GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Journal of Scientific Computing
Discretization of free surface flows and other moving boundary problems
Journal of Computational Physics
Journal of Computational Physics
Three-Dimensional Front Tracking
SIAM Journal on Scientific Computing
Journal of Electronic Materials - Special issue on the 1997 U.S. workshop on the physics and chemistry of II-VI materials
Multigrid methods for incompressible heat flow problems with an unknown interface
Journal of Computational Physics
Computation of solid-liquid phase fronts in the sharp interface limit on fixed grids
Journal of Computational Physics
Numerical Methods for Problems with Moving Fronts
Numerical Methods for Problems with Moving Fronts
Computational Fluid Dynamics with Moving Boundaries
Computational Fluid Dynamics with Moving Boundaries
The Diffuse Interface Approach in Materials Science: Thermodynamic Concepts and Applications of Phase-Field Models
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A Schur complement formulation that utilizes a linear iterative solver is derived to solve a free-boundary, Stefan problem describing steady-state phase change via the Isotherm-Newton approach, which employs Newton's method to simultaneously and efficiently solve for both interface and field equations. This formulation is tested alongside more traditional solution strategies that employ direct or iterative linear solvers on the entire Jacobian matrix for a two-dimensional sample problem that discretizes the field equations using a Galerkin finite-element method and employs a deforming-grid approach to represent the melt-solid interface. All methods demonstrate quadratic convergence for sufficiently accurate Newton solves, but the two approaches utilizing linear iterative solvers show better scaling of computational effort with problem size. Of these two approaches, the Schur formulation proves to be more robust, converging with significantly smaller Krylov subspaces than those required to solve the global system of equations. Further improvement of performance are made through approximations and preconditioning of the Schur complement problem. Hence, the new Schur formulation shows promise as an affordable, robust, and scalable method to solve free-boundary, Stefan problems. Such models are employed to study a wide array of applications, including casting, welding, glass forming, planetary mantle and glacier dynamics, thermal energy storage, food processing, cryosurgery, metallurgical solidification, and crystal growth.