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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 391918, 11 pages
A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations
1Department of Mathematics and Information Science, Yantai University, Yantai 264005, China
2Schools of Mathematics and Quantitative Economics, Shandong University of Finance, Jinan 250014, China
Received 5 July 2012; Accepted 24 August 2012
Academic Editor: Xinguang Zhang
Copyright © 2012 Chuanjun Chen and Wei Liu. 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.
A two-grid method is presented and discussed for a finite element approximation to a nonlinear parabolic equation in two space dimensions. Piecewise linear trial functions are used. In this two-grid scheme, the full nonlinear problem is solved only on a coarse grid with grid size . The nonlinearities are expanded about the coarse grid solution on a fine gird of size , and the resulting linear system is solved on the fine grid. A priori error estimates are derived with the -norm which shows that the two-grid method achieves asymptotically optimal approximation as long as the mesh sizes satisfy . An example is also given to illustrate the theoretical results.
Let be a bounded convex domain with smooth boundary and consider the initial-boundary value problem for the following nonlinear parabolic equations: where denotes . , is a given real-valued function on . We assume that is a symmetric, uniformly positive definite second-order diagonal tensor. and satisfy the Lipschitz continuous condition with respect to . where is a positive constant. We also assume that each element of is twice continuously differentiable in space and where is a positive constant.
It is assumed that the functions have enough regularity, and they satisfy appropriate compatibility conditions so that the initial-boundary value problem (1.1) has a unique solution satisfying the regularity results as demanded by our subsequent analysis.
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 one Newton-like iteration on the fine grid. This method involves a nonlinear solve on the coarse grid with grid size and a linear solve 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 wheerler  and Dawson et al.  have applied this method combined with mixed finite element method and finite difference method to nonlinear parabolic equations. Wu and Allen  have applied two-grid method combined with mixed finite element method to reaction-diffusion equations. Chen et al. [6–9] have constructed two-grid methods for expanded mixed finite element solution of semilinear and nonlinear reaction-diffusion equations and nonlinear parabolic equations. Bi and Ginting  have studied two-grid finite volume element method for linear and nonlinear elliptic problems. Chen et al.  and Chen and Liu [12, 13] have studied two-grid method for semilinear parabolic and second-order hyperbolic equations using finite volume element.
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 nonlinear parabolic problems. With the proposed techniques, solving the nonlinear problems on the fine space is reduced to solving a linear problems on the fine space and a nonlinear problems 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 parabolic problem (1.1). 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. A numerical example is presented to confirm the theoretical results in the last section.
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. 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, . We denote by the subspace of of functions vanishing on the boundary .
The corresponding variational form of (1.1) is to find such that where denotes the -inner product and the bilinear form is defined by
Henceforth, it will be assumed that the problem (2.2) 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 quasi-uniform triangulation of with , where is the diameter of the triangle . Denote the continuous piecewise linear finite element space associated with the triangulation by With 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 with . Since we have discretized only in the space variables, this is referred to as a spatially semidiscrete problem. By means of Brouwer fixed-point iteration, Li  has proved the existence and uniqueness of the solution of this problem.
3. Error Analysis for the Finite Element Method
To describe the error estimates for the finite element scheme (2.5), we will give some useful lemmas. In , 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 3.1. For sufficiently small , there exist two positive constants such that, for all , the coercive property and the boundedness property hold true.
Lemma 3.2. Let be the standard Ritz projection such that Thus is the finite element approximation of the solution of the corresponding elliptic problem whose exact solution is . From [16–18], we have for some positive constant independent of and . In addition, Yuan and Wang  have proved that and are bounded by a positive constant.
Now we turn to describe the estimates for the finite element method. We give the error estimates in the -norm between the exact solution and the semidiscrete finite element solution.
Theorem 3.3. Let and be the solutions of problem (1.1) and the semidiscrete finite element scheme (2.5), respectively. Under the assumptions given in Section 1, if , for , we have where independent of .
Proof. For convenience, let . Then from (1.1), (2.5), and (3.3), we get the following error equation: Choosing in (3.7) to get For the first term of (3.8), we have Integrating (3.8) from 0 to , by (3.9) and noting that , we have Now let's estimate the right-hand terms of (3.10), for , there is For , by (1.2), we obtain with a small positive constant. By Lemma 3.1, from (3.10)–(3.12), we get Choosing proper and kicking the last term into the left side of (3.13), and applying Gronwall lemma, for , we have By (3.4) and (3.5), we obtain which yields (3.6).
4. Two-Grid Finite Element Method
In this section, we will present two-grid finite element algorithm for problem (1.1) 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 a fine grid by solving a nonlinear problem on a coarse grid. The basic mechanisms are two quasi-uniform triangulations of , , and , with two different mesh sizes and (), and the corresponding piecewise linear finite element spaces and which satisfies and will be called the coarse-grid space and the fine-grid space, respectively.
To solve problem (1.1), we introduce a two-grid algorithm into finite element method. This method involves a nonlinear solve on the coarse-grid space and a linear solve on the fine-grid space. We present the two-grid finite element method as two steps.
Algorithm 4.1. Step 1. On the coarse grid , find , such that
where is defined by (3.3).
Step 2. On the fine grid , find , such that
The main feature of this method is that it replaces the resolution of a nonlinear problem on the fine grid with the resolution of a nonlinear problem on the coarse grid coupled with the resolution of a linear system on the fine grid. Now we consider the error estimates in -norm for the two-grid finite element method Algorithm 4.1.
Proof. Once again, we set and choose . Then for Algorithm 4.1, we get the error equation
For the terms of (4.4), we have
Integrating (4.4) from 0 to , combining with (4.4)–(4.6) and noting that and , we have
Now let's estimate the right-hand terms of (4.7), for , there is with a small positive constant. For , by (1.4), we obtain where we used the fact is bounded by a positive constant .
For –, by (1.2) and Theorem 3.3, we get with a small positive constant. By Lemma 3.1, from (4.7)–(4.12), we get Choosing proper and and kicking the last term into the left side of (4.13), by (3.5) and the Gronwall lemma, for , we have Together with (3.4), this yields (4.3).
Remark 4.3. We consider the spatial discretization to focus on the two-grid method. Algorithm 4.1 is only a semidiscrete two-grid finite element method. In practical computations, the method should be combined with a time-stepping algorithm. We consider a time step and approximate the solutions at , , . Denote , , we can get an implicit backward Euler two-grid finite element algorithm as follows.(1) On the coarse grid , find , such that
where is defined by (3.3). (2) On the fine grid , find , such that Higher order temporal discretization methods such as Crank-Nicolson method or Runge-Kutta method can also be used. For the space discretization, from a practical point of view, we just need to choose space grid to obtain a considerable error reduction in spite of the demanding requirement .
5. Numerical Example
In this section, we consider the following nonlinear parabolic problem: where , and is decided by the exact solution of (5.1).
Let the exact solution of (5.1) be
Our main interest is to verify the performances of the two-grid finite element method Algorithm 4.1. Choose the space step and obtain the coarse grids. Let and then obtain the fine grids. We further discretize time of Algorithm 4.1 and consider a time step and approximate the solutions at , , . The mesh consists of triangular elements and the backward Euler scheme is used for the time discretization. We use Newton iteration method for the solutions on the coarse grid. In order to prove the efficiency of the two-grid finite element method, we compare this method with the standard finite element method. Computational results are shown in Tables 1 and 2.
From Tables 1 and 2, we can see that the numerical results coincide with the theoretical analysis, and the two-grid finite element method spends less time than the standard finite element method, that is to say, the two-grid algorithm is effective for saving a large amount of computational time and still keeping good accuracy.
This work is supported in part by Shandong Province Natural Science Foundation (ZR2010AQ010, ZR2011AQ021) and a Project of Shandong Province Higher Educational Science and Technology Program (J11LA09, J10LA01).
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