal on Scientific Computing
A multigrid tutorial: second edition
A multigrid tutorial: second edition
A review of algebraic multigrid
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
Multigrid
Quasi-Orthogonal Grids with Impedance Matching
SIAM Journal on Scientific Computing
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Multilevel First-Order System Least Squares for Elliptic Grid Generation
SIAM Journal on Numerical Analysis
Jacobian-free Newton-Krylov methods: a survey of approaches and applications
Journal of Computational Physics
A finite element method for unstructured grid smoothing
Journal of Computational Physics
A finite element method for three-dimensional unstructured grid smoothing
Journal of Computational Physics
A fully implicit, nonlinear adaptive grid strategy
Journal of Computational Physics
A Computational Differential Geometry Approach to Grid Generation (Scientific Computation)
A Computational Differential Geometry Approach to Grid Generation (Scientific Computation)
Hi-index | 0.09 |
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; instead its action on a vector is approximated through two nonlinear function evaluations. The preconditioning required by GMRES is also discussed. Two different preconditioners are developed, both of which employ existing Algebraic Multigrid (AMG) methods. Further, the most efficient preconditioner, overall, for the problems considered is based on a Picard linearization. Numerical examples demonstrate that these solvers are significantly faster than a standard Newton-Krylov approach; a speedup factor of approximately 26 was obtained for the Picard preconditioner on the largest grids studied here. In addition, these JFNK solvers exhibit good algorithmic scaling with increasing grid size.