About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 761242, 16 pages
http://dx.doi.org/10.1155/2012/761242
Research Article

A Discontinuous Finite Volume Method for the Darcy-Stokes Equations

School of Mathematical Sciences, Shandong Normal University, Jinan, Shandong 250014, China

Received 12 June 2012; Accepted 7 December 2012

Academic Editor: Claudio Padra

Copyright © 2012 Zhe Yin 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.

Abstract

This paper proposes a discontinuous finite volume method for the Darcy-Stokes equations. An optimal error estimate for the approximation of velocity is obtained in a mesh-dependent norm. First-order -error estimates are derived for the approximations of both velocity and pressure. Some numerical examples verifying the theoretical predictions are presented.

1. Introduction

The study of discontinuous Galerkin methods has been a very active research field since they were proposed by Reed and Hill [1] in 1973. Discontinuous Galerkin methods use discontinuous functions as finite element approximation and enforce the connections of the approximate solutions between elements by adding some penalty terms. The flexibility of discontinuous functions gives discontinuous Galerkin methods many advantages, such as high parallelizability and localizability. Arnold et al. [2] provided a framework for the analysis of a large class of discontinuous Galerkin methods for second-order elliptic problems.

Based on the advantages of using discontinuous functions for approximation in discontinuous Galerkin methods, it is natural to consider using discontinuous functions as trial functions in the finite volume method, which is called the discontinuous finite volume method. Such a method has the flexibility of the discontinuous Galerkin method and the simplicity and conservative properties of the finite volume method. Ye [3] developed a new discontinuous finite volume method and analyzed it for the second-order elliptic problem. Bi and Geng [4] proposed the semidiscrete and the backward Euler fully discrete discontinuous finite volume element methods for the second-order parabolic problems. Ye [5] considered the discontinuous finite volume method for solving the Stokes problems on both triangular and rectangular meshes and derived an optimal order error estimate for the approximation of velocity in a mesh-dependent norm and first-order -error estimates for the approximations of both velocity and pressure.

The Darcy-Stokes problem is interesting for a variety of reasons. Apart from being a modeling tool in its own right, it also appears, less obviously, in time-stepping methods for Stokes and for high Reynolds number flows (where of course the convective term causes additional difficulties). In [6], the nonconforming Crouzeix-Raviart element is stabilized for the Darcy-Stokes problem with terms motivated by a discontinuous Galerkin approach. In [7], a new stabilized mixed finite element method is presented for the Darcy-Stokes equations.

In this paper, we will extend the discontinuous finite volume methods to solve the Darcy-Stokes equations. In our methods, velocity is approximated by discontinuous piecewise linear functions on triangular meshes and by discontinuous piecewise rotated bilinear functions on rectangular meshes. Piecewise constant functions are used as the test functions for velocity in the discontinuous finite volume methods. We obtained an optimal error estimate for the approximation of velocity in a mesh-dependent norm. First-order -error estimates are derived for the approximations of both velocity and pressure. For the sake of simplicity and easy presentation of the main ideas of our method, we restrict ourselves to the model problem.

We consider the Darcy-Stokes equationswhere is a bounded polygonal domain in with boundary . is the velocity, is the pressure, and is a given force term. We assume .

2. Discontinuous Finite Volume Formulation

Let be a triangular or rectangular partition of . The triangles or rectangles in are divided into three or four subtriangles by connecting the barycenter of the triangle or the center of the rectangles to their corner nodes, respectively. Then we define the dual partition of the primal partition to be the union of the triangles shown in Figures 1 and 2 for both triangular and rectangular meshes.

761242.fig.001
Figure 1: Element for triangular mesh.
761242.fig.002
Figure 2: Element for rectangular mesh.

Let consist of all the polynomials with degree less than or equal to defined on . We define the finite dimensional trial function space for velocity on a triangular partition by and on rectangular partition by where denotes the space of functions of the form on .

