We investigate an -Galerkin expanded mixed finite element approximation of nonlinear second-order hyperbolic equations, which model a wide variety of phenomena that involve wave motion or convective transport process. This method possesses some features such as approximating the unknown scalar, its gradient, and the flux function simultaneously, the finite element space being free of LBB condition, and avoiding the difficulties arising from calculating the inverse of coefficient tensor. The existence and uniqueness of the numerical solution are discussed. Optimal-order error estimates for this method are proved without introducing curl operator. A numerical example is also given to illustrate the theoretical findings.

1. Introduction

The objective of this paper is to present and analyze an -Galerkin expanded mixed finite element method for the following second-order nonlinear hyperbolic equation:where is a bounded convex polygonal domain in with boundary and with . denotes the sound pressure, is the external force, and is the coefficient, which is supposed to satisfy the following conditions.() There exist positive constants , such that (), , and are Lipschitz continuous with respect to .

The primary interests in engineering application for the mathematical model (1a)–(1d) are the sound pressure , the gradient of sound pressure , and the acceleration of sound transmission . Extensive research has been carried out on the numerical methods and corresponding numerical analysis for model (1a)–(1d), including finite difference methods, finite element methods, and mixed finite element methods. One can refer to [14] and the references cited herein.

The standard finite difference or finite element methods solve the sound pressure directly, then differentiate it to determine , and multiply the gradient of by to determine the acceleration of sound transmission . Therefore, the resulting acceleration of sound transmission and the gradient of sound pressure are often inaccurate, which then reduces the accuracy of the prediction, as well as the accuracy of the adjoint vector . The mixed finite element method can approximate both and simultaneously and yields an accurate . However, the mixed formulation has to face numerical difficulties arising in a low permeability zone because the inversion and the finite element spaces need to satisfy the LBB conditions.

In order to overcome the above problems, we propose an -Galerkin expanded mixed finite element method for model (1a)–(1d) which can solve the sound pressure , the gradient of sound pressure , and the acceleration of sound transmission directly and avoid inverting explicitly. In this formulation the finite element spaces are free of LBB conditions as required by the standard mixed finite element methods. Another feature of the new procedure we have found so far is that it avoids the trouble which resulted from representation of the time derivatives for nonlinear problems and leads to optimal error estimates without introducing curl operator. We prove the equivalence of the problem (1a)–(1d), the -Galerkin expanded mixed variational formulation and the existence and uniqueness of the semidiscrete -Galerkin expanded mixed finite element procedure. By introducing some projection and interpolation operators as well as lemmas, optimal-order error estimates for this formulation are deduced. The theoretical findings are verified by one numerical example. In recent years, there exist lots of work in the literature on the development and analysis of -Galerkin mixed finite element method. One can refer to [58] for linear parabolic type equations, [9] for regularized long wave equation, and [10] for linear second-order hyperbolic equation.

The rest of the paper is organized as follows. In Section 2, we describe the -Galerkin expanded mixed finite element variational form and prove the equivalence between primal problem and the variational formulation. In Section 3, the -Galerkin expanded mixed finite element procedure is presented, and the existence and uniqueness of the solution are proved. In Section 4, we prove the main error estimates. A numerical example is given in Section 5 to illustrate the theoretical findings.

Throughout this paper, denotes a generic constant which does not depend on mesh parameter . We use and to denote the inner product and the norm, respectively, in or . Also we will denote the norms in usual Sobolev spaces by and the norms in by with being omitted.

2. An -Galerkin Expanded Mixed Variational Formulation

In order to derive an -Galerkin expanded mixed variational formulation we split (1a)–(1d) into a first-order system by introducing and :Define the following spaces:

Multiplying (3a) by for and integrating lead to the weak form (5a). Multiplying (3b) by for leads to the weak form (5b). Multiplying (3c) by for and integrating on result in the weak form (5c). Then the -Galerkin expanded variational problem is to find such that

In order to prove the equivalence of problem (3a)–(3e) and variational problem (5a)–(5e), we need the following lemmas (see [11] or Theorem 3.5, Chapter 1 of [12]).

Lemma 1. For , there exist a and a satisfying , such that .

Lemma 2. For , there exists a , such that .

With the help of these lemmas we can prove the following theorem.

Theorem 3. is a solution to the system (3a)–(3e) if and only if it is a solution to the variational formulation (5a)–(5e).

Proof. The proof of the “only if” part is pretty straightforward. It remains to prove the “if” part. We insert into (5c) to get (3c). By (3a)–(3e) and taking in (5b), we conclude Then we have which, together with (5a) and (3c), yields the following equation: Note that . We obtain that . Setting in (8) we obtain , which implies Then by we can conclude that By (5c) we obtain Here we select the initial value as in (5d) to have . Then, we derive and (7) reduces to Further we get By lemma 2, there exists a such that . Therefore, we have which implies That is, This completes the proof.

3. -Galerkin Expanded Mixed Finite Element Procedure

In this section we will present the numerical scheme for (5a)–(5e). Let be a quasiuniform partition of domain ; that is, with . Let and be the finite dimensional subspaces of and defined by where denotes the set of polynomials of degree at most . Assume that , and satisfy the following approximation properties. For integers , Here when is one of the Raviart-Thomas elements or the Nedelec elements, and , when is one of the other classical mixed elements, such as Breezi-Douglas-Fortin-Marini elements and Breezi-Douglas-Marini elements.

