We present a two-grid finite element scheme for the approximation of a second-order nonlinear hyperbolic equation in two space dimensions. In the two-grid scheme, the full nonlinear problem is solved only on a coarse grid of size H. The nonlinearities are expanded about the coarse grid solution on the fine gird of size h. The resulting linear system is solved on the fine grid. Some a priori error estimates are derived with the H1-norm O(h+H2) for the two-grid finite element method. Compared with the standard finite element method, the two-grid method achieves asymptotically same order as long as the mesh sizes satisfy h=O(H2).

1. Introduction

Let Ω⊂ℝ2 be a bounded convex domain with smooth boundary Γ, and consider the initial-boundary value problem for the following second-order nonlinear hyperbolic equation
(1)utt-∇·(A(u)∇u)=f(x,t),x∈Ω,0<t≤T,u(x,t)=0,x∈Γ,0<t≤T,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,
where utt and ut denote ∂2u/∂t2 and ∂u/∂t, respectively. x=(x1,x2). We assume that A(u) is a symmetric positive definite matrix. A(u) and Au(u) satisfy the Lipschitz continuous condition with respect to u, where Au=∂A/∂u and
(2)|A(u1)-A(u2)|≤L|u1-u2|,(3)|Au(u1)-Au(u2)|≤L|u1-u2|,u1,u2∈ℝ,
where L is a positive constant.

Two-grid method is a discretization technique for nonlinear equations based on two grids of different sizes. The main idea is to use a coarse-grid space to produce a rough approximation of the solution of nonlinear problems and then use it as the initial guess for the solution on the fine grid. This method involves a nonlinear solution on the coarse grid with grid size H and a linear solution on the fine grid with grid size h<H. Two-grid method was first introduced by Xu [1, 2] for linear (nonsymmetric or indefinite) and especially nonlinear elliptic partial differential equations. Later on, two-grid method was further investigated by many authors. Dawson and Wheeler [3, 4], Chen and Liu [5] have constructed the two-grid method by using finite difference method, mixed finite element method, and piecewise linear finite element method for nonlinear parabolic equations, respectively. Wu and Allen [6] have applied two-grid method combined with mixed finite element method to reaction-diffusion equations. Chen et al. [7–10] have constructed two-grid methods for expanded mixed finite-element solution of semilinear and nonlinear reaction-diffusion equations. Bi and Ginting [11] have studied two-grid finite volume element method for linear and nonlinear elliptic problems. Chen et al. [12], Chen and Liu [13, 14] have studied two-grid methods for semilinear parabolic and second-order hyperbolic equations using finite volume element method.

The finite element analysis for the second-order linear hyperbolic equations was discussed by Dupont [15] and Baker [16]. They have obtained optimal L∞(L2) estimates for the error, O(hr), using subspaces of piecewise polynomial functions of degree ≤r-1, for r≥1. Then Yuan and Wang [17, 18] have studied error estimates for the finite element method of the second-order nonlinear hyperbolic equations and proved the optimal error estimates in the L2 and H1 norm. Kumar et al. [19] presented and discussed semidiscrete piecewise linear finite volume approximations for a second-order wave equation and obtained optimal error estimates in L2, H1, and L∞ norms. For second-order hyperbolic equations with a nonlinear reaction term, Chen and Liu [14] have presented a two-grid method using finite volume element method and obtained error estimate in the H1-norm.

However, as far as we know there is no two-grid finite element convergence analysis for the second-order nonlinear hyperbolic equations (1). In this paper, based on two conforming piecewise linear finite element spaces SH and Sh on one coarse grid with grid size H and one fine grid with grid size h, respectively, we consider the two-grid finite element discretization techniques for the second-order nonlinear hyperbolic problems. With the proposed techniques, solving the nonlinear problems on the fine-grid space is reduced to solving a linear system on the fine-grid space and a nonlinear system on a much smaller space. This means that solving a nonlinear problem is not much more difficult than solving one linear problem, since dimSH≪dimSh and the work for solving the nonlinear problem is relatively negligible. A remarkable fact about this simple approach is, as shown in [1], that the coarse mesh can be quite coarse and still maintain a good accuracy approximation.

The rest of this paper is organized as follows. In Section 2, we describe the finite element scheme for the nonlinear second-order hyperbolic problem (1). Section 3 contains the error estimates for the finite element method. Section 4 is devoted to the two-grid finite element and its error analysis. Throughout this paper, the letter C or with its subscript denotes a generic positive constant which does not depend on the mesh parameters and may be different at its different occurrences.

2. Standard Finite Element Method

We adopt the standard notation for Sobolev spaces Ws,p(Ω) with 1≤p≤∞ consisting of functions that have generalized derivatives of order s in the space Lp(Ω). The norm of Ws,p(Ω) is defined by
(4)∥u∥s,p,Ω=∥u∥s,p=(∫Ω∑|α|≤s|Dαu|pdx)1/p,
with the standard modification for p=∞. In order to simplify the notation, we denote Ws,2(Ω) by Hs(Ω) and omit the index p=2 and Ω whenever possible; that is, ∥u∥s,2,Ω=∥u∥s,2=∥u∥s. Let H01(Ω) be the subspace of H1(Ω) of functions vanishing on the boundary Γ.

For the variational formulation we multiply (1) by a smooth function v, which vanishes on Γ and find, after integration over Ω and using Green’s formula, that u(·,t)∈H01(Ω), 0<t≤T such that
(5)(utt,v)+a(u;u,v)=(f,v),∀v∈H01(Ω),u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,
where (·,·) denotes the L2(Ω)-inner product and the bilinear form a(·;·,·) is defined by
(6)a(w;u,v)=∫ΩA(w)∇u·∇vdx.

Henceforth, it will be assumed that the problem (5) has a unique solution u, and in the appropriate places to follow, additional conditions on the regularity of u which guarantee the convergence results, will be imposed.

Let 𝒯h be a quasiuniform triangulation of Ω with h=maxhK, where hK is the diameter of the triangle K∈𝒯h. With the triangulation 𝒯h, we associate the function space Sh consisting of continuous, piecewise linear functions on 𝒯h, vanishing on Γ; that is,
(7)Sh={v∈C(Ω¯):vlinearinKforeachK∈𝒯h,v=0onΓ}.
Using the above assumptions on 𝒯h, it is easy to see that Sh is a finite-dimensional subspace of the Hilbert space H01(Ω) [20].

Thus, the continuous-time finite element approximation is defined as to find a solution uh(t)∈Sh, 0<t≤T, such that
(8)(uh,tt,vh)+a(uh;uh,vh)=(f,vh),∀vh∈Sh,uh(0)=u0,uh,t(0)=u1,
where uh,tt=∂2uh/∂t2. Since we have discretized only in the space variables, this is referred to as a spatially semidiscrete problem. The existence and uniqueness of the solution of (8) have been proved by Yuan and Wang [17].

3. Error Analysis for the Finite Element Method

To describe the error estimates for the finite element scheme (8), we will give some useful lemmas. In [17, 21] it was shown that the bilinear form a(·;·,·) is symmetric and positive definite and the following lemma was proved, which indicates that the bilinear form a(·;·,·) is continuous and coercive on Sh.

Lemma 1.

For h sufficiently small, there exist two positive constants C1,C2>0 such that, for all uh,vh,wh∈Sh, the coercive property
(9)a(wh;uh,uh)≥C1∥uh∥12
and the boundedness property
(10)|a(wh;uh,vh)|≤C2∥uh∥1∥vh∥1
hold true.

Lemma 2.

Let u~∈Sh be the standard Ritz projection such that
(11)a(u(x,t);(u~-u)(x,t),vh)=0,∀vh∈Sh.
Thus u~ is the finite element approximation of the solution of the elliptic problem whose exact solution is u. From [21–23], we have
(12)∥u-u~∥+h∥u-u~∥1≤Ch2∥u∥2,(13)∥(u-u~)t∥+h∥(u-u~)t∥1≤Ch2∥ut∥2,
for some positive constant C independent of h and u.

And there exists a positive constant C0 independent of h, such that [21]
(14)∥∇u~∥∞+∥∇u~t∥∞≤C0,fort≤T.

We now turn to describe the estimates for the finite element method. We give the error estimates in the H1-norm and L2-norm between the exact solution and the semidiscrete finite element solution.

Theorem 3.

Let u and uh be the solutions of problem (1) and the semidiscrete finite element scheme (8), respectively. Under the assumptions given in Section 1, if uh(0)=u~0 and uh,t(0)=u~1, for 0<t≤T, one has
(15)∥u(t)-uh(t)∥+h∥u(t)-uh(t)∥1≤𝒞h2,∥(u(t)-uh(t))t∥≤𝒞h2,
where 𝒞=C(∥u∥L2(H2),∥u∥L∞(H2),∥ut∥L2(H2),∥utt∥L2(H2)) independent of h.

Proof.

For convenience, let u-uh=(u-u~)+(u~-uh)=:η+ξ. Then from (1), (8), and (11), we get the following error equation:
(16)(ξtt,vh)+a(uh;ξ,vh)=-(ηtt,vh)+a(uh;u,vh)-a(u;u,vh),∀vh∈Sh.
Choosing vh=ξt in (16) and by (11), we get
(17)(ξtt,ξt)+a(uh;ξ,ξt)=-(ηtt,ξt)+a(uh;u~,ξt)-a(u;u~,ξt).
For the terms of (17), we have
(18)(ξtt,ξt)=12ddt(ξt,ξt)=12ddt∥ξt∥2.(19)a(uh;ξ,ξt)=∫ΩA(uh)∇ξ·∇ξtdx=12ddta(uh;ξ,ξ)-12∫Ω∂A(uh)∂u∂uh∂t∇ξ·∇ξdx.(20)a(uh;u~,ξt)-a(u;u~,ξt)=∫Ω[A(uh)-A(u)]∇u~·∇ξtdx=ddt(∫Ω[A(uh)-A(u)]∇u~·∇ξdx)-∫Ω[A(uh)-A(u)]∇∂u~∂t·∇ξdx-∫Ω(∂A(uh)∂u∂uh∂t-∂A(u)∂u∂u∂t)∇u~·∇ξdx.
Integrating (17) from 0 to t, combining with (18)–(20), and noting that ξ(0)=0 and ξt(0)=0, we have
(21)12∥ξt∥2+12a(uh;ξ,ξ)=-∫0t(ηtt,ξt)dt+12∫0t(∫Ω∂A(uh)∂u∂uh∂t∇ξ·∇ξdx)dt+∫Ω[A(uh)-A(u)]∇u~·∇ξdx-∫0t(∫Ω[A(uh)-A(u)]∇∂u~∂t·∇ξdx)dt-∫0t(∫Ω(∂A(uh)∂u∂uh∂t-∂A(u)∂u∂u∂t)∇u~·∇ξdx)dt≡∑i=15Ti.
Now let us estimate the right-hand side terms of (21); for T1, there is
(22)|T1|≤C∫0t∥ηtt∥∥ξt∥dt≤C∫0t(∥ηtt∥2+∥ξt∥2)dt.
For T2, by (2), we obtain
(23)|T2|≤C∫0t|∂A(uh)∂u∂uh∂t|∞∥∇ξ∥2dt≤CL∫0t|∂uh∂t|∞∥∇ξ∥2dt≤C∫0t∥ξ∥12dt,
where we used the fact that |∂uh/∂t|∞ is bounded by a positive constant [17].

For T3 by (14), Schwarz inequality, and (3), we get
(24)|T3|≤C∥∇u~∥∞∥A(uh)-A(u)∥∥∇ξ∥≤C∥∇u~∥∞L∥ξ+η∥∥ξ∥1≤C(∥η∥2+∥ξ∥2)+ϵ∥ξ∥12≤C(∥η∥2+∫0t∥ξt∥2dt)+ϵ∥ξ∥12,
with ϵ being a small positive constant. For T4, similarly we have
(25)|T4|≤C∫0t∥∇u~t∥∞∥A(uh)-A(u)∥∥∇ξ∥dt≤C∫0t∥∇u~t∥∞L∥ξ+η∥∥ξ∥1dt≤C∫0t(∥ξ∥2+∥η∥2+∥ξ∥12)dt≤C∫0t(∥η∥2+∥ξ∥12)dt.

For T5, by Lemma 2, we obtain
(26)|T5|≤C∫0t∥∇u~∥∞∥∂uh∂t-∂u∂t∥∥∇ξ∥dt≤C∫0t(∥ξt∥+∥ηt∥)∥ξ∥1dt≤C∫0t(∥ξt∥2+∥ηt∥2+∥ξ∥12)dt.
By Lemma 1, from (21)–(26), we get
(27)∥ξt∥2+C0∥ξ∥12≤C1[∫0t(∥ηtt∥2+∥ηt∥2+∥η∥2)dt+∥η∥2]+C2∫0t(∥ξt∥2+∥ξ∥12)dt+ϵ∥ξ∥12.
Choosing proper ϵ and kicking the last term into the left-hand side of (27), and applying Gronwall’s lemma, for t≤T, we have
(28)∥ξt∥2+∥ξ∥12≤C1[∫0T(∥ηtt∥2+∥ηt∥2+∥η∥2)dt+∥η∥2]≤Ch4[∫0T(∥utt∥22+∥ut∥22+∥u∥22)dt+∥u∥22].
Together with (12) and (13), this yields (15).

4. Two-Grid Finite Element Method

In this section, we will present a two-grid finite element algorithm for problem (1) based on two different finite element spaces. The idea of the two-grid method is to reduce the nonlinear problem on a fine grid into a linear system on the fine grid by solving a nonlinear problem on a coarse grid. The basic mechanisms are two quasiuniform triangulations of Ω, 𝒯H and 𝒯h, with two different mesh sizes H and h (H>h), and the corresponding piecewise linear finite element spaces SH and Sh which will be called the coarse-grid and the fine-grid spaces, respectively.

To solve problem (1), we introduce two-grid algorithms into finite element method. This method involves a nonlinear solution on the coarse grid space and a linear solution on the fine grid space. We present the two-grid finite element method with two steps.

Algorithm 4.

Consider the following.

Step 1. On the coarse grid 𝒯H, find uH∈SH, such that
(29)(uH,tt,vH)+a(uH;uH,vH)=(f,vH),∀vH∈SH,uH(0)=u~0,uH,t(0)=u~1.Step 2. On the fine grid 𝒯h, find uh∈Sh, such that
(30)(uh,tt,vh)+a(uH;uh,vh)=(f,vh),∀vh∈Sh,uh(0)=u~0,uh,t(0)=u~1.

Now we consider the error estimates in the H1-norm for the two-grid finite element method Algorithm 4.

Theorem 5.

Let u and uh be the solutions of problem (1) and the two-grid scheme Algorithm 4, respectively. Under the assumptions given in Section 1, if uh(0)=u~0 and uh,t(0)=u~1, for 0<t≤T, we have
(31)∥u(t)-uh(t)∥1≤𝒞(h+H2),
where 𝒞=C(∥u∥L2(H2),∥u∥L∞(H2),∥u∥L∞(W1,∞),∥ut∥L2(H2),∥ut∥L∞(W1,∞),∥utt∥L2(H2)) independent of h.

Proof.

