SUPERCLOSENESS ANALYSIS OF STABILIZER FREE WEAK GALERKIN FINITE ELEMENT METHOD FOR CONVECTION-DIFFUSION EQUATIONS

Recently, a stabilizer free weak Galerkin (SFWG) method is proposed in [14], which is easier to implement and more efficient. In this paper, we developed an SFWG scheme for solving the general second-order elliptic problem on triangular meshes in 2D. This new SFWG method will dramatically reduce the error between the L 2 -projection of the exact solution and the numerical solution.

A stabilizer free weak Galerkin finite element method is proposed by Ye and Zhang in [14] for the solution of Poisson equation on polytopal meshes in 2D or 3D, where P k (T ), P k (e), [P j (T )] d elements are used.It is shown that there is a j 0 > 0 so that the SFWG method converges with optimal order for any j ≥ j 0 .However, when j is too large, the magnitude of the weak gradient can be extremely large, causing numerical instability.In [1], the optimal j 0 is given to improve the efficiency and to avoid unnecessary numerical difficulties.In this setting, if P k (T ), P k (e), [P j (T )] 2 elements are used for a triangular mesh, j 0 = k + 1, where k ≥ 1.Recently, the SFWG finite element method has been developed to solve the parabolic equations [3], Stokes equations [15] and the second order elliptic problem [4].In [2], the authors proposed a scheme using P 0 (T ), P 1 (e), [P 1 (T )] 2 elements for triangular meshes with the optimal order of convergence.In [5], a SFWG finite element (P k (T ), P k+1 (e), [P k+1 (T )] 2 ) is investigated for second order elliptic problems.
In this paper, we will develop the theoretical foundation of the SFWG scheme using the general P k (T ), P k+1 (e), [P k+1 (T )] 2 elements for solving the convectiondiffusion equation (1.1)-(1.2) on a triangle mesh in 2D.As one of the main contributions of this paper, it is shown that by replacing P k (T ), P k (e), [P k+1 (T )] 2 elements with P k (T ), P k+1 (e), [P k+1 (T )] 2 elements, the error between L 2 -projection of the exact solution and the numerical solution will be dramatically reduced.
The rest of this paper is organized as follows.In Section 2, the notations and finite element spaces are introduced.Section 3 is devoted to investigating the error equations and several other required inequalities.The error analysis for the SFWG solutions in an energy norm is studied in Section 4. In Section 5, we will derive the L 2 error estimates for the SFWG finite element method for solving (1.1)-(1.2).Numerical test results are presented in Section 6. Concluding remarks are given in Section 7.

Notations
In this section, we shall introduce some notations and definitions.Suppose T h is a quasi uniform triangular partition of Ω.For every element T ∈ T h , denote by h T its diameter and h = max T ∈T h h T .Let E h be the set of all the edges in T h .For k ≥ 0, the weak Galerkin finite element space is defined as follows: where the component v 0 symbolizes the interior value of v, and the component v b symbolizes the edge value of v on each T and e, respectively.Let V 0 h be the subspace of V h defined as: For each element T ∈ T h , let Q i , i = 0, 1, k + 1, be the L 2 -projection onto P ti (T ) and let Q h be the L 2 -projection onto [P k+1 (T )] 2 , where t 0 = k, t 1 = 1, and t k+1 = k + 1, respectively.Also, let Qh and Q0 be the L 2 -projection onto [P 1 (T )] 2 and P 0 (T ), respectively, for each T ∈ T h .On each edge e, denote by Q b the L 2projection operator onto P k+1 (e).Combining Q 0 and Q b , denote by 2 is defined on T as the unique polynomial satisfying where n is the unit outward normal vector of ∂T .
For simplicity, we adopt the following notations: We are ready to introduce an SFWG finite element scheme for the problems (1.1)-(1.2).

