A Moving-Mesh Finite Element Method and Its Application to the Numerical Solution of Phase-Change Problems

A distributed Lagrangian moving-mesh finite element method is applied to
problems involving changes of phase. The algorithm uses a distributed conservation
principle to determine nodal mesh velocities, which are then used to move the nodes.
The nodal values are obtained from an ALE (Arbitrary Lagrangian-Eulerian) equation,
which represents a generalization of the original algorithm presented in Applied
Numerical Mathematics, 54:450–469 (2005). Having described the details of the generalized
algorithm it is validated on two test cases from the original paper and is then
applied to one-phase and, for the first time, two-phase Stefan problems in one and two
space dimensions, paying particular attention to the implementation of the interface
boundary conditions. Results are presented to demonstrate the accuracy and the effectiveness
of the method, including comparisons against analytical solutions where
available.