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Advances in Numerical Analysis
Volume 2012 (2012), Article ID 309871, 14 pages
Two-Level Stabilized Finite Volume Methods for Stationary Navier-Stokes Equations
1École Nationale Supérieure des Arts et Métiers-Casablanca, Université Hassan II, B.P. 150, Mohammedia, Morocco
2Department of Mathematics and Computing Sciences, Faculty of Sciences and Technology, University Hassan 1st, B.P. 577, Settat, Morocco
Received 17 December 2011; Accepted 17 February 2012
Academic Editor: Weimin Han
Copyright © 2012 Anas Rachid et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We propose two algorithms of two-level methods for resolving the nonlinearity in the stabilized finite volume approximation of the Navier-Stokes equations describing the equilibrium flow of a viscous, incompressible fluid. A macroelement condition is introduced for constructing the local stabilized finite volume element formulation. Moreover the two-level methods consist of solving a small nonlinear system on the coarse mesh and then solving a linear system on the fine mesh. The error analysis shows that the two-level stabilized finite volume element method provides an approximate solution with the convergence rate of the same order as the usual stabilized finite volume element solution solving the Navier-Stokes equations on a fine mesh for a related choice of mesh widths.
We consider a two-level method for the resolution of the nonlinear system arising from finite volume discretizations of the equilibrium, incompressible Navier-Stokes equations: where is the velocity vector, is the pressure, is the body force, is the viscosity of the fluid, and , the flow domain, is assumed to be bounded, to have a Lipschitz-continuous boundary , and to satisfy a further condition stated in (H1).
Finite volume method is an important numerical tool for solving partial differential equations. It has been widely used in several engineering fields, such as fluid mechanics, heat and mass transfer, and petroleum engineering. The method can be formulated in the finite difference framework or in the Petrov-Galerkin framework. Usually, the former one is called finite volume method [1, 2], MAC (marker and cell) method , or cell-centered method , and the latter one is called finite volume element method (FVE) [5–7], covolume method , or vertex-centered method [9, 10]. We refer to the monographs [11, 12] for general presentations of these methods. The most important property of FVE is that it can preserve the conservation laws (mass, momentum, and heat flux) on each control volume. This important property, combined with adequate accuracy and ease of implementation, has attracted more people to do research in this field.
On the other hand, the two-level finite element strategy based on two finite element spaces on one coarse and one fine mesh has been widely studied for steady semilinear elliptic equations [13, 14] and the Navier-Stokes equations [15–22]. For the finite volume element method, Bi and Ginting  have studied two-grid finite volume element method for linear and nonlinear elliptic problems; Chen et al.  have applied two-grid methods for solving a two-dimensional nonlinear parabolic equation using finite volume element method. Chen and Liu  have also studied this method for semilinear parabolic problems. However, to the best of our knowledge, there is no two-level finite volume convergence analysis for the Navier-Stokes equations in the literature.
In this paper we aim to combine FVE method based on macroelement with two-level strategy to solve the two-dimensional Navier-Stokes (1.1)–(1.3). The heart of the analysis is the use of a transfer operator to connect finite volume and finite element estimations which will lead to more difficult term to estimate. We choose the two-grid spaces as two conforming finite element spaces and on one coarse grid with mesh size and one fine grid with mesh size . We propose two algorithms of two-level method for resolving the nonlinearity in the stabilized finite volume approximation of the problem (1.1)–(1.3): the simple and Newton algorithms. First we prove that the simple two-level stabilized finite volume solution is the following error estimate: Second we prove that the Newton two-level stabilized finite volume solution is the following error estimate: where denotes some generic constant which may stand for different values at its different occurrences.
Hence, the two-level algorithms achieve asymptotically optimal approximation as long as the mesh sizes satisfy for the simple two-level stabilized finite volume solution and for the Newton two-level stabilized finite volume solution. As a result, solving the nonlinear Navier-Stokes equations will not be much more difficult than solving one single linearized equation.
The rest of this paper is organized as follows. In the next section, we introduce some notations and construct a FVE scheme. In Section 3 we recall same preliminary estimates of the stabilized finite volume approximations. Finally the two-level FVE algorithms and the improved error estimates are presented and established in Section 4.
2. Finite Volume Scheme
We will use and to denote the norm and seminorm of the Sobolev space , . Let be the standard Sobolev subspace of of functions vanishing on . We introduce the following notations: The scalar product and norm in are denoted by the usual inner product and , respectively. As mentioned above, we need a further assumption on .
Assume that is regular so that the unique solution of the steady Stokes problem for a prescribed exists and satisfies where is a constant depending on .
The weak formulation of the problem (1.1)–(1.3) is to find such that where the bilinear forms , and the trilinear form are given by Introducing the generalized bilinear form on by we can rewrite (2.4) in a compact form: find such that
Let be a quasi-uniform triangulation of with , where is the diameter of the triangle . We assume that the partition has been obtained from a macrotriangular partition by joining the sides of each element of . Every element must lie in exactly one macroelement , which implies that macroelements do not overlap. For each , the set of interelement edges which are strictly in the interior of will be denoted by , and the length of an edge is denoted by .