Algorithm 1 Stabilizer Free Weak Galerkin Algorithm
A numerical approximation for (1.1)-(1.2) can be obtained by finding u h = {u 0 , u b } ∈ V 0 h , such that the following equation holds We define the following energy norm ||| • ||| on V h : An H 1 semi norm on V h is defined as: It is easy to show that ∥v∥ 1,h defines a norm on V 0 h .The following lemmas will be needed.Lemma 2.1 (Lemma 3.2, [5]).There exist C 1 > 0 and C 2 > 0 such that (2.7)

Error equation
In this section, we derive an error estimate of Algorithm 1.For simplicity, we will confine our attention to the case where α in (1.1) is a piecewise constant matrix with respect to the finite element partition T h .
Lemma 3.1 (see [13]).For any function ψ ∈ H 1 (T ), the following trace inequality holds true: Lemma 3.2 (Inverse Inequality see [13]).There exists a constants C such that for any piecewise polynomial ψ| T ∈ P k (T ), (Ω), and T h be a finite element partition of Ω satisfying the shape regularity assumptions.Then, the L 2 projections Q 0 and Q h satisfy Then for each element T ∈ T h , we have Proof.By definition (2.3) and integration by parts, for each q ∈ [P k+1 (T )] 2 we have which implies (3.5).

Lemma 3.5. Let u be the solution of the convection-diffusion problem (1.1)-(1.2) and then for
where Proof.From definition 2.1 we get where in the last equality we have used the fact that , and thus completes the proof.Lemma 3.6.For any T ∈ T h and q ∈ [P k+1 (T )] 2 , (∇ w v, q) T = (∇v 0 , q) T + ⟨v b − v 0 , q • n⟩ ∂T . (3.7) Proof.(3.7) follows directly from the definition of ∇ w v and integration by parts: This completes the proof of the lemma. (3.8) Then (3.8) follows from Lemma 3.4 and (3.9).

Lemma 3.8. Let e
where ℓ α (u, v), ℓ β β β (u, v) and ℓ c (u, v) are defined as follows: Using integration by part and the fact that It follows from Lemma 3.7 that Combining (3.12) and (3.13) gives

Error Estimates
We will derive error estimates in this section. ) Proof.By using Cauchy-Schwarz inequality, the trace inequality (3.1), Lemmas 3.3 and 2.1, we obtain To prove (4.2), we need to estimate ℓ β1 β1 β1 (u, v) and ℓ β2 β2 β2 (u, v) first.It follows from Cauchy-Schwarz inequality and 3.3 that It follows from the definition of Q b and the Cauchy-Schwarz inequality that ≤ Ch k+2 ∥u∥ k+2 |||v|||.
This completes the proof.
From the definition 2.1 and integration by parts, we obtain and here we have used that the fact that ⟨β β β • nv b , w b ⟩ ∂T h = 0 in the last equality.By summing (4.8) and (4.9), and setting v = w, we have Then, there exists a constant γ > 0, such that Proof.Let γ = 1 2 min{α m , c 0 } and v = e h .From (4.7) and the definition of ||| • ||| in (2.6), we get if Ch < γ, which completes the proof.
Lemma 4.4.The weak Galerkin scheme 2.5 has one and only one solution when h is small enough.
Proof.It suffices to verify the uniqueness for the homogeneous equation.Assume that u (1) h are two solutions of (2.5).Then e h = u (1) h would satisfy the forthcoming equation Note that e h ∈ V 0 h .Suppose that v = e h , in the equation (4.11) we obtain From Lemma 4.3, we have h and thus completes the proof.Theorem 4.1.Let u h ∈ V h be the SFWG finite element solution of (2.5).In addition, assuming the regularity of exact solution u ∈ H k+2 0 (Ω), then there exists a constant C such that (4.12) Proof.It follows from (4.10) that Letting v = e h in (3.10), yields Then (4.12) follows from Lemma 4.1.

Numerical Experiments
In this section, various numerical examples in 2D uniform triangular meshes are presented to validate the theoretical results derived in previous sections.We will compare our SFWG method (2.5) with the older version of the SFWG method proposed in [1,14] and the WG method in [10].

Example 1 (Constant diffusion α, convection β β β and reaction c)
In this example, we use the SFWG scheme The source term f (x, y) and the boundary conditions are computed accordingly.We applied the SFWG algorithm 1 with (P k (T ), P k+1 (e), [P k+1 (T )] 2 ) elements and the older version of the SFWG algorithm with (P k (T ), P k (e), [P k+1 (T )] 2 ) elements in the computation.As we can see in Table 1 that the error between u 0 and Q 0 u, the numerical solution obtained by using the SFWG method (2.5) and the If the WG is used with P k (T ), P k+1 (e), [P k+1 (T )] 2 elements, ∥Q 0 u − u 0 ∥ = O(h k+1 ), as can be see from Figure 1.Thus our new SFWG method is much more accurate.Since we have increased the complexity, the new SFWG method is slower than the older version of the SFWG method.However, in comparison with the WG method, the new SFWG method is faster as can be seen from Figure 2.

Example 2 (L-shaped domain)
In this example, we consider the problem (1.1)-(1.2) posed on an L-shaped domain  elements, converge at rate of k in H 1 -norm and k+1 in L 2 -norm, as can be see from Figure 3.The numerical solutions for the SFWG (2.5) are plotted in Figure 4.

Example 3
Interior layer-continuous boundary condition.This example is adopted from [10].Let Ω = (0, 1) × (0, 1) with the following data: β β β = (1, 0) ⊤ , c = 1, and the exact solution is given by   where the parameters η and γ control the location and thickness of the interior layer.Figures 5 shows that the error between u 0 and Q 0 u, the numerical results obtained by using the SFWG method (2.5) and the L 2 -projection of u, respectively, is

Example 4
In this example, we use the SFWG scheme (2.5) to solve the convection-diffusion equations (1.1)-(1.2) posed on the unit square Ω = (0, 1) 2 with the following data:  Table 4 shows that the numerical performance of the (P k (T ), P k+1 (e), [P k+1 (T )] 2 ) elements on the uniform triangular partition.It can be observed in Table 4 that the numerical solutions obtained by our SFWG algorithm 2.5 converge at rate of k + 2 in H 1 -norm and k + 3 L 2 -norm.We observe from Table 4 that the numerical performance is the same as those in Tables 1-2 and 3, two orders of superconvergence in both L 2 -norm and three-bar norm.
Table 5 shows that the numerical solutions obtained by our SFWG algorithm 2.5 converge at rate of k + 2 in H 1 -norm and k + 3 L 2 -norm.We can see from Table 5 that we do have two orders of superconvergence in both three-bar norm and L 2 norms.

Example 6(L-shaped domain)
In this example, we perform the SFWG scheme (2.

Concluding remarks
In this paper, we presented an SFWG finite element method for solving the general second-order elliptic problem on triangular meshes in 2D.We have shown both theoretically and numerically that on a triangular mesh, using (P k (T ), P k+1 (e) , [P k+1 (T )] 2 ) elements instead of (P k (T ), P k (e), [P k+1 (T )] 2 ) elements, the accuracy of the approximation to the L 2 -projection of the exact solution can be greatly improved.
2) follows from combining (4.4) and (4.5) and where we used the fact ∥u − Q k+1 u∥ e ≥ ∥u − Q b u∥ e in the inequality (4.5).The last estimate (4.3) is resulting from the Cauchy-Schwarz inequality and Lemma 3.3:

Table 2
lists errors and convergence rates in ||| • |||-norm and L 2 -norm.It can be observed from Table2that the numerical solutions obtained by our SFWG algorithm 2.5 converge at rate of k + 2 and k + 3 in H 1 -norm and L 2 -norm, respectively, while the corresponding rates are k and k + 1, respectively, by using the older version of SFWG algorithm.If the WG method is used with P k (T ), P k+1 (e), [P k+1 (T )] 2

Table 4 .
Example 4:Errors and numerical rates of convergence for the SFWG (2.5).

Table 5 .
Example 5:Errors and numerical rates of convergence for the SFWG (2.5).

Table 6 .
Example 6:Errors and numerical rates of convergence for the SFWG(2.5).k 1/h |||Q h u − u h ||| Rate ∥Q0u − u0∥ RateTable6shows that the numerical solution obtained by our SFWG algorithm (2.5) converge at rate of O(h k+2 ) in H 1 -norm and O(h k+3 ) in L 2 -norm.As one can observe from Table6that we can capture two order of superconvergence in both L 2 -norm and H 1 -norm by using SFWG algorithm (2.5).