Let be the finite dimensional space for pressure where Define the finite dimensional test function space for velocity associated with the dual partition as

Multiplying (1.1a) and (1.1b) by and , respectively, we have where is the unit outward normal vector on .

Let be the triangles in , where for triangular meshes and for rectangular meshes, as shown as Figures 3 and 4. Then we have where .

761242.fig.003
Figure 3: Triangular partition and its dual.
761242.fig.004
Figure 4: Rectangular partition and its dual.

For vectors and , let denote the matrix whose th component is as in [5]. For two matrix valued variables and , we define . Let . Let be an interior edge shared by two elements and in . We define the average and jump on for scalar , vector , and matrix , respectively. If , where and are unit normal vectors on e pointing exterior to and , respectively. We also define a matrix valued jump for a vector as If , define A straightforward computation gives Let . Using (2.7), (2.12), and the fact that for on , (2.7) becomes Since for on , we also have Let . Define a mapping , where is the length of the edge .

We define two norms for as follows: where ,  ,, and diameter of .

As in [5], the standard inverse inequality implies that there is a constant such that

Lemma 2.1. There exists a positive constant independent of such that

Proof. As in [4], where . Since , we have . Note that is a piecewise linear function, and . By Lemma 3.6 in [4], , we have .

Lemma 2.2 (see [4]). The operator is self-adjoint with respect to the -inner product, . Define . Then and are equivalent; here the equivalence constants are independent of . And .

Let It is clear that the solutions of the Darcy-Stokes equations (1.1a)–(1.1c) satisfy the following: Define the following bilinear forms: Then systems (2.21) are equivalent to

We propose two discontinuous finite volume formulations based on modification of the weak formulation (2.23) for Darcy-Stokes problem (1.1a)–(1.1c). Let us introduce the bilinear forms as follows: where is a parameter to be determined later. For the exact solution of (1.1a)–(1.1c), we have Therefore, it follows from (2.23) that The corresponding discontinuous finite volume scheme seeks , such that

Let be an edge of element . It is well known (see [2]) that there exists a constant such that for any function , where depends only on the minimum angle of .

Let and be the functions whose restriction to each element is equal to and , respectively.

Lemma 2.3. For , there exists a positive constant independent of such that

Proof. Let , By Lemma 3.1 in [5],

Lemma 2.4 (see [5]). For any , one has

Lemma 2.5 (see [5]). For , there exists a positive constant independent of such that If , then

Lemma 2.6. For any , there is a constant independent of such that for large enough

Proof. Using the proof of Lemmas 3.1 and 3.5 in [5], for , we have when is large enough.

The value of depends on the constant in the inverse inequality. Therefore, the value of for which is coercive is mesh dependent. We introduce a second discontinuous finite volume scheme which is parameter insensitive. Define a bilinear form as follows: Similar to the bilinear form , for the exact solution of the Darcy-Stokes problem we have Consequently, the solution of the Darcy-Stokes problem satisfies the following variational equations: Our second discontinuous finite volume scheme for (1.1a)–(1.1c) seeks , such that

For any value of , we have Similarly, we can prove that Let or . In the rest of the paper, we assume that the following is true: If , (2.45) holds for any . If , (2.45) holds for only large enough.

3. Error Estimates

We will derive optimal error estimates for velocity in the norm and for pressure in the -norm. A first-order error estimate for velocity in -norm will be obtained.

Let be an interior edge shared by two elements and in . If , we say that is continuous on . We say that is zero at if . Define a subspace of by for rectangular meshes and by for triangular mesh.

It has been proven in [8, 9] that the following discrete inf-sup condition is satisfied; that is, there exists a positive constant such that

Lemma 3.1. The bilinear form satisfies the discrete inf-sup condition where is a positive constant independent of the mesh size .

Proof. For and , we have , and . By Poincare-Friedrichs , with (3.3), and (2.17) we get for any With , we have proven (3.4).

