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 $L^2$-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 e...

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