Original articleFully discrete finite element method based on pressure stabilization for the transient Stokes equations☆
Introduction
Let Ω be a bounded domain in assumed to have a Lipschitz continuous boundary ∂Ω. We consider the following time-dependent Stokes problem
where u = (u1(x, t), u2(x, t))T is the velocity, p = p(x, t) the pressure, f = f(x, t) ∈ L2(Ω)2 the prescribed body force, ν > 0 the viscosity, u0 the initial velocity, and T > 0 a finite time.
The development of efficient mixed finite element methods for the Stokes equations is an important but challenging problem in incompressible flow simulations. The importance of ensuring the compatibility of the component approximations of velocity and pressure by satisfying the so-called inf-sup condition is widely known. Although some stable mixed finite element pairs have been studied over the years [13], [23], the P1–P1 pair not satisfying the inf-sup condition may also works well. The P1–P1 pair is computationally convenient in a parallel processing and multigrid context because this pair holds the identical distribution for both the velocity and pressure. Moreover, The P1–P1 pair is of practical importance in scientific computation with the lowest computational cost. Therefore, much attention has been attracted by the P1–P1 pair for simulating the incompressible flow, we can refer to [1], [3], [16], [20] and the references therein.
In order to use the P1–P1 pair, various stabilized techniques have been proposed and studied. For example, the polynomial pressure projection method [3], [10], the Brezzi–Pitkaranta method [5], the stream upwind Petrov–Galerkin (SUPG) method [8], the Douglas–Wang method [11], the macro-element method [17] and the method of local Gauss integrations [20], [21]. Most of these stabilized methods necessarily introduce the stabilization parameters either explicitly or implicitly. In addition, some of these techniques are conditionally stable and are of suboptimal accuracy. Therefore, the development of mixed finite element methods free from stabilization parameters has become increasingly important.
Recently, based on polynomial pressure projection, a new family of stabilized methods for the stationary Stokes equations have been proposed and studied in [2]. Based on the research of the instabilities of the equal-order velocity–pressure pairs (i.e., the P1–P1 and P2–P2 pairs), these new methods add terms to the continuity equation particularly suited to stabilize these instabilities. These added terms depend on the projection operators of pressure. In the implementation, by applying the difference between a consistent mass matrix M and an underintegrated mass matrix as the complement for the pressure, namely
where the coefficients of the discrete pressure ph ∈ Mh (Mh will be defined in the following section) in the Lagrange basis Ψi of the pressure space: . The matrix M can be obtained by local Gauss integration, while the matrix is a diagonal matrix, which can be obtained by adding the corresponding row of M. With the help of (1.2), Becker and Hansbo have provided the optimal error estimates for the velocity in H1-norm and pressure in L2-norm. This new stabilized method is characterized by the following features. (i) The method does not require approximation of the pressure derivatives and the mesh-dependent parameters. (ii) The method is unconditionally stable and parameter-free. (iii) The method can be applied to existing codes with a little additional effort. This paper aims to extend the work of Becker and Hansbo [2] to the two-dimensional transient Stokes problem. We only confine our attention to the P1–P1 triangular element in the theoretical analysis. However, for comparison, numerical results for the stable MINI element are also provided. This work can be considered as a complement to the work of [2] in the sense that it demonstrates the high efficiency of the pressure projection stabilized method not only for the steady problem, but also for the unsteady equations.
The rest of this paper is organized as follows. In the next section, some basic notations and results for the time-dependent Stokes problem are stated. In Section 3, the stabilized finite element method is introduced. Section 4 is devoted to derive the error estimates for the stabilized finite element solution. In Section 5, some numerical results are given to illustrate the established theoretical results. Finally, some conclusions are given in Section 6.
Section snippets
Function setting of the transient Stokes problem
In this paper, the standard notations for Sobolev spaces and the associated norms are used. The spaces L2(Ω)i(i = 1, 2) are endowed with the standard L2-scalar product (· , ·) and norm ∥ · ∥ 0. The spaces are equipped with the scalar product and norm . Due to the equivalence between ∥u ∥ 1 and ∥ ∇ u ∥ 0 on , we use the same notations for them.
For the mathematical setting of problem (1.1), we introduce the following Hilbert spaces.
Stabilized finite element method
Let be a regular partition of into triangles with mesh size h > 0. The mesh parameter h is defined by . This work focus on the following mixed finite element spacesandwhere P1(K) is the set of all polynomials on K of degree less than or equal to one.
For the above finite element spaces Xh and Mh, it is well-known that the following approximate estimates hold: for all , there exist
Error estimates
This section is devoted to derive the error estimates for the problem (3.5) in both semidiscrete and fully discrete formulations. First of all, we define the Galerkin projection (Rh, Qh) : (X, M) → (Xh, Mh) as followswhich is well defined by Theorem 3.3.
By using the techniques to one used in [17], [20] and the Bramble–Hilbert lemma (see [12]), we have the following theorem.
Theorem 4.1 If ∂Ω is of C2 or Ω is a two-dimensional convex
Numerical examples
In this section, we present a series of numerical results to confirm the established theoretical analysis in Section 4. For comparison, we also give the numerical results for the MINI element and the results for the P1–P1 element using the stabilization technique proposed in [3], [20], [21]. We consider the problem (1.1) on the unit square in all experiments. The experimental rates of convergence with respect to the mesh size h are calculated by the formula ,
Conclusions
We have developed and studied a new stabilized Galerkin finite element method for the unsteady Stokes equations in both semidiscrete and fully discrete formulations. The spatial discretization is based on the linear equal-order interpolations for the velocity and pressure, error estimates for the numerical solutions have been derived. Numerical tests have revealed that the novel stabilized method has high efficiency for the time-dependent Stokes problem.
Acknowledgements
The authors would like to thank the editor and referees for valuable constructive comments and suggestions which helped to improve the results of this paper.
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