Optimal convergence of arbitrary Lagrangian–Eulerian finite element methods for the Stokes equations in an evolving domain

The numerical solution of the Stokes equations in an evolving domain with a moving boundary is studied based on the arbitrary Lagrangian–Eulerian finite element method along the trajectories of the evolving mesh, where the Taylor–Hood finite elements of degree r ≥ 2 is employed for spatial discretization. The error of the semidiscrete arbitrary Lagrangian–Eulerian method is proved to be O(h^r+1) for velocity approximation in L^∞(0, T; L²) norm and O(h^r) for pressure approximation in L²(0, T; L²) norm, respectively. The analyses are based on a Nitsche’s duality argument adapted to an evolving mesh, by proving that the material derivative and the Stokes–Ritz projection commute up to terms that have optimal-order convergence in the L² norm. Numerical examples are provided to support the theoretical analysis.