AnH-Galerkin Expanded Mixed Finite Element Approximation of Second-Order Nonlinear Hyperbolic Equations

and Applied Analysis 3 we can conclude that ψ t (x, t) ≡ 0. (11) By (5c) we obtain σ (x, 0) = A (u 0 ) ∇u (x, 0) + A (u 0 )ψ (x, 0) . (12) Here we select the initial value σ(x, 0) as in (5d) to have ψ(x, 0) = 0. Then, we derive ψ(x, t) = 0 and (7) reduces to p = ∇u. (13) Further we get (u tt , ∇ ⋅ q) − (∇ ⋅ (A (u) ∇u) , ∇ ⋅ q) = (f, ∇ ⋅ q) , ∀q ∈ H. (14) By lemma 2, there exists a F ∈ H(div, Ω) such that ∇ ⋅ F = u tt − f. Therefore, we have (∇ ⋅ F, ∇ ⋅ q) − (∇ ⋅ (A (u) ∇u) , ∇ ⋅ q) = 0, (15) which implies F = A (u) ∇u = σ. (16) That is, u tt − f = ∇ ⋅ σ. (17) This completes the proof. 3. H1-Galerkin Expanded Mixed Finite Element Procedure In this section we will present the numerical scheme for (5a)– (5e). Let T h be a quasiuniform partition of domain Ω; that is, Ω = ⋃ K∈Th K with h = max{diam(K); K ∈ T h }. Let H h and V h be the finite dimensional subspaces of H(div; Ω) and H 1 0 (Ω) defined by H h = {q h ∈ H (div; Ω) ; q h | K ∈ (P k (K)) d , ∀K ∈ T h } , V h = {V h ∈ H 1 0 (Ω) ; V h | K ∈ P m (K) , ∀K ∈ T h } , (18) where P j (K) denotes the set of polynomials of degree at most j. Assume thatH h , andV h satisfy the following approximation properties. For integers k ≥ 0, m ≥ 1, inf qh∈Hh 󵄩󵄩󵄩󵄩q − qh 󵄩󵄩󵄩󵄩 ≤ Ch k+1 ‖q‖ k+1,Ω , q ∈ (Hk+1 (Ω)) d ∩H, inf qh∈Hh 󵄩󵄩󵄩󵄩∇ ⋅ (q − qh) 󵄩󵄩󵄩󵄩 ≤ Ch k1 ‖q‖ k1+1,Ω , q ∈ (Hk+1 (Ω)) d ∩H, inf Vh∈Vh { 󵄩󵄩󵄩󵄩V − Vh 󵄩󵄩󵄩󵄩 + h 󵄩󵄩󵄩󵄩V − Vh 󵄩󵄩󵄩󵄩1,Ω } ≤ Ch m+1 ‖V‖m+1,Ω, V ∈ Hm+1 (Ω) ∩ V. (19) Here k 1 = k + 1 when H h is one of the RaViart-Thomas elements or the Nedelec elements, and k 1 = k ≥ 1, when H h is one of the other classical mixed elements, such as Breezi-Douglas-Fortin-Marini elements and Breezi-Douglas-Marini elements. Then the H-Galerkin expanded mixed finite element procedure for the system (3a)–(3e) is to find (u h , p h ,σ h ) ∈ V h ×H h ×H h such that (p htt , q h ) + (∇ ⋅ σ h , ∇ ⋅ q h ) = − (f, ∇ ⋅ q h ) , ∀q h ∈ H h , (20a) (p h , ∇V h ) = (∇u h , ∇V h ) , ∀V h ∈ V h , (20b) (σ h ,w h ) = (A (u h ) p h ,w h ) , ∀w h ∈ H h , (20c) σ h (x, 0) = Π h σ (x, 0) , ∀x ∈ Ω, (20d) p h (x, 0) = Π h p (x, 0) , p ht (x, 0) = Π h p t (x, 0) , ∀x ∈ Ω, (20e) where Π h 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 (u h , p h ,σ h ) ∈ V h × H h × H h to the H-Galerkin expanded mixed finite element procedure (20a)–(20e). Proof. Let H h = span{ψ i } M i=1 and V h = span{φ i } N i=1 ; then σ h ∈ H h , p h ∈ H h , and u h ∈ V h have the following expressions: p h = M ∑ i=1 p i ψ i , σ h = M ∑ i=1 λ i ψ i , u h = N ∑ i=1 u i φ i . (21) Then the scheme (20a)–(20e) can be written in the following matrix form: AP tt + BΛ = F, (22a) DU = CP, (22b) AΛ = G (U)P, (22c) P (0) ,P t (0) are given, (22d) 4 Abstract and Applied Analysis where