A Novel Characteristic Expanded Mixed Method for Reaction-Convection-Diffusion Problems

Anovel characteristic expandedmixed finite elementmethod is proposed and analyzed for reaction-convection-diffusion problems. The diffusion term ∇ ⋅ (a(x, t)∇u) is discretized by the novel expanded mixed method, whose gradient belongs to the square integrable space instead of the classical H(div; Ω) space and the hyperbolic part d(x)(∂u/∂t) + c(x, t) ⋅ ∇u is handled by the characteristic method. For a priori error estimates, some important lemmas based on the novel expanded mixed projection are introduced. The fully discrete error estimates based on backward Euler scheme are obtained. Moreover, the optimal a priori error estimates in LandH-norms for the scalar unknown u and a priori error estimates in (L)2-norm for its gradient λ and its flux σ (the coefficients times the negative gradient) are derived. Finally, a numerical example is provided to verify our theoretical results.


Introduction
In this paper, we consider the following reaction-convectiondiffusion problems:
In 1994, Chen [17,18] proposed an expanded mixed finite element method for second-order linear elliptic equation.Compared to standard mixed element methods, the expanded mixed method can approximate three variables simultaneously, namely, the scalar unknown, its gradient, and its flux (the tensor coefficient times the negative gradient).From then on, the expanded mixed method was applied Introducing the two auxiliary variables  = ∇,  = −((x, )∇) = −((x, )), we obtain the following first-order system for ( 1 Let the characteristic direction corresponding to the hyperbolic part / + c ⋅ ∇ be denoted by  = (x), which is subject to where Then the first-order system (4) can be rewritten as Using the similar method to the one in [29], the expanded mixed weak formulation for problem (1) is to find {, , } : [0, ]  →  × V × X such that where In this paper, we propose and discuss a novel characteristic expanded mixed method.The new characteristic expanded mixed weak formulation is to find {, , } : For approximating the solution at time   = Δ, the characteristic derivative will be approximated by and then we have the following approximation: Let ( ℎ , W ℎ ) ⊂ ( 1 0 , ( 2 (Ω)) 2 ) be defined by the following finite element pair  1 −  2 0 [25,26]: Now the novel characteristic expanded mixed finite element procedure for (8a), (8b), and (8c) is to find where Remark 1. From [25,26], we find that ( ℎ , W ℎ ) satisfies the so-called discrete Ladyzhenskaya-Babuska-Brezzi condition.
Remark 2. Compared to the scheme (7a), (7b), and (7c) based on Chen's expanded mixed element method, the gradient in the scheme (8a), (8b), and (8c) belongs to the simple square integrable space instead of the classical H(div; Ω) space.For H(div; Ω) ⊂ ( 2 (Ω)) 2 , we easily find that our method reduces the regularity requirement on the gradient solution  = ∇.Remark 3. Based on finite element space ( ℎ , W ℎ ) in (11), the number of total degrees of freedom for our scheme ((12a), (12b), and (12c)) is less than that for the scheme in [29].By the same discussion as Remark 1 in [30], we can obtain the detailed analysis for degrees of freedom.

Some Lemmas and Error Estimates
3.1.Novel Expanded Mixed Projection and Lemmas.We first introduce the novel expanded mixed elliptic projection [24] associated with our equations to derive a priori error estimates for the proposed method.
Let (ũ ℎ , λℎ , σℎ ) : [0, ] →  ℎ × W ℎ × W ℎ be given by the following mixed relations: In the following discussion, we will give some important lemmas based on the novel expanded mixed projection.

A Priori Error Estimates.
In the following discussion, we will derive a priori error estimates based on fully discrete backward Euler method.Let 0 =  0 <  1 <  2 < ⋅ ⋅ ⋅ <   =  be a given partition of the time interval [0, ] with step length Δ = / and nodes   = Δ, for some positive integer .
Remark 9. Compared to the results in [29], we can obtain the optimal a priori error estimates in  1 -norm for the scalar unknown  in Theorem 8.

Numerical Experiment
In this section, in order to confirm our theoretical results for the novel characteristic expanded mixed finite element method, we consider the following test problem: with boundary condition and initial condition where , and (x) = 1 and (x, ) is chosen so that the exact solution for the scalar unknown function is The corresponding exact gradient function is and its exact flux function is ] . (49) We divide the domain Ω = [0, 1] × [0, 1] into the triangulations of spatial mesh parameter ℎ uniformly and use the backward Euler procedure with uniform time discretization parameter Δ.We consider the piecewise linear space  ℎ with index  = 1 and the corresponding piecewise constant space W ℎ with index  = 0.
In Table 1, we get the optimal a priori error estimate in  2 -norm for the scalar unknown  with ℎ = 2 √ 2Δ = √ 2/16, √ 2/32, √ 2/64.At the same time, we also obtain the optimal a priori error estimate in  1 -norm for the scalar unknown .
In Table 2, we obtain some convergence results in ( 2 (Ω)) 2 -norm for the gradient  with ℎ = 2 √ 2Δ = √ 2/16, √ 2/32, √ 2/64 in Table 2.The similar results are obtained for the flux  in Table 2. From the data obtained in Table 2, we can   find that the numerical results confirm the theoretical results of Theorem 8.
Figure 1 shows the surface for the exact solution  at  = 1, and Figure 2 describes the corresponding surface for the numerical solution  ℎ with ℎ = 2 √ 2Δ = √ 2/16 at  = 1.In Figures 1 and 2, we can find easily that the exact solution  is approximated very well by the numerical solution  ℎ .
Figures 3 and 4 show the surface of the exact gradient function  and the surface of the numerical gradient function  ℎ with ℎ = 2 √ 2Δ = √ 2/16 at , respectively.A similar comparison between the surface of the exact flux function  and the surface of the numerical flux function  ℎ in Figures 5  and 6 also is made.
From the convergence results for in Tables 1 and 2 and Figures 1 and 2, we find that the numerical results confirm our theoretical analysis.

Concluding Remarks
In this paper, we propose and study a novel characteristic expanded mixed finite element method, which combines the novel expanded mixed method [24] applied to approximating the diffusion term and the characteristic method that handled the hyperbolic part, for reaction-convection-diffusion equations.Compared to Chen's expanded mixed method, the  gradient for our method belongs to the square integrable space instead of the classical H(div; Ω) space.We derive a priori error estimates based on backward Euler method.Moreover, we prove the optimal a priori error estimates in  2 -and  1 -norms for the scalar unknown  and a priori error estimates in ( 2 ) 2 -norm for its gradient  and its flux .Finally, we choose a test problem to confirm our theoretical results.In the near future, the proposed characteristic expanded mixed scheme will be applied to other linear/nonlinear evolution equations, such as nonlinear reaction-diffusion equations, linear/nonlinear convection-dominated Sobolev equations, and time-dependent convection-diffusion optimal control problems.Talents of Inner Mongolia University (125119, Z200901004, and 30105-125132).
The exact solution  at  = 1

Figure 1 :
Figure 1: The surface for the exact solution .
The numerical solution  ℎ at  = 1

Figure 2 :
Figure 2: The surface for the numerical solution  ℎ .