Define an operator or . For all , where , are the sides of the element . if is a triangle and if is a rectangle. It was proven in [8] that For all , define by Using the definition of and integration by parts, we can show that The Cauchy-Schwarz inequality implies Equations (2.28) and (3.8) imply that The definitions of the norm , (3.7), and (3.11) give

Theorem 3.2. Let be the solution of (2.27), and let ) be the solution of (1.1a)–(1.1c). Then there exists a constant independent of such that

Proof. Let , where is projection from . Then . Subtracting (2.26) from (2.27) and using Lemma 2.4, we get error equationsBy letting in (3.15a) and in (3.15b), the sum of (3.15a) and (3.15b) gives Thus, it follows from the coercivity (2.45), the boundedness (2.30), (2.44), and (2.34) that which implies the following: The previous estimate can be rewritten as Now using the triangle inequality, (3.7), the definition of , and the inequality mentioned previously, we get which completes the estimate for the velocity approximation.
Discrete inf-sup condition (3.4), (3.15a), (3.15b), Lemmas 2.5, 2.4, and inverse inequality give Using the previous inequality and the triangle inequality, we have completed the proof of (3.13).
Using Lemma 2.1, (3.12), and (3.13), we have Equations (3.22) and (3.7) and the triangle inequality imply (3.14). We have completed the proof.

4. Numerical Experiments

In this section, we present a numerical example for solving the problems (1.1a)–(1.1c) by using the discontinuous finite volume element method presented with (2.27) and (2.42). Let , be the Delaunay triangulation generated by EasyMesh [10] over with mesh size as shown in Figure 5. We consider the case of , the exact velocity , and the pressure . Denote the numerical solution as and with step which is used to generate the mesh data in the EasyMesh input file, and . For , the numerical results are presented in Tables 1 and 2. It is observed from the tables that the numerical results support our theory.

tab1
Table 1: Error behavior for scheme (2.27).
tab2
Table 2: Error behavior for scheme (2.42).
761242.fig.005
Figure 5: Triangular and its dual partition of .

Acknowledgments

This paper is supported by the Excellent Young and Middle-Aged Scientists Research Fund of Shandong Province (2008BS09026), National Natural Science Foundation of China (11171193), and National Natural Science Foundation of Shandong Province (ZR2011AM016).

References

  1. W. H. Reed and T. R. Hill, “Triangular mesh methods for the neutron transport equation,” Tech. Rep. LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM, USA, 1973.
  2. D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, “Unified analysis of discontinuous Galerkin methods for elliptic problems,” SIAM Journal on Numerical Analysis, vol. 39, no. 5, pp. 1749–1779, 2002. View at Publisher · View at Google Scholar · View at Scopus
  3. X. Ye, “A new discontinuous finite volume method for elliptic problems,” SIAM Journal on Numerical Analysis, vol. 42, no. 3, pp. 1062–1072, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. C. Bi and J. Geng, “Discontinuous finite volume element method for parabolic problems,” Numerical Methods for Partial Differential Equations, vol. 26, no. 2, pp. 367–383, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. X. Ye, “A discontinuous finite volume method for the Stokes problems,” SIAM Journal on Numerical Analysis, vol. 44, no. 1, pp. 183–198, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. E. Burman and P. Hansbo, “Stabilized Crouzeix-Raviart element for the Darcy-Stokes problem,” Numerical Methods for Partial Differential Equations, vol. 21, no. 5, pp. 986–997, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. Masud, “A stabilized mixed finite element method for Darcy-Stokes flow,” International Journal for Numerical Methods in Fluids, vol. 54, no. 6–8, pp. 665–681, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. R. Rannacher and S. Turek, “Simple nonconforming quadrilateral Stokes element,” Numerical Methods for Partial Differential Equations, vol. 8, no. 2, pp. 97–111, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. Crouzeix and P.-A. Raviart, “Conforming and nonconforming finite element methods for solving the stationary Stokes equations,” RAIRO. Modélisation Mathématique et Analyse Numérique, vol. 7, pp. 33–75, 1973. View at MathSciNet
  10. EasyMesh, http://www-dinma.univ.trieste.it/nirftc/research/easymesh/.