A New Characteristic Nonconforming Mixed Finite Element Scheme for Convection-Dominated Diffusion Problem

A characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusion problem based on a new mixed variational formulation. The optimal order error estimates for both the original variable u and the auxiliary variable σ with respect to the space are obtained by employing some typical characters of the interpolation operator instead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis. At last, we give some numerical results to confirm the theoretical analysis.

As for the characteristic MFEM or expanded characteristic MFEM, the convergence rates of  and  in existing literature were suboptimal [11,18,21,22] and the convergence analysis was valid only to the case of the lowest order MFE approximation [10,17].So far, to our best knowledge there are few studies on the optimal order error estimates except for [23], in which a family of characteristic MFEM with arbitrary degree of Raviart-Thomas-Nédélec space in [24,25] for transient convection diffusion equations was studied.
Recently, based on the low regularity requirement of the flux variable in practical problems, a new mixed variational form for second elliptic problem was proposed in [26].It has two typical advantages: the flux space belongs to the square integrable space instead of the traditional ( div ; Ω), which makes the choices of MFE spaces sufficiently simple and easy; the LBB condition is automatically satisfied when the gradient of approximation space for the original variable is included in approximation space for the flux variable.Motivated by this idea, this paper will construct a characteristic nonconforming MFE scheme for (1) with a new mixed variational formulation.Similar to the expanded characteristic MFEM, the coefficient  of (1) in this proposed scheme does not need to be inverted; therefore, it is also suitable for the case when  is small.By employing some distinct characters of the interpolation operators on the element instead of the mixed or expanded mixed elliptic projection used in [1,17,20] which is an indispensable tool in the traditional characteristic MFEM analysis, the (ℎ 2 ) order error estimate in  2 -norm for original variable , which is one order higher than [1,20] and half order higher than [18], is derived, and the optimal error estimates with order (ℎ) for auxiliary variable  in  2norm and for  in broken  1 -norm are obtained, respectively.It seems that the result for  in broken  1 -norm has never been seen in the existing literature by making full use of the high-accuracy estimates of the lowest order Raviart-Thomas element proved by the technique of integral identities in [27] and the special properties of nonconforming  rot 1 element (see Lemma 1 below).
The paper is organized as follows.Section 2 is devoted to the introduction of the nonconforming FE approximation spaces and their corresponding interpolation operators.In Section 3, we will give the construction of the new characteristic nonconforming MFE scheme and two important lemmas, and the existence and uniqueness of the discrete scheme solution will be proved.In Section 4, the convergence analysis and optimal error estimates for both the original variable  and the flux variable  are obtained.In Section 5, some numerical results are provided to illustrate the effectiveness of our proposed method.
Throughout this paper,  denotes a generic positive constant independent of the mesh parameters ℎ and Δ with respect to domain Ω and time .

Construction of Nonconforming MFEs
As in [28], we frequently employ the space  2 (Ω) of square integrable functions with scalar product and norm We also employ the Sobolev space   (Ω),  ≥ 1, of functions V such that   V ∈  2 (Ω) for all || ≤ , equipped with the norm and seminorm The space  1 0 (Ω) denotes the closure of the set of infinitely differentiable functions with compact supports in Ω.For any Sobolev space ,   (0, ; ) is the space of measurable valued functions Φ of  ∈ (0, ), such that We now introduce the nonconforming MFE space described in [29] for and summarize it as follows.
By introducing  = −(, )∇ and using Green's formula, we obtain the new characteristic mixed form of (11).
Remark 1.In [1], the expanded characteristic MFE scheme was presented by introducing two new auxiliary variables which avoided the inversion of the coefficient  when  is small.The new mixed schemes (15a), (15b), and (15c) not only keep the advantage of expanded characteristic MFE scheme, but also donot need to solve three variables.Now, we prove the existence and uniqueness of the solution of (15a), (15b), and (15c).
Proof.The linear system generated by (15a), (15b), and (15c) is square, so the existence of the solution is implied by its uniqueness.From (15a), (15b), and (15c), we have Let   ℎ and  be zero, and thus   ℎ is zero too; taking 16) and adding them together, we have Thus assumption (A3) implies that   ℎ =   ℎ = 0.The proof is complete.
To get error estimates, we state the following two important lemmas.

Convergence Analysis and Optimal Order Error Estimates
In this section, we aim to analyze the convergence analysis and error estimates of characteristic nonconforming MFEM.
In order to do this, let Taking  =   in ( 12) yields From (23a), (23b), (15a), (15b), and (15c) we get We are now in a position to prove the optimal order error estimates.
Using the method similar to [3], we have ( (Err) 2 can be estimated as On the other hand, the left hand of (28) can be bounded by where the inequality ‖ −1 ‖ 2 ≤ (1 + Δ)‖ −1 ‖ 2 proved in [3] is used in the last step.
Remark 2. From (37), we have This byproduct can be regarded as the superclose result between  1 ℎ  and  ℎ in mean broken  1 -norm.It seems that both (25) and (50) have never been seen in the existing studies.At the same time, by employing the new characteristic nonconforming MFE scheme, we can also obtain the same error estimate of (27) as traditional characteristic MFEM [10].
Remark 3. From the analysis of Theorem 2 in this paper, we may see that Lemma 1 is the key result leading to the successful optimal order error estimations.If we want to get higher order accuracy, similar to Lemma 1, the nonconforming finite elements for approximating  should also possess a very special property, that is, the consistency error estimates with (ℎ 2 ) order, and satisfy (18).For the famous nonconforming Wilson element [32] whose shape function is span{1, , ,  2 ,  2 }, by a counter-example, it has been proven in [32] that its consistency error estimate is of (ℎ) order and cannot be improved any more.For the rotated bilinear  1 element [33] whose shape function is span{1, , ,  2 −  2 }, although its consistency error with (ℎ 2 ) order and (∇( −  1 ℎ ), ∇V ℎ ) ℎ = 0 on square meshes is satisfied, the second term of ( 18) is not valid.Thus when they are applied to (1) on new characteristic mixed finite element scheme, up to now, the optimal order error estimates of ( 25), (26), and ( 27) cannot be obtained directly.

Numerical Example
In order to verify our theoretical analysis in previous sections, we consider the convection-dominated diffusion problem (1) as follows:    We first divide the domain Ω into  and  equal intervals along -axis and -axis and the numerical results at different times are listed in Tables 1, 2, and 3 and pictured in Figures 1, 2, 3, and 4, respectively.( ℎ , p ℎ ) denotes the characteristic nonconforming MFE solution of the problem (15a), (15b), and (15c).Δ represents the time step and the experiment is done with Δ = ℎ 2 . stands for the convergence order.
It can be seen from the above Tables 1, 2, and 3 that ‖ −  ℎ ‖ 1,ℎ and ‖− ℎ ‖ are convergent at optimal rate of (ℎ) and ‖ −  ℎ ‖ is convergent at optimal rate of (ℎ 2 ), respectively, which coincide with our theoretical investigation in Section 4.