Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 510241, 11 pages

http://dx.doi.org/10.1155/2015/510241

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

^{1}School of Mathematics and Information Sciences, Yantai University, Yantai 264005, China^{2}School 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 , 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.