In this paper, two efficient two-grid algorithms for the convection-diffusion problem with a modified characteristic finite element method are studied. We present an optimal error estimate in -norm for the characteristic finite element method unconditionally, while all previous works require certain time-step restrictions. To linearize the characteristic method equations, two-grid algorithms based on the Newton iteration approach and the correction method are applied. The error estimate and the convergence result of the two-grid method are derived in detail. It is shown that the coarse space can be extremely coarse and achieve asymptotically optimal approximations as long as the mesh sizes in the first algorithm and in the second algorithm, respectively. Finally, two numerical examples are presented to demonstrate the theoretical analysis.

1. Introduction

In this paper, we consider the following convection-diffusion equations:where is a bounded polygonal domain in with boundary , , is some final time, , , is a twice continuously differentiable function with bounded derivatives through the third order, is a given function, is the diffusion coefficient, and is the convection field. To state some results about the convergence of the present method, we assume the following conditions:(i)The function satisfies the Lipschitz continuous condition about .(ii)We assume that the coefficient is bounded above and below by positive constants:(iii)We also assume that the coefficient is bounded, and we obtainwhere and are the positive constants.

Convection-diffusion problems have been investigated extensively due to their wide applications in various fields of sciences and technologies, such as solutes in moving fluids and transport problems of heat [112]. In these areas, numerical solutions of problems (1)–(3) play a key role. A number of numerical methods have been developed for the simulation of convection-diffusion equations, including finite element methods [3, 4], finite volume methods [5], characteristic methods [68], multilevel adaptive particle methods [9], variational multiscale method [10], and discontinuous Galerkin method [11]. One of the most popular methods is the characteristic finite element method (CFEM) [6]. The characteristic method is a combination of a characteristic approximation to handle the convection part in time and a finite element spatial approximation to deal with the diffusion part. Compared with the standard finite element method, the CFEM has a much smaller time truncation error, because the solution of the above method changes more slowly in the characteristic direction than in the direction. Thus, this method will allow one to use a large time step in practical computations. In the past several decades, many authors have contributed to developing, analyzing, and applying CFEM (see [1316]).

As we know, the two-grid method is a highly efficient and accurate method, which was first formulated for nonsymmetric linear and nonlinear elliptic problems by Xu [17, 18]. Due to the better practical performance, the two-grid method was applied and investigated for other types of problems by many authors (see [12, 1925]). Among them, Bi and Ginting [19] have given the two-grid discontinuous Galerkin method for a quasi-linear elliptic problem, Li and Tan [20] have used the two-grid method for nonlinear hyperbolic equations, Dawson and Wheeler [21] have used two-grid mixed finite element methods for nonlinear parabolic equations, and Chen et al. [22, 23] have considered the two-grid method for nonlinear reaction-diffusion equations by the expanded mixed finite element method. In particular, Qin and Ma [12] have constructed and analysed a two-step algorithm by using a two-grid idea for the characteristic finite element solution of nonlinear convection-diffusion equations. Recently, we have proposed the two-grid algorithms based on the mixed finite element approximation for nonlinear hyperbolic equations [24] and studied the unconditional error estimates of two-gird finite element methods for parabolic integrodifferential equations [25]. In the abovementioned works, the theoretical study and numerical experiments show that the two-grid technique is computationally more efficient than the original method.

In this paper, we propose two efficient algorithms of the two-grid method for the approximation of semilinear convection-diffusion equations (1)–(3) by using CFEM and present a new analysis to obtain unconditionally optimal error estimates. The analysis is based on a temporal-spatial error splitting technique [26], with which the error function is split into two parts (temporal error and spatial error) by introducing a new characteristic time-discrete system. The previous analyses required certain time-step conditions to obtain optimal error estimates. For example, a two-scale product approximation for the semilinear parabolic problem with the mixed finite element method was analysed, and optimal error estimates were obtained under the condition [27]. The restriction was also required for the Galerkin alternating-direction method for a kind of three-dimensional nonlinear hyperbolic problem [28]. However, the certain restriction conditions between spatial mesh size and temporal step size are not necessary and can be removed. Our results show clearly that the characteristic temporal discrete scheme is effective, and the time-step conditions in previous analyses were required mainly due to the error estimate of quadratic terms arising from linearization by using the inverse inequality. Moreover, compared with [12], the feature of our algorithms mainly lies in step 3 where a further coarse/fine grid correction is performed. It is of great theoretical interest that a further coarse/fine grid correction after the fine grid correction can actually improve the accuracy. A remarkable fact about this simple approach is, as shown in [17, 18], that the coarse mesh can be quite coarse and still maintain a good accuracy approximation.

The paper is organized as follows: In Section 2, we introduce some notations and approximation results that are used throughout the paper and construct the CFEM for (1)–(3). In Section 3, by using the splitting technique, we analyze the temporal and spatial errors, respectively, and prove the optimal error estimate of CFEM in -norm unconditionally. In Section 4, we present the two-grid algorithms and analyze their error estimates. In Section 5, the numerical results are given to confirm our theoretical analysis. Finally, some conclusions and further works are presented in the last section.

2. CFEM for the Convection-Diffusion Equations

Let be the standard Sobolev space (see [29] for details). The norm of is denoted by , for and , for . We set with norm (the notation ) and . In addition, let denote a generic positive constant that does not depend on the spatial mesh and temporal discretization parameters.

Let be a quasi-uniform family of finite element partitions of , where is the maximal element diameter. Let be the piecewise continuous finite element spaces and .

Letand let the characteristic direction associated with the operator be denoted by , where

Then, problem (1)–(3) can be written in the equivalent form as follows:

Let be a uniform partition in the time direction with the time step . For a sequence of functions , we denote

The characteristic derivative is approximated in the following way at :

That is, a backtracking algorithm is used to approximate the characteristic derivative. is the foot (at level ) of the characteristic corresponding to at the head (at level ) (see Figure 1).

