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

Abstract

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 TaylorHood Pm/P m−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(Δt k + h m+1 ). When the domain is varying with respect to time and when h/Δt = O(1), the convergence rate reduces to O(Δt k + h m ). 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.

Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations

“…The generalization of the scheme (2.7)-(2.8) to higher order BDF type schemes is straightforward with a general BDF operator D τ [16,25], e.g.,…”

“…Lemma 3.2 Telescope formula for D τ (see [25]): With the definition of the BDF temporal discrete operator D τ in (2.6), there exists c i , i = 1, . .…”

“…It is worth mentioning that the BDF type schemes achieve high order accuracy without increasing the computation significantly, which is a major advantage of the BDF type schemes compared with high order Runge-Kutta schemes. The BDF schemes also have been widely used for the time evolution in numerical PDEs, see [6,11,15,25] and references therein. We note that in a recent work of Liu [25], energy argument for higher order (> 3) BDF schemes for the time-dependent Stokes equation was established to prove the convergence and stability of the scheme.…”

“…The BDF schemes also have been widely used for the time evolution in numerical PDEs, see [6,11,15,25] and references therein. We note that in a recent work of Liu [25], energy argument for higher order (> 3) BDF schemes for the time-dependent Stokes equation was established to prove the convergence and stability of the scheme. Energy argument greatly simplifies the analysis of BDF schemes for parabolic equations.…”

“…Thus, with the help of energy method argument, the results obtained for the first order schemes in [13,23] and the second order scheme in [22] can be extended to higher order BDF type schemes. Combining the Li-Sun error splitting argument [23,24] and the telescope formula for the BDF temporal discretization operator [25], we are able to derive a primary error estimate unconditionally which directly implies uniform boundedness of the FEM solutions in certain strong norms. Then we apply a traditional approach to obtain an optimal L 2 error estimate for r -th order Lagrange element methods (r ≥ 1) unconditionally.…”