  The numerical solution of the Stokes equations on an evolving domain with a
moving boundary is studied based on the arbitrary Lagrangian-Eulerian finite
element method and a second-order projection method along the trajectories of
the evolving mesh for decoupling the unknown solutions of velocity and
pressure. The error of the semidiscrete arbitrary Lagrangian-Eulerian method is
shown to be $O(h^{r+1})$ for the Taylor--Hood finite elements of degree $r\ge
2$, using Nitsche's duality argument adapted to an evolving mesh, by proving
that the material derivative and the Stokes--Ritz projection commute up to
terms which have optimal-order convergence in the $L^2$ norm. Additionally, the
error of the fully discrete finite element method, with a second-order
projection method along the trajectories of the evolving mesh, is shown to be
$O(\ln(1/\tau+1)\tau^{2}+\ln(1/h+1)h^{r+1})$ in the discrete $L^\infty(0,T;
L^2)$ norm using newly developed energy techniques and backward parabolic
duality arguments that are applicable to the Stokes equations with an evolving
mesh. To maintain consistency between the notations of the numerical scheme in
a moving domain and those in a fixed domain, we introduce the equivalence class
of finite element spaces across time levels. Numerical examples are provided to
support the theoretical analysis and to illustrate the performance of the
method in simulating Navier--Stokes flow in a domain with a rotating propeller.
