A priori error analysis for finite element approximations of the Stokes problem on dynamic meshes

In this article we study finite element approximations of the time-dependent Stokes system on dynamically changing meshes. Applying the backward Euler method for time discretization we use the discrete Helmholtz or Stokes projection to evaluate the solution at time t(n-1) on the new spatial mesh at time t(n). The theoretical results consist of a priori error estimates that show a dependence on the time step size not better than O(1/Delta t). These surprisingly pessimistic upper bounds are complemented by numerical examples giving evidence for a negative convergence rate, at least for a large range of time step sizes, and in this sense backing our theory. These observations imply that using adaptive meshes for incompressible flow problems is delicate and requires further investigation.