Then the -Galerkin expanded mixed finite element procedure for the system (3a)–(3e) is to find such thatwhere denotes the Raviart-Thomas projection. We next prove the existence and uniqueness of solutions of the scheme (20a)–(20e).

Theorem 4. There exists a unique solution to the -Galerkin expanded mixed finite element procedure (20a)–(20e).

Proof. Let and ; then , , and have the following expressions: Then the scheme (20a)–(20e) can be written in the following matrix form:where Noting that and are positive definite. We can rewrite (22b) and (22c) as Then the system (22a)–(22d) can be characterized as follows: Recalling the assumptions on , we can deduce that the coefficients of and are all Lipschitz continuous with respect to . By the standard theory for the initial-value problems of nonlinear ordinary differential equations, we can deduce that there exists a unique solution to the -Galerkin expanded mixed finite element scheme (20a)–(20e).

4. Convergence Analysis

In this section we will prove the error estimates for the -Galerkin expanded mixed finite element discretization scheme. We begin by reviewing some preliminary knowledge that will be used in the following theoretical analysis.

Let be the Raviart-Thomas projection defined by The following error estimates [1315] hold for and : Let denote the elliptic projection defined by which satisfies the following error estimates (see Theorems and , Chapter 3 of [16]):

To derive the main error estimates we also need the following lemma.

Lemma 5. Suppose that , , and satisfy Then there exists a constant such that

Proof. Assume that is the solution of the following equation with : Recalling that is convex, we have Then by (31) and (32) we deduce Using the estimate of we have By (31), we obtain Inserting the estimates of , , and into (36) leads to By (32) we derive Further we have which implies Further, taking in (31) and by Hölder inequalities as well as inverse property of the finite element spaces and yield Therefore we obtain which, together with (42), yields the desired result.

Theorem 6. Let and be the solutions of (5a)–(5e) and (20a)–(20e), respectively. Assume that satisfies the following regularities: and , , and . Then there exists a positive constant independent of such that with , , for , and , , for .

Proof. In order to derive the error estimates, we decompose the errors as follows: Subtracting the numerical scheme (20a)–(20e) from the weak formulation (5a)–(5e), we can derive the following error equations: Choosing in the second equation of (48) leads to By setting in the third equation of (48) and using the assumption on we deduce In the following we will estimate . Differentiating the third equation in (48) gives Taking in the first equation of (48) and in (51) and then subtracting the resulting equations lead to The left terms can be dealt with as follows: The terms on the right side can be rewritten as follows by integral formula by parts: Combining all the terms mentioned above we arrive at Now we are in the position to estimate the terms . By Lemma 5 we can deduce where was used. Notice that Then using the assumption , (30), and Cauchy-Schwartz inequality gives Integrating from 0 to and using (50) as well as Cauchy-Schwartz inequality yield Note that for . Then we have Similarly, we can estimate the other terms. By inequality we deduce For we can rewrite it as Then by Cauchy-Schwartz inequality and inequality we derive Here the boundedness of and the Ritz projection were used. For we have Therefore by Cauchy-Schwartz inequality and inequality we obtain Here we used the boundedness of to obtain the above estimate. Similarly, we can deduce Further for and by Cauchy-Schwartz inequality and inequality we have Combining the above estimates leads to To prove the main result we need to make the following induction hypothesis: there exists a constant such that the following estimate holds for : Then by setting small enough and using Gronwall’s inequality we obtain the following estimate which holds for constant : Further, using (27) and (29) gives where constant is independent of . We are now in position to prove the inductive hypothesis (70) which holds on . Suppose that there exists a constant such that Let Then we know that By the same arguments for (72) we can prove Moreover, we can also deduce By inverse inequality of finite element spaces we can conclude Choose satisfying which implies This contradicts with (75). Therefore the induction hypothesis (70) holds.
By Poincaré’s inequality and (49) we have Combining (50), (60), (72), (82), the estimates of projections (27), (29), and triangle inequality leads to the desired theorem result.

5. Numerical Examples

The goal of this section is to carry out two numerical experiments to illustrate our theoretical findings. We consider the following second-order nonlinear hyperbolic problem: where .

Example 7. In this example the exact solution is chosen as
We set . Inserting the above functions into the governing equation we can derive the corresponding right term .

In the first example, we investigate the order of convergence for the -Galerkin expanded mixed finite element method proposed in this paper. Piecewise linear polynomial is used to approximate the unknown function , while the gradient function and the flux function are approximated by the vector function space of the lowest Raviart-Thomas spaces, respectively. For time discretization we adopt backward Euler method. Here we couple the time step with spatial mesh as .

The errors of , , and in norm at different times and the order of convergence for , , and are presented in Tables 1, 2, and 3, respectively. We can observe that the order of convergence for approaches 2, and those for and approach 1, which are in agreement with our theoretical results proposed in the previous section.

The figures of the exact solutions , and the numerical solutions , at are shown in Figures 1 and 2, respectively. We can see that the numerical solutions are accurate and without oscillation compared with the exact solutions.

Example 8. In this example we consider problem (83) with prescribed data and .

The profiles of the numerical solutions for and are shown in Figure 3, respectively. From these figures we can see that our method works well for this kind of problems.


This work is supported by the National Natural Science Foundation of China under Grants 10971254 and 11171193 and Young Scientists Fund of Shandong Province under Grant 2008BS01008. The authors would like to thank the referee for his or her suggestions which have considerably improved the quality of their paper.