The Arbitrary Lagrangian Eulerian (ALE) method has been developed by [49,50,51,52,53] and applied in to quadrilateral spectral elements in [54].
So far we have considered domains that are unchanging in time. The
next step is to allow parts of the domain to move in a time dependent
manner. This is not a trivial generalization because we now have to
discretize time dependent operators as well as time dependent
fields. We have made an initial step here, however the implementation
we have used is far from ideal. The principle we wish to demonstrate
here is that the ALE approach using unstructured spectral elements is
robust and has great potential. We must admit that a lot of work (and
questions) remains in order to efficiently implement the time
dependent operators. For the moment we reform the local elemental
operators at every time step. We do this because as yet we do not have
an effective preconditioner for the full
implicit solver. By experiment we have determined that it is more
efficient in two-dimensional simulations to use the fully direct Schur
complement method for low (
) simulations and the
iterative-boundary direct-interior Schur complement method
[30] for higher order. Bearing in mind the
aforementioned caveats we will demonstrate the possible advantages of
using the ALE method with the spectral element method. Namely, the
usual advantages of minimal dispersion problems, high accuracy and the
novel fact that since the spectral elements are larger and less
numerous than finite element or finite volume elements they can
support larger deformations without becoming entangled. We have also
shown that the triangles and tetrahedra can support deformation
without loosing excessive resolution.