A = ((ψ i ,ψ j )) M×M , B = ((∇ ⋅ ψ i , ∇ ⋅ ψ j )) M×M , C = ((ψ i , ∇φ j )) N×M , D = ((∇φ i , ∇φ j )) N×N , G (U) = ((A (U)ψ i ,ψ j )) M×M , F = ((−f,ψ j )) M×1 , P = (p 1 , p 2 , . . . , p M ) T , Λ = (λ 1 , λ 2 , . . . , λ M ) T , U = (u 1 , u 2 , . . . , u N ) T . (23) Noting that A and D are positive definite. We can rewrite (22b) and (22c) as U = D−1CP, Λ = A −1 G (U)P. (24) Then the system (22a)–(22d) can be characterized as follows: AP tt + BA −1 G(D −1 CP)P = F, (25a) P (0) ,P t (0) are given. (25b) Recalling the assumptions on A(u), we can deduce that the coefficients of P tt and P are all Lipschitz continuous with respect to P(t). By the standard theory for the initial-value problems of nonlinear ordinary differential equations, we can deduce that there exists a unique solution (u h , p h ,σ h ) ∈ V h × H h × H h to the H-Galerkin expanded mixed finite element scheme (20a)–(20e). 4. Convergence Analysis In this section we will prove the error estimates for the H -Galerkin expanded mixed finite element discretization scheme.We begin by reviewing some preliminary knowledge that will be used in the following theoretical analysis. Let Π h : H → H h be the RaViart-Thomas projection defined by (∇ ⋅ (q − Π h q) , ∇ ⋅ q h ) = 0, ∀q h ∈ H h . (26) The following error estimates [13–15] hold forΠ h and 2 ≤ p ≤ ∞: 󵄩󵄩󵄩󵄩q − Πhq 󵄩󵄩󵄩󵄩p,Ω ≤ Ch k+1 ‖q‖ k+1,p,Ω , 󵄩󵄩󵄩󵄩∇ ⋅ (q − Πhq) 󵄩󵄩󵄩󵄩p,Ω ≤ Ch k1 ‖q‖ k1+1,p,Ω . (27) Let R h : V → V h denote the elliptic projection defined by (∇ (w − R h w) , ∇V h ) = 0, ∀V h ∈ V h (28) which satisfies the following error estimates (see Theorems 3.2.2 and 3.2.5, Chapter 3 of [16]): 󵄩󵄩󵄩󵄩w − Rhw 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩(w − ht 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩(w − htt 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩(w − httt 󵄩󵄩󵄩󵄩 + h 󵄩󵄩󵄩󵄩∇ (w − Rhw) 󵄩󵄩󵄩󵄩 ≤ Ch m+1 , (29) max {󵄩󵄩󵄩󵄩Rhw 󵄩󵄩󵄩󵄩0,∞ , 󵄩󵄩󵄩󵄩(Rhw)t 󵄩󵄩󵄩0,∞ , 󵄩󵄩󵄩󵄩(Rhw)tt 󵄩󵄩󵄩0,∞ } ≤ C. (30) To derive the main error estimates we also need the following lemma. Lemma 5. Suppose that ξ, ζ ∈ H h , θ ∈ H, and β tt ∈ V h satisfy (ξ tt + θ tt , q h ) + (∇ ⋅ ζ, ∇ ⋅ q h ) = 0, ∀q h ∈ H h , (31) (ξ tt + θ tt , ∇V h ) = (∇β tt , ∇V h ) , ∀V h ∈ V h . (32) Then there exists a constant C such that 󵄩󵄩󵄩󵄩 β tt 󵄩󵄩󵄩󵄩 ≤ C (‖∇ ⋅ ζ‖ + h 󵄩󵄩󵄩󵄩 θ tt 󵄩󵄩󵄩󵄩) . (33) Proof. Assume thatφ ∈ H(Ω) is the solution of the following equation with ψ ∈ L(Ω):


Introduction
The objective of this paper is to present and analyze an 1 -Galerkin expanded mixed finite element method for the following second-order nonlinear hyperbolic equation: where Ω is a bounded convex polygonal domain in 2 with boundary Ω and = [0, ] with < ∞. (x, ) denotes the sound pressure, (x, ) is the external force, and ( ) is the coefficient, which is supposed to satisfy the following conditions.
The primary interests in engineering application for the mathematical model (1a)-(1d) are the sound pressure , the gradient of sound pressure p, 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 [1][2][3][4] 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 1 -Galerkin expanded mixed finite element method for model (1a)-(1d) which can solve the sound pressure , the gradient of sound pressure p, and the acceleration of sound 2 Abstract and Applied Analysis 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 1 -Galerkin expanded mixed variational formulation and the existence and uniqueness of the semidiscrete 1 -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 1 -Galerkin mixed finite element method. One can refer to [5][6][7][8] 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 1 -Galerkin expanded mixed finite element variational form and prove the equivalence between primal problem and the variational formulation. In Section 3, the 1 -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.
Then by Abstract and Applied Analysis 3 we can conclude that By (5c) we obtain Here we select the initial value (x, 0) as in (5d) to have (x, 0) = 0. Then, we derive (x, ) = 0 and (7) reduces to Further we get By lemma 2, there exists a ∈ H(div, Ω) such that ∇ ⋅ = − . Therefore, we have which implies That is, This completes the proof.

1 -Galerkin Expanded Mixed Finite Element Procedure
In this section we will present the numerical scheme for (5a)-(5e). Let T ℎ be a quasiuniform partition of domain Ω; that is, Ω = ⋃ ∈T ℎ with ℎ = max{diam( ); ∈ T ℎ }. Let H ℎ and ℎ be the finite dimensional subspaces of H(div; Ω) and 1 0 (Ω) defined by where P ( ) denotes the set of polynomials of degree at most . Assume that H ℎ , and ℎ satisfy the following approximation properties. For integers ≥ 0, Here 1 = + 1 when ℎ is one of the V ℎ elements or the elements, and 1 = ≥ 1, when ℎ is one of the other classical mixed elements, such as ---elements and ---elements. Then the 1 -Galerkin expanded mixed finite element procedure for the system (3a)-(3e) is to find where Π ℎ 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
Proof. Let H ℎ = span{ } =1 and ℎ = span{ } =1 ; then ℎ ∈ H ℎ , p ℎ ∈ H ℎ , and ℎ ∈ ℎ 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 (24) Then the system (22a)-(22d) can be characterized as follows: Recalling the assumptions on ( ), we can deduce that the coefficients of P and P are all Lipschitz continuous with respect to P( ). By the standard theory for the initial-value problems of nonlinear ordinary differential equations, we can deduce that there exists a unique solution ( ℎ , p ℎ , ℎ ) ∈ ℎ × H ℎ × H ℎ to the 1 -Galerkin expanded mixed finite element scheme (20a)-(20e).

Convergence Analysis
In this section we will prove the error estimates for the 1 -Galerkin expanded mixed finite element discretization scheme. We begin by reviewing some preliminary knowledge that will be used in the following theoretical analysis.
