Unified Analysis of Finite Element Methods for Problems with Moving Boundaries

We present a unified analysis of finite element methods for problems with prescribed moving boundaries. In particular, we study an abstract parabolic problem posed on a moving domain with prescribed evolution, discretized in space with a finite element space that is associated with a moving mesh that conforms to the domain at all times. The moving mesh is assumed to evolve smoothly in time, except perhaps at a finite number of remeshing times where the solution is transferred between finite element spaces via a projection. A key result of our analysis is an abstract estimate for the L-norm of the error between the exact and semidiscrete solutions at a fixed positive time, expressed in terms of the total variation in time of a quantity that measures the difference between the exact solution at time t and its elliptic projection onto the finite element space at time t. Specializing the abstract estimate to particular choices of the mesh motion strategy, finite element space, and projector leads to error estimates in terms of the mesh spacing for various semidiscrete schemes. In particular, the estimate can be specialized to conventional arbitrary Lagrangian-Eulerian (ALE) schemes with remeshing as well as schemes based upon universal meshes, where the mesh motion is derived from small deformations of a periodically updated reference subtriangulation of a background mesh that contains the moving domain. We demonstrate such an application by deducing error estimates of optimal order in the mesh spacing for ALE schemes under mild assumptions on the nature of the mesh deformation and the regularity of the exact solution and the moving domain.