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 , satisfyingwhere 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)
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 thatDefine 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 thatwhere By setting and , the notation , (1) can equivalently be written aswhere .
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 haveSince , by (2) and (20), for , we getAdding the term into the two hand sides of (21), we get So on each interval , we haveWe defineThen 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 getIn order to estimate , we choose in (24); thenUsing 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 getSumming (50) from to , we obtainFor , we have Noting that , thenBy the fact that , we haveIn view of the definition of the operator , we haveSubtracting (56) from (55), we get Integrating (58) from to , we obtainSumming (59) from to , we obtainThus, we haveThen,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 haveFor , with Cauchy-Schwarz inequality and Lemma 5, we getBy Cauchy-Schwarz inequality and Lemma 5, we obtainCombining (71)–(73), we obtainFor (69), by triangle inequality, similarly we have By Lemma 5 and Green’s formulation, we getNow we will estimate the right-hand terms of (76). By Lemma 5 and the Cauchy-Schwarz inequality, we obtainCombining (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.
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).