The following error estimates [13][14][15] hold for Π ℎ and 2 ≤ ≤ ∞: Let ℎ : → ℎ denote the elliptic projection defined by To derive the main error estimates we also need the following lemma.
Lemma 5. Suppose that , ∈ H ℎ , ∈ H, and ∈ ℎ satisfy Then there exists a constant such that Proof. Assume that ∈ 2 (Ω) is the solution of the following equation with ∈ 2 (Ω): Recalling that Ω is convex, we have Then by (31) Using the estimate of ℎ we have By (31), we obtain Further we have which implies Further, taking ℎ = in (31) and bÿinequalities as well as inverse property of the finite element spaces ℎ and H ℎ yield Therefore we obtain which, together with (42), yields the desired result.
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 V ℎ = in the second equation of (48) leads to By setting ℎ = 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 q ℎ = in the first equation of (48) and w ℎ = in (51) and then subtracting the resulting equations lead to (∇ ⋅ , ∇ ⋅ ) + ( ( ℎ ) , ) The left terms can be dealt with as follows: (∇ ⋅ , ∇ ⋅ ) = 1 2 (∇ ⋅ , ∇ ⋅ ) , 6 Abstract and Applied Analysis 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 , = 0, 1, 2, . . . , 7. By Lemma 5 we can deduce where (0) = 0 was used. Notice that Then using the assumption 1 , (30), and Cauchy-Schwartz inequality gives Integrating from 0 to and using (50) as well as Cauchy-Schwartz inequality yield Note that for (0) = 0. Then we have Similarly, we can estimate the other terms. By inequality we deduce Abstract and Applied Analysis 7 For 3 we can rewrite it as Then by Cauchy-Schwartz inequality and inequality we derive Here the boundedness of ‖ ‖ 0,∞ and the Ritz projection ‖( ℎ ) ‖ 0,∞ were used. For 4 we have Therefore by Cauchy-Schwartz inequality and inequality we obtain Here we used the boundedness of ‖ ‖ ∞ (0, ; ∞ (Ω)) to obtain the above estimate. Similarly, we can deduce Further for 6 and 7 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 0 < ℎ 0 < 1 such that the following estimate holds for 0 < ℎ ≤ ℎ 0 : Then by setting small enough and using Gronwall's inequality we obtain the following estimate which holds for constant > 0: Further, using (27) and (29) gives where constant̃> 0 is independent of ℎ. We are now in position to prove the inductive hypothesis (70) which holds on ∈ . Suppose that there exists a constant 0 < ℎ * ≤ ℎ 0 such that Then we know that By the same arguments for (72) we can prove ∇ ⋅ * 2 + * 2 ≤̃1ℎ 2 min{ +1, +1} , 0 < ≤ * . (77) Moreover, we can also deduce * 2 ≤ ∫ 0 * 2 By inverse inequality of finite element spaces we can conclude * ∞ (0, * ; ∞ (Ω)) ≤̃1ℎ min{ +1, +1}− /2 , which implies This contradicts with (75). Therefore the induction hypothesis (70) holds. By Poincaré's inequality and (49) we have Combining (50)

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 Ω = [0, 1] × [0, 1].
We set ( ) = 2 + 1. 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 1 -Galerkin expanded mixed finite element method proposed in this paper. Piecewise linear polynomial is used to approximate the unknown function , while the gradient function p 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 ℎ = Δ .
The errors of − ℎ , p − p ℎ , and − ℎ in 2 norm at different times and the order of convergence for , p, and are presented in Tables 1, 2, and 3, respectively. We can observe that the order of convergence for approaches 2, and those for p and approach 1, which are in agreement with our theoretical results proposed in the previous section.
The figures of the exact solutions , p and the numerical solutions ℎ , p ℎ at = 1.0 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.
The profiles of the numerical solutions for and p are shown in Figure 3, respectively. From these figures we can see that our method works well for this kind of problems.