The fully discrete CFEM for (1)–(3) is defined. For , we seek such thatwhere , , and is the Lagrange interpolation operator over . Hence, it can be seen in [30, 31] that if , there holds

In this paper, we assume that the initial boundary value problem (1)–(3) admits a unique solution as follows:

Let denote the Ritz projection that satisfies

Then, the following estimates hold [32] for and :

The discrete Gronwall’s lemma [33] is recalled.

Lemma 1. Assume that is a non-negative sequence and that the sequence satisfies

Then, also satisfies

Moreover, if and for , it follows that

3. Unconditional Error Estimate for the CFEM Solution

In this section, we provide a primary error estimate for the CFEM (13) unconditionally by the Li–Sun error splitting technique [26]. For , we introduce a characteristic time-discrete system as follows:

For convenience, we split the error into the following manner:

Now, we start to prove the regularity and error estimation of the solution of the above time-discrete system.

Theorem 1. Suppose that the time-discrete system (22)–(24) has a unique solution . Then, for and , we have

Proof. On level , we rewrite (8) asFrom (8) and (22), we observe that the error function satisfies the following equation:We multiply both sides of (28) by to getObviously, we have the following equality:First, we estimate the left-hand side of (29), and it is easy to derive thatNow, we estimate the right-hand side terms of (29). For , using the Cauchy–Schwarz inequality and -Cauchy inequality, we obtainfor any small constant . For , with the result in [6], we haveand therefore,In the inequality, by using the result by Russell (see Theorem 1 in [13]), can be bounded byChoosing proper and using the equivalence of and , inequalities (31)–(34) can be combined with (29) to getNote that . If we multiply (36) by , sum for , and apply the discrete Gronwall’s lemma, it follows thatOn the other hand, we multiply (28) by to getSimilarly to the proof of (36), we see thatMultiplying (39) by , summing for , noting that , and applying the discrete Gronwall’s lemma, we getwhere we have used the result (see [34, 35]). It follows from (37) and (40) thatIn particular, by (15) and (41), we further derive thatThe proof of Theorem 1 is complete.

Remark 1. Based on the result of Theorem 1, we use the Sobolev embedding inequality to getwhere .
In the following, the unconditional convergence result is deduced through the analysis of the spatial error.

Theorem 2. Let be the solution of (22) and be the fully discrete solution of (13). Then, for and , we have

Proof. We rewrite the time-discrete system (22)–(24) in a weak form byFrom (13) and (45), we see that the error function satisfies the equation as follows:for . Substituting into (46), the following estimate holds for the left-hand terms of (46):The terms of the right-hand side of (46) are estimated byCombining the estimates above, we can bound (46) asMultiplying both sides of the above equation by and summing for from 1 to , for , since , we see thatHere, is sufficiently small, and we have used (17) and (18). It follows the discrete Gronwall’s lemma thatIf , by (51), and similar to the proof of Theorem 3.4 in [18], we can obtainWe define the elliptic operator asObviously, is isomorphism. Corresponding to , we define a bilinear form, for , asTo prove (44), we use a duality argument by considering the auxiliary problem. We find so thatGiven , let . It is uniquely solvable for and has the following regularity (Grisvard [36]):for . If , it follows from the Sobolev embedding theorem thatTherefore, with , by Lemma 3.1 of Xu [18], we obtainConsequently, we obtainFor , if , thenFor , note that , and we deduce thatThe desired result (44) can be derived by using the triangle inequality, (17), (60), and (61).
Therefore, we combine (17) with the results in Remark 1 and Theorem 2 to obtain the following theorem.

Theorem 3. Suppose that (15) holds. Let and be the solutions of (1)–(3) and (13) at , respectively. Then, for and , we have

4. Two-Grid Algorithms and Error Estimates

In this section, we propose two two-grid algorithms for (1)–(3) based on two finite element spaces. The basic ingredient in our approach is another finite element space defined on a coarser quasi-uniform triangulation (with mesh size ) of . Let us now present our first two-grid algorithm.

Step 1: On the coarse grid , for , we find and solve the following nonlinear system:
Step 2: On the fine grid , for , given , we find to satisfy the following linear system:
Step 3: On the coarse grid , for , given and , we find to satisfy the following linear system:

In the following analysis, we first introduce the following lemma [6].

Lemma 2. Let , , and and be bounded functions on . If the time step is sufficiently small, then there exists a positive integer such thatwhere is only dependent on and , and the norm of is defined as follows:

Next, we need to prove .

Lemma 3. If be the solution of (27) and be the solution of Algorithm 1, thenfor and .

Proof. We set . From (27) and Algorithm 1, it is easy to derive the following equation:where we employ the Taylor expansion , means that is evaluated at a point between and . By taking in (66), then the left-hand side of (66) becomesIn the inequality, the estimates of can be obtained similarly as in the proof of Theorem 2:By (17), we estimate for as follows:With the estimate (62) for , we haveIt follows from (66)–(70) thatNoting that . Multiplying (71) by and summing for , we see thatApplying the discrete Gronwall’s lemma, we getSimilarly, as in Theorem 2, we havefor . Thus, (65) follows from (17) and (74), and the inequality is obtained as follows:Now, we give the error estimate for two-grid Algorithm 1.

Theorem 4. Assume is obtained by Algorithm 1, and then, for , we have

Proof. Let . From (27), Algorithm 1, and (16), we see thatwhere . For the last term of the right-hand side of (77), a Taylor expansion of about yieldsfor some function between and . Then,We rewrite (77) as follows: