Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 510241, 11 pages
http://dx.doi.org/10.1155/2015/510241

Research Article

A Posteriori Error Estimate for Finite Volume Element Method of the Second-Order Hyperbolic Equations

Chuanjun Chen,¹ Xin Zhao,² and Yuanyuan Zhang¹

¹School of Mathematics and Information Sciences, Yantai University, Yantai 264005, China
²School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China

Received 6 September 2015; Accepted 18 November 2015

Academic Editor: Xinguang Zhang

Copyright © 2015 Chuanjun Chen 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.

We establish a posteriori error estimate for finite volume element method of a second-order hyperbolic equation. Residual-type a posteriori error estimator is derived. The computable upper and lower bounds on the error in the -norm are established. Numerical experiments are provided to illustrate the performance of the proposed estimator.

1. Introduction

The finite volume element method is a class of important numerical tools for solving partial differential equations. Due to the local conservation property and some other attractive properties, it is wildly used in many engineering fields, such as heat and mass transfer, fluid mechanics, and petroleum engineering, especially for those arising from conservation laws including mass, momentum, and energy. For the second-order hyperbolic equations, Li et al. [1] have proved the optimal order of convergence in -norm. In [2], Kumar et al. have proved optimal order of convergence in and -norm for the semidiscrete scheme and quasi-optimal order of convergence in maximum norm.

Since the pioneering work of Babuvška and Rheinboldt [3], the adaptive finite element methods based on a posteriori error estimates have become a central theme in scientific and engineering computations. Adaptive algorithm is among the most important means to boost accuracy and efficiency of the finite element discretization. The main idea of adaptive algorithm is to use the error indicator as a guide which shows whether further refinement of meshes is necessary. A computable a posteriori error estimator plays a crucial role in an adaptive procedure. A posteriori error analysis for the finite volume element method has been studied in [4–6] for the second-order elliptic problem, in [7–9] for the convection-diffusion equations, in [10] for the parabolic problems, in [11] for a model distributed optimal problem governed by linear parabolic equations, in [12] for the Stokes problem in two dimensions, and in [13] for the second-order hyperbolic equations.

However, to the best of our knowledge, there are few works related to the a posteriori error estimates of the finite volume element method for the second-order hyperbolic problems. The aim of this paper is to establish residual-type a posteriori error estimator of the finite volume element method for the second-order hyperbolic equation. We first construct a computable a posteriori error estimator of the finite volume element method. Then we analyze the residual-type a posteriori error estimates and obtain the computable upper and lower bounds on the error in the -norm.

The organization of this paper is stated as follows. In Section 2, we present the framework of the finite volume element method for the second-order hyperbolic equation. In Section 3, we establish the residual-type a posteriori error estimator of the finite volume element method and derive the upper and lower bounds on the error in the -norm. We provide some numerical experiments to illustrate the performance of the error estimator in Section 4.

2. Finite Volume Element Formulation

We use standard notation for Sobolev spaces with the norm [14]. In order to simplify the notation, we denote by and omit the index and whenever possible.

In this paper, we consider the following second-order hyperbolic problem:

where is a polygonal bounded cross section, possessed with a Lipschitz boundary . For simplicity, the right-hand side is assumed to be measurable and square-integrable on and to be continuous with respect to time. The initial datum and are assumed to be measurable and square-integrable on . is a real-valued smooth matrix function, uniformly symmetric, and positive definite in .

The corresponding variational problem is to find , for , satisfying

where the bilinear form is defined by

Denote by the primal quasi-uniform triangulation of with , where is the diameter of the triangle . Let be the standard conforming finite element space of piecewise linear functions, defined on the triangulation :

Denote by the dual partition which is constructed in the same way as in [1, 15]. Let be the barycenter of . We connect with the midpoints of the edges of by straight line, 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 (see Figure 1). Finally, we obtain a group of control volumes covering the domain , which is called the dual partition of the triangulation . Denote by the set of interior vertices of and denote by the set of all interior edges of , respectively.

(a)

(b)

Figure 1: (a) The dotted line shows the boundary of the corresponding control volume with , a common vertex. (b) A triangle is partitioned into three subregions .

The partition is regular or quasi-uniform, if there exists a positive constant such that

The dual partition will also be quasi-uniform [5] if the finite element triangulation is quasi-uniform. The test function space is defined by

For any , we define an interpolation operator , such that

where is the characteristic function of the control volume .

According to [16], for each , there exists a positive constant independent of , such that satisfies the following inequality:

Introduce the following adjoint elliptic problem:

Denote by the solution operator of this problem, so that

Define negative norms by

In fact, by Cauchy-Schwarz inequality, we obtain

For our error analysis in the next section, it will be convenient to introduce such a norm defined by

According to Thomée [17], we have the following lemma.

Lemma 1. The norm is equivalent to and , where is a nonnegative integer. Particularly, is equivalent to when .
In order to get the fully discrete finite volume element method of (1), we give a partition of the time interval : . Let , , , and . With the help of , we obtain the fully discrete finite volume element method of (1): to find , for , such that

where

By setting and , the notation , (1) can equivalently be written as

where .
Let ; we define

The residual system, with , is defined as follows:

where the quantities in and in are affine functions on each interval , that

And the quantities are defined as follows.
From the fully discrete algorithm (14), for any , we have

Since , by (2) and (20), for , we get

Adding the term into the two hand sides of (21), we get

So on each interval , we have

We define

Then the term on the interval can be written as

When ,

3. Residual-Type A Posteriori Error Estimates

In this section, we will construct the residual-type a posteriori error estimates of the finite volume element method for (1). We introduce the jump of a vector-valued function across the edge which will be used in the residual-type a posteriori error estimates. Let be an interior edge shared by elements and . Define the unit normal vectors and on pointing exterior to and , respectively. Let be a vector-valued function that is smooth inside each of the elements and . and denote the traces of on taken from within the interior of and , respectively. Then the jump of on the edge is defined by . We denote space refinement indicator by defined by

We define time refinement indicator as

3.1. Upper Bound

The Scott-Zhang interpolation function is introduced in the following lemma [18].

Lemma 2. For each , a positive constant is independent of and such that, for any ,

where and .

We also introduce the trace theorem [14].

Lemma 3 (trace theorem). There exists a positive constant independent of such that

Then we can get the following theorem for the upper bound of the error.

Theorem 4. The following a posteriori error estimate holds between the solution of (1) and the solution of (14), for :

Proof. Taking the inner product of (18) with and setting

we obtain, for ,

hence,

Integrating the inequality from to (), we have

Using Lemma 1, we obtain

By the definition of , we get

In order to estimate , we choose in (24); then

Using Green’s formula, we have

By the definition of , we get

From Cauchy-Schwarz inequality and Lemma 2, we can get

For , since is a constant in , we have

Since and are continuous inside each element , we have

Thus,

Then we get

By (8) and Cauchy-Schwarz inequality, we obtain

Since is a piecewise constant function, by Lemma 3 and (8), we get

Substituting the estimate of into (38) and by the definition of , we have

hence

Substituting the estimation of into (37), we get

Summing (50) from to , we obtain

For , we have

Noting that , then

By the fact that , we have

In view of the definition of the operator , we have

Subtracting (56) from (55), we get

Integrating (58) from to , we obtain

Summing (59) from to , we obtain

Thus, we have

Then,

By (62) and (54), we have

3.2. Lower Bound

In order to derive the local lower bounds on the error, we will introduce some properties of the bubble functions. For each triangle , denote by the barycentric coordinates. Define the element-bubble function by

Assume that and share the edge . Let the barycentric coordinates with respect to the end points of be and . Define the edge-bubble function by

For properties of the bubble functions, we have the following lemma [19].

Lemma 5. For each of the elements and , functions and have the following properties:

We define the average of on () and the average of on () by

Then we have the following local lower bounds.

Theorem 6. For any , the following local posteriori lower bounds on the error hold for a positive constant independent of and :

Proof. By triangle inequality, we have

By the properties of , the definition of , and Green’s formulation, we have

For , with Cauchy-Schwarz inequality and Lemma 5, we get

By Cauchy-Schwarz inequality and Lemma 5, we obtain

Combining (71)–(73), we obtain

For (69), by triangle inequality, similarly we have

By Lemma 5 and Green’s formulation, we get

Now we will estimate the right-hand terms of (76). By Lemma 5 and the Cauchy-Schwarz inequality, we obtain

Combining (77) with (76), we get

With (74), we obtain

4. Numerical Examples

Now we present some numerical examples to show the performance of the proposed error estimator. We consider problem (1) in . We discretize into number of rectangles in each direction and then each rectangle is divided into two triangles, resulting in a mesh with size . Discretize time by taking time step . We consider the following two cases.

Case 1. Consider

Case 2. Consider

Define

We present the results of the above cases when at Tables 1 and 2.

Table 1: Error estimates for Case 1.

Rate

Rate

—

—
0.4589
21.3382
46.4952

1.7610

0.9669
0.4417
21.1967
47.9920

2.1321

0.9919
0.4307
21.0618
48.9005

2.1787

0.9979
0.4247
20.9680
49.3734

2.1308

0.9996
0.4215
20.9138
49.6128

Table 2: Error estimates for Case 2.

Rate

Rate

—
2.3532
—
6.6633
66.7292
10.0144

1.9232
1.1822
0.9931
6.3537
64.6603
10.1768

2.0227

0.9967
6.1806
63.5799
10.2870

2.1383

0.9991
6.0904
63.0179
10.3471

2.1894

0.9999
6.0444
62.7297
10.3782

From Tables 1 and 2 we can see that the global a posteriori error estimator can predict the exact global error. The error estimator is reliable as evidenced by the ratio listed on the tables. This list shows that the ratio is converging to a constant when the mesh size is decreased by half. This shows that the proposed global a posteriori error estimator is robust for predicting the error in the finite volume element method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China (nos. 11301456, 11426193, and 11571297) and Shandong Province Natural Science Foundation (nos. ZR2014AP003 and ZR2014AM003).

References

R. Li, Z. Chen, and W. Wu, Generalized Difference Methods for Differential Equations Numerical Analysis of Finite Volume Methods, Marcel Dekker, New York, NY, USA, 2000.

S. Kumar, N. Nataraj, and A. K. Pani, “Finite volume element method for second order hyperbolic equations,” International Journal of Numerical Analysis and Modeling, vol. 5, no. 1, pp. 132–151, 2008.

I. Babuvška and W. C. Rheinboldt, “Error estimates for adaptive finite element computations,” SIAM Journal on Numerical Analysis, vol. 15, no. 4, pp. 736–754, 1978.

M. Afif, A. Bergam, Z. Mghazli, and R. Verfürth, “A posteriori estimators for the finite volume discretization of an elliptic problem,” Numerical Algorithms, vol. 34, no. 2–4, pp. 127–136, 2003.

C. Bi and V. Ginting, “A residual-type a posteriori error estimate of finite volume element method for a quasi-linear elliptic problem,” Numerische Mathematik, vol. 114, no. 1, pp. 107–132, 2009.

X. Ye, “A posterior error estimate for finite volume methods of the second order elliptic problem,” Numerical Methods for Partial Differential Equations, vol. 27, no. 5, pp. 1165–1178, 2011.

D. Kröner and M. Ohlberger, “A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions,” Mathematics of Computation, vol. 69, no. 229, pp. 25–39, 2000.

R. Lazarov and S. Tomov, “A posteriori error estimates for finite volume element approximations of convection-diffusion-reaction equations,” Computational Geosciences, vol. 6, no. 3-4, pp. 483–503, 2002.

M. Ohlberger, “A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations,” Numerische Mathematik, vol. 87, no. 4, pp. 737–761, 2001.

Z. Chen and J. Feng, “An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems,” Mathematics of Computation, vol. 73, no. 247, pp. 1167–1193, 2004.

W. Liu and N. Yan, “A posteriori error estimates for optimal control problems governed by parabolic equations,” Numerische Mathematik, vol. 93, no. 3, pp. 497–521, 2003.

P. Chatzipantelidis, C. Makridakis, and M. Plexousakis, “A posteriori error estimates for a finite volume method for the Stokes problem in two dimensions,” Applied Numerical Mathematics, vol. 46, no. 1, pp. 45–58, 2003.

C. Bernardi and E. Süli, “Time and space adaptivity for the second-order wave equation,” Mathematical Models and Methods in Applied Sciences, vol. 15, no. 2, pp. 199–225, 2005.

S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, NY, USA, 2008.

R. E. Ewing, T. Lin, and Y. P. Lin, “On the accuracy of the finite volume element method based on piecewise linear polynomials,” SIAM Journal on Numerical Analysis, vol. 39, no. 6, pp. 1865–1888, 2002.

P. Chatzipantelidis, R. D. Lazarov, and V. Thomée, “Error estimates for a finite volume element method for parabolic equations in convex polygonal domains,” Numerical Methods for Partial Differential Equations, vol. 20, pp. 650–674, 2004.

V. Thomée, Galerkin Finite Element Method for Parabolic Equations, Springer, Berlin, Germany, 2006.

L. R. Scott and S. Zhang, “Finite element interpolation of nonsmooth functions satisfying boundary condition,” Mathematics of Computation, vol. 54, no. 190, pp. 483–493, 1990.

R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-refinement Techniques, John Wiley & Sons, New York, NY, USA, 1996.