Once again, we set u-uh=(u-u~)+(u~-uh)=:η+ξ and choose vh=ξt. Then for Algorithm 4, we get the error equation
(32)(ξtt,ξt)+a(uH;ξ,ξt)=-(ηtt,ξt)+a(uH;u~,ξt)-a(u;u~,ξt).
Similarly as the proof of Theorem 3, we get
(33)12∥ξt∥2+12a(uH;ξ,ξ)=-∫0t(ηtt,ξt)dt+12∫0t(∫Ω∂A(uH)∂u∂uH∂t∇ξ·∇ξdx)dt+∫Ω[A(uH)-A(u)]∇u~·∇ξdx-∫0t(∫Ω[A(uH)-A(u)]∇∂u~∂t·∇ξdx)dt-∫0t(∫Ω(∂A(uH)∂u∂uH∂t-∂A(u)∂u∂u∂t)∇u~·∇ξdx)dt≡∑i=15Ti′.
For T1′ and T2′, we can estimate them similarly as in Theorem 3. So our main task is to deal with T3′–T5′. By (3), we have
(34)|T3′|≤C∥∇u~∥∞∥A(uH)-A(u)∥∥∇ξ∥≤C∥∇u~∥∞L∥uH-u∥∥ξ∥1≤C∥uH-u∥2+ϵ∥ξ∥12,
with ϵ being a small positive constant
(35)|T4′|≤C∫0t∥∇u~t∥∞∥A(uH)-A(u)∥∥∇ξ∥dt≤C∫0t∥∇u~t∥∞L∥uH-u∥∥ξ∥1dt≤C∫0t(∥uH-u∥2+∥ξ∥12)dt,(36)|T5′|≤C∫0t∥∇u~∥∞∥∂uH∂t-∂u∂t∥∥∇ξ∥dt≤C∫0t∥(uH-u)t∥∥ξ∥1dt≤C∫0t(∥(uH-u)t∥2+∥ξ∥12)dt.
Substituting the estimates of Ti′ in (33) and by Lemma 1, we obtain
(37)∥ξt∥2+C0∥ξ∥12≤C1∫0t(∥ηtt∥2+∥uH-u∥2+∥(uH-u)t∥2)dt+C2∥uH-u∥2+C3∫0t(∥ξt∥2+∥ξ∥12)dt+ϵ∥ξ∥12.
Choosing proper ϵ and kicking the last term into the left-hand side of (33), and applying Gronwall’s lemma, for t≤T, we have
(38)∥ξt∥2+∥ξ∥12≤C1∫0t(∥ηtt∥2+∥uH-u∥2+∥(uH-u)t∥2)dt+C2∥uH-u∥2.
By Theorem 3, we obtain
(39)∥ξt∥2+∥ξ∥12≤𝒞(h4+H4),
where 𝒞=C(∥u∥L2(H2),∥u∥L∞(H2),∥u∥L∞(W1,∞),∥ut∥L2(H2),∥ut∥L∞(W1,∞),∥utt∥L2(H2)) independent of h. Thus,
(40)∥ξt∥+∥ξ∥1≤𝒞(h2+H2).
By (12) and the triangular inequality, the proof is complete.

Remark 6.

In order to give the fully discrete scheme, we further discretize time t of the semidiscrete two-grid finite element method in this section. We consider a time step Δt and approximate the solutions at tn=nΔt, Δt=T/N, n=0,1,…,N. Denote uhn=uh(tn), uh,ttn=(uhn+1-2uhn+uhn-1)/(Δt)2, uh,tn=(uhn+1-uhn)/Δt, we can get the fully discrete two-grid finite element scheme for (1). For simplicity and convenience, we only give the fully discrete scheme for Algorithm 4.

Algorithm 4^{
′}. Consider the following.

Step 1. On the coarse grid 𝒯H, find uHn∈SH(n=1,2,…), such that
(41)(uH,ttn,vH)+a(uHn+1;uHn+1,vH)=(fn+1,vH),∀vH∈SH,uH0=u~0,uH,t0=u~1.

Step 2. On the fine grid 𝒯h, find uhn∈Sh(n=1,2,…), such that
(42)(uh,ttn,vh)+a(uHn+1;uhn+1,vh)=(fn+1,vh),∀vh∈Sh,uh0=u~0,uh,t0=u~1.

We can get the same kind of estimate as Theorem 5 with the result ∥un-uhn∥1≤𝒞((Δt)2+h+H2).

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 the National Natural Science Foundation of China (no. 11301456) and Shandong Province Natural Science Foundation (nos. ZR2010AQ010 and ZR2011AQ021).

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