The Postprocessed Mixed Finite-Element Method for the Navier--Stokes Equations

Abstract

Abstract.A postprocessing technique for mixed finite-element methods for the incompressible Navier-Stokes equations is studied. The technique was earlier developed for spectral and standard finite-element methods for dissipative partial differential equations. The postprocessing amounts to solving a Stokes problem on a finer grid (or higher-order space) once the time integration on the coarser mesh is completed. The analysis presented here shows that this technique increases the convergence rate of both the velocity and the pressure approximations. Numerical experiments are presented that confirm both this increase in the convergence rate and the corresponding improvement in computational efficiency.Key words. Navier-Stokes equations, mixed finite-element methods AMS subject classifications. 65M60, 65M20, 65M15, 65M12 DOI. 10.1137/0406028211. Introduction. This paper in a sense culminates the development of a postprocessing technique to increase the accuracy and computational efficiency of Galerkin methods for dissipative partial differential equations introduced in [18]. We turn to the equations which gave rise to this postprocessing technique, the incompressible Navier-Stokes equations, and we address those Galerkin methods for these equations which, when complex-shaped bodies are present, are acknowledged to be of wider applicability, mixed finite-element (MFE) methods.The postprocessing technique we study here was originally developed for spectral methods [18], [19]. At that moment, either its analysis and understanding or its development seemed to depend heavily on the properties of the Fourier modes, although this was not a shortcoming to prove its usefulness in the study of nonlinear shell vibrations [27]. In later works [13], [14], the dependence on the Fourier modes was overcome. Of particular importance to the present work, besides [14], has been the development of the postprocessing technique for finite-element methods in [20], [15]. In [20], it was devised how to carry out the postprocessing without the help of an approximate inertial manifold [11], [12], a concept more suited to spectral methods and eigenfunction expansion. In [15], it is shown what gains can be expected when postprocessing low-order elements.As is usually the case with MFE methods, it is the experience and understanding gained in previous works (see [14] [20], and the references cited therein) with simpler equations and methods which has allowed the present one to be written. Furthermore, although for simplicity we focus on Hood-Taylor [26] elements, the postprocessing technique can be easily adapted to other kinds of mixed

The Postprocessed Mixed Finite-Element Method for the Navier–Stokes Equations: Refined Error Bounds

“…By writing Proof. The proof follows the same steps as [6,Theorem 3.14]. Let S h (u(t)) ∈ V be the Stokes projection of the solution of (1.1)-(1.2) at time t. We decompose u(t) −ũ h (t) l ≤ u(t) − S h (u(t)) l + S h (u(t)) −ũ h (t) l , l = 0, 1.…”

“…Numerical experiments in [6], [16], [19], [17], [26], [28] have repeatedly shown, for the different discretizations considered, that the increase in accuracy and convergence rate predicted by the theory is also seen in practice (provided errors arising from the time discretization are kept sufficiently small). Nevertheless, in the present paper we give an explanation of this fact; that is, the gain in (spatial) accuracy in the postprocessing step takes place independently of errors arising from the temporal discretization being present or not.…”

“…In the case of low-order LBB-stable elements, the analysis is much simpler; see Remark 4.2 below. However, since to simplify the analysis we make use of several results from [6] which are stated and proved for Hood-Taylor elements, we will restrict ourselves to these elements in the present paper.…”

“…The proof is reached by reasoning as in [6,Theorem 3.15]. Let us denote byq h (t) the approximation to the pressure p(t) such that ( S h (u(t)),q h (t)) ∈ ( X, Q) solves (2.11)-(2.12) with g = f − u t − u · ∇u.…”

“…These are not with respect to the Stokes projection (as in [17], [6], [4]) but with respect to the solution of a certain linear evolution problem. In this process, we also obtain error bounds for the pressure that improve those originally proved in [37,Theorem 3.1] by a factor of t 1/2 .…”