Simple and Efficient ALE Methods with Provable Temporal Accuracy up to Fifth Order for the Stokes Equations on Time Varying Domains

We present a class of semi-implicit finite element (FE) schemes that uses arbitrary Lagrangian Eulerian methods (ALE) to solve the incompressible Navier–Stokes equations (NSE) on time varying domains. We use the kth order backward differentiation formula (BDFk) and Taylor– Hood Pm/Pm−1 finite elements. The well-known telescope formulas of BDFk have been extended from k = 1, 2 to k = 3, 4, 5. They enable us to prove that when k ≤ 5, for Stokes equations on a fixed domain, our schemes converge at rate O(Δtk + hm+1). When the domain is varying with respect to time and when h/Δt = O(1), the convergence rate reduces to O(Δtk + hm). For analysis, we assume that meshes at different time levels have the same topology. Consequently, our methods do not require the computation of characteristic paths and are Jacobian-free. Numerical tests for NSE on time varying domains are presented. They indicate that our schemes may have full accuracy on time varying domains and can handle meshes with large aspect ratio. The benchmark test of flow past an oscillating cylinder is also performed.