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A Two-Grid Finite Element Method for a Second-Order Nonlinear Hyperbolic Equation
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 . The nonlinearities are expanded about the coarse grid solution on the fine gird of size . The resulting linear system is solved on the fine grid. Some a priori error estimates are derived with the -norm 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 .
Let be a bounded convex domain with smooth boundary , and consider the initial-boundary value problem for the following second-order nonlinear hyperbolic equation where and denote and , respectively. . We assume that is a symmetric positive definite matrix. and satisfy the Lipschitz continuous condition with respect to , where and where 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 and a linear solution on the fine grid with grid size . 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  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  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  have studied two-grid finite volume element method for linear and nonlinear elliptic problems. Chen et al. , 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  and Baker . They have obtained optimal estimates for the error, , using subspaces of piecewise polynomial functions of degree , for . 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 and norm. Kumar et al.  presented and discussed semidiscrete piecewise linear finite volume approximations for a second-order wave equation and obtained optimal error estimates in , , and norms. For second-order hyperbolic equations with a nonlinear reaction term, Chen and Liu  have presented a two-grid method using finite volume element method and obtained error estimate in the -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 and on one coarse grid with grid size and one fine grid with grid size , 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 and the work for solving the nonlinear problem is relatively negligible. A remarkable fact about this simple approach is, as shown in , 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 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 with consisting of functions that have generalized derivatives of order in the space . The norm of is defined by with the standard modification for . In order to simplify the notation, we denote by and omit the index and whenever possible; that is, . Let be the subspace of of functions vanishing on the boundary .
For the variational formulation we multiply (1) by a smooth function , which vanishes on and find, after integration over and using Green’s formula, that , such that where denotes the -inner product and the bilinear form is defined by
Henceforth, it will be assumed that the problem (5) has a unique solution , and in the appropriate places to follow, additional conditions on the regularity of which guarantee the convergence results, will be imposed.
Let be a quasiuniform triangulation of with , where is the diameter of the triangle . With the triangulation , we associate the function space consisting of continuous, piecewise linear functions on , vanishing on ; that is, Using the above assumptions on , it is easy to see that is a finite-dimensional subspace of the Hilbert space .
Thus, the continuous-time finite element approximation is defined as to find a solution , , such that where . 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 .
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 is symmetric and positive definite and the following lemma was proved, which indicates that the bilinear form is continuous and coercive on .
Lemma 1. For sufficiently small, there exist two positive constants such that, for all , the coercive property and the boundedness property hold true.
Lemma 2. Let be the standard Ritz projection such that Thus is the finite element approximation of the solution of the elliptic problem whose exact solution is . From [21–23], we have for some positive constant independent of and .
And there exists a positive constant independent of , such that 
We now turn to describe the estimates for the finite element method. We give the error estimates in the -norm and -norm between the exact solution and the semidiscrete finite element solution.
Theorem 3. Let and be the solutions of problem (1) and the semidiscrete finite element scheme (8), respectively. Under the assumptions given in Section 1, if and , for , one has where independent of .
Proof. For convenience, let . Then from (1), (8), and (11), we get the following error equation:
Choosing in (16) and by (11), we get
For the terms of (17), we have
Integrating (17) from 0 to , combining with (18)–(20), and noting that and , we have
Now let us estimate the right-hand side terms of (21); for , there is
For , by (2), we obtain
where we used the fact that is bounded by a positive constant .
For by (14), Schwarz inequality, and (3), we get with being a small positive constant. For , similarly we have
For , by Lemma 2, we obtain By Lemma 1, from (21)–(26), we get Choosing proper and kicking the last term into the left-hand side of (27), and applying Gronwall’s lemma, for , we have 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 , and , with two different mesh sizes and (), and the corresponding piecewise linear finite element spaces and 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.
Consider the following.
Step 1. On the coarse grid , find , such that Step 2. On the fine grid , find , such that
Now we consider the error estimates in the -norm for the two-grid finite element method Algorithm 4.
Proof. Once again, we set and choose . Then for Algorithm 4, we get the error equation Similarly as the proof of Theorem 3, we get For and , we can estimate them similarly as in Theorem 3. So our main task is to deal with . By (3), we have with being a small positive constant Substituting the estimates of in (33) and by Lemma 1, we obtain Choosing proper and kicking the last term into the left-hand side of (33), and applying Gronwall’s lemma, for , we have By Theorem 3, we obtain where independent of . Thus, By (12) and the triangular inequality, the proof is complete.
Remark 6. In order to give the fully discrete scheme, we further discretize time of the semidiscrete two-grid finite element method in this section. We consider a time step and approximate the solutions at , , . Denote , , , 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 , find , such that
Step 2. On the fine grid , find , such that
We can get the same kind of estimate as Theorem 5 with the result .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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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