Based on this triangulation, let be the standard conforming finite element subspace of piecewise linear velocity, and let be the piecewise constant pressure subspace It is well known that the standard element does not satisfy the inf-sup condition and cannot be applied to problem (1.1)–(1.3) directly. But a locally stabilized method based on the macroelement can be used to yield adequate approximations .
In order to describe the FVE method for solving problem (1.1)–(1.3), we will introduce a dual partition based upon the original partition whose elements are called control volumes. We construct the control volumes in the same way as in [5, 26]. Let be the barycenter of . We connect with line segments to the midpoints of the edges of , thus partitioning into three quadrilaterals , , where are the vertices of . Then with each vertex , we associate a control volume , which consists of the union of the subregions , sharing the vertex . Thus we finally obtain a group of control volumes covering the domain , which is called the dual partition of the triangulation . We denote by the set of interior vertices.
We call the partition regular or quasi-uniform if there exists a positive constant such that If the finite element triangulation is quasi-uniform, then the dual partition is also quasi-uniform .
2.2. Construction of the FVE Scheme
We formulate the FVE method for the problem (1.1)–(1.3) as follows: given a and , integrating (1.1) over the associated control volume and (1.1) over the element and applying Green’s formula, we obtain an integral conservation form where denotes the unit outer normal vector to (Figure 1).
Let be the transfer operator defined by where and is the characteristic function of the control volume . The operator satisfies  Now for an arbitrary , we multiply (2.11) by and sum over all to get Here , and are defined by We also define the trilinear forms and on by To formulate the discrete problem so as to eliminate any such potential difficulties, we rewrite (2.16) as follows: We multiply (2.12) by and sum over all : then, we obtain Now we rewrite (2.19) and (2.20) to a variational form similar to finite element problems. The locally stabilized FVE scheme is to find such that where is a stabilized form defined on , is the jump operator across the edge , and is the local stabilization parameter. It is trivial that , , .
A general framework for analyzing the locally stabilized formulation (2.21) can be developed using the notion of equivalence class of macroelements. As in Stenberg  each equivalence class, denoted by , contains macroelements which are topologically equivalent to a reference macroelement .
Let We can rewrite (2.21) in a compact form: find such that
3. Technical Preliminaries
This section considers preliminary estimates which will be very useful in the error estimates of two-level finite volume solution .
The following lemma gives the boundedness of the trilinear form .
Lemma 3.1 (see ). The following estimates hold: Here and after , , and are positive constants depending only on the data .
Theorem 3.2. Assume that and satisfy the following uniqueness condition: where Then the problem (2.7) admits a unique solution such that
In  the following lemma was proved, which shows that the finite volume element bilinear forms and are equal to the finite element ones, respectively.
Lemma 3.3. For any , and , one has
Theorem 3.4. Given a a stabilization parameter , suppose that every macroelement belongs to one of the equivalence classes and that the following macroelement connectivity condition is valid: for any two neighboring macroelements and with , there exists such that Then for all , and where are constants independent of and , is some fixed positive constant, and is the out-normal vector.
Next, we establish the existence and the uniqueness of FVE scheme (2.25), by the fixed-point theorem, in the following.
Theorem 3.5 (see ). Suppose the assumptions of Theorems 3.2 and 3.4 hold, and the body force satisfies the following uniqueness condition Then the variation problem (2.25) admits a unique solution such that where
Then the optimal error estimates can be obtained as follows.
4. Two-Level FVE Algorithms and Its Error Analysis
In this section, we will present two-level stabilized finite volume element algorithm for (1.1)–(1.3) and derive some optimal bounds for errors. The idea of the two-level method is to reduce the nonlinear problem on a fine mesh into a linear system on a fine mesh by solving a nonlinear problem on a coarse mesh. The basic mechanisms are two quasi-uniform triangulations of , , and , with two different mesh sizes and , and the corresponding solutions spaces and , which satisfy and will be called the coarse and the fine spaces, respectively. Now find as follows.
Algorithm 4.1 (Simple two-level stabilized FVE approximation). We have the following steps:
Step 1. On the coarse mesh , solve the stabilized Navier-Stokes problem.
Find such that, ,
Step 2. On the fine mesh , solve the stabilized linear Stokes problem.
Find such that, ,
Next, we study the convergence of to in some norms. For convenience, we set and .
Algorithm 4.3 (The Newton two-level stabilized FVE approximation). We have the following steps:
Step 1. On the coarse mesh , solve the stabilized Navier-Stokes problem.
Find by (4.1).
Step 2. On the fine mesh , solve the stabilized linear Stokes problem.
Find such that, ,
Now, we will study the convergence of the Newton two-level stabilized finite element solution to in some norms. To do this, let us set , , and .
Proof. Subtracting (4.7) from (2.25), using the Galerkin projection (3.12) and taking , we get Using Lemma 3.1 and Theorems 3.2, 3.5, and 3.7, we obtain Combining (4.10)–(4.15) with (4.9) yields Thanks to (3.7), (4.9), Theorems 3.2 and 3.5, and estimates (4.10)–(4.14) and (4.16), we have Combining (4.16) and (4.17) with (3.14) and Theorem 3.2 yields (2.13).
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