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

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 includingmass,momentum, and energy. For the secondorder hyperbolic equations, Li et al. [1] have proved the optimal order of convergence in H-norm. In [2], Kumar et al. have proved optimal order of convergence in L andHnorm 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 residualtype a posteriori error estimates and obtain the computable upper and lower bounds on the error in theH-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 H-norm. We provide some numerical experiments to illustrate the performance of the error estimator in Section 4.


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 secondorder hyperbolic equations, Li et al. [1] have proved the optimal order of convergence in  1 -norm.In [2], Kumar et al. have proved optimal order of convergence in  2 and  1norm 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][5][6] for the second-order elliptic problem, in [7][8][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 residualtype a posteriori error estimates and obtain the computable upper and lower bounds on the error in the  1 -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  1 -norm.We provide some numerical experiments to illustrate the performance of the error estimator in Section 4.
In this paper, we consider the following second-order hyperbolic problem: where Ω ⊂ R 2 is a polygonal bounded cross section, possessed with a Lipschitz boundary Ω.For simplicity, the right-hand side  is assumed to be measurable and squareintegrable on Ω × (0, ] and to be continuous with respect to time.The initial datum  0 and V 0 are assumed to be measurable and square-integrable on Ω. (, ) = (  (, )) 2  ,=1 is a real-valued smooth matrix function, uniformly symmetric, and positive definite in Ω.
The corresponding variational problem is to find  ∈  1 0 (Ω), for  > 0, satisfying where the bilinear form (⋅, ⋅) is defined by Denote by  ℎ the primal quasi-uniform triangulation of Ω with ℎ = max ℎ  , where ℎ  is the diameter of the triangle  ∈  ℎ .Let U ℎ be the standard conforming finite element space of piecewise linear functions, defined on the triangulation  ℎ : Denote by  * ℎ 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   ,  ∈  ℎ (), where  ℎ () are the vertices of .Then with each vertex  ∈  ℎ = ∪ ∈ ℎ  ℎ (), 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 Ω, which is called the dual partition  * ℎ of the triangulation  ℎ .Denote by  0 ℎ the set of interior vertices of  ℎ and denote by E ℎ the set of all interior edges of  ℎ , respectively.The partition  * ℎ is regular or quasi-uniform, if there exists a positive constant  > 0 such that The dual partition  * ℎ will also be quasi-uniform [5] if the finite element triangulation  ℎ is quasi-uniform.The test function space V ℎ is defined by For any  ℎ ∈ U ℎ , we define an interpolation operator Π ℎ : where Ψ  is the characteristic function of the control volume   .
According to [16], for each  ℎ ∈ U ℎ , there exists a positive constant  independent of ℎ, such that Π ℎ satisfies the following inequality: Introduce the following adjoint elliptic problem: Denote by T :  2 (Ω) →  2 (Ω) ⋂  1 0 (Ω) the solution operator of this problem, so that  (T, ) = (, ) , ∀ ∈ 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.

Mathematical Problems in Engineering
We define Then the term  V on the interval [ −1 ,   ] (2 ≤  ≤ ) can be written as

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  ∈ E ℎ 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 n  + and n  − on  pointing exterior to  + and  − , respectively.Let k be a vector-valued function that is smooth inside each of the elements  + and  − .k + and k − denote the traces of k on  taken from within the interior of  + and  − , respectively.
Then the jump of k 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 where We also introduce the trace theorem [14].

Lemma 3 (trace theorem).
There exists a positive constant  independent of ℎ  such that (30) 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 (  ℎ ) 1≤≤ of ( 14), for 2 ≤  ≤ : Proof.Taking the inner product of ( 18) with ( ) and setting Using Lemma 1, we obtain By the definition of    , we get In order to estimate ‖  ‖ −1 , we choose V = I ℎ  in (24); then Using Green's formula, we have By the definition of R   , R   , we get From Cauchy-Schwarz inequality and Lemma Since ()∇  ℎ and V are continuous inside each element  ∈  ℎ , we have Thus, Then we get By (8) and Cauchy-Schwarz inequality, we obtain 1 .

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Since Π ℎ V is a piecewise constant function, by Lemma 3 and (8), we get In view of the definition of the operator T, we have Subtracting ( 56) from (55), we get Integrating (58) from  −1 to   , we obtain Summing (59) from  = 1 to  = , we obtain Assume that  and   share the edge  ∈ E ℎ .Let the barycentric coordinates with respect to the end points of  be  ,1 and  ,2 .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  ∈  ℎ and  ∈ E ℎ , functions   and   have the following properties: We define the average of R   on  (R   ) and the average of R   on  (R   ) by Then we have the following local lower bounds.Theorem 6.For any  ∈  ℎ ,  ∈ E ℎ , the following local posteriori lower bounds on the error   −   ℎ hold for a positive constant  independent of ℎ  and ℎ  : Proof.By triangle inequality, we have By the properties of   , the definition of R   , and Green's formulation, we have Now we will estimate the right-hand terms of (76

Numerical Examples
Now we present some numerical examples to show the performance of the proposed error estimator.We consider problem (1) in Ω × [0, ] = [0, 1; 0, 1] × [0, 1].We discretize Ω into  number of rectangles in each direction and then each rectangle is divided into two triangles, resulting in a mesh with size ℎ = √ 2/.Discretize time by taking time step   = Δ = ℎ.We consider the following two cases.
We present the results of the above cases when  =  at Tables 1 and 2.
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 R listed on the tables.This list shows that the ratio R 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.

2 MathematicalFigure 1 :
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   .

Table 2 :
Error estimates for Case 2.