Supercloseness Result of Higher Order FEM / LDG Coupled Method for Solving Singularly Perturbed Problem on S-Type Mesh

and Applied Analysis 3 2.2.TheWeak Formulation of the FEM/LDGCoupledMethod. The S-type mesh defined in Section 2.1 is fine on Ω 1 = [0, λ] and coarse onΩ 2 = [λ, 1]. We discretize problem (1) by using the FEM on Ω 1 where the mesh is fine enough and strong stable LDG method is used on coarse mesh part Ω 2 . The derived method is the so-called FEM/LDG coupled method. The motivation to this coupled approach is to construct a numerical scheme with strong stability property but has less degrees of freedom than LDG method. Let ui = u| Ωi , i = 1, 2, and q = (u)󸀠 in Ω 2 . Rewrite problem (1) as the following equivalent transmission problem: − ε(u 1 ) 󸀠󸀠 − b(u 1 ) 󸀠 + cu 1 = f in Ω 1 , q − (u 2 ) 󸀠 = 0 in Ω 2 , − εq 󸀠 − b(u 2 ) 󸀠 + cu 2 = f in Ω 2 ,


Introduction
In this paper we are interested in the construction and validation of high-order finite element approximations to problems of type −  −   +  =  in Ω = (0, 1) , where 0 <  ≪ 1 is a small positive parameter and , , and  are sufficiently smooth functions with the following properties: for some constants  and  0 .This assumption guarantees that (1) has a unique solution in  2 (Ω) ∩  1 0 (Ω) for all  ∈  2 (Ω) [1].Typically the solution of (1) has an exponential boundary layer at  = 0.
Problem ( 1) is a simple model problem that helps understanding the behavior of numerical methods in presence of layers in more complex problems like the Navier-Stokes equations in fluid dynamics or convection diffusion equations in chemical reaction processes.
The smallness of  causes global unphysical oscillations if standard discretization schemes on general meshes are applied.To obtain accurate results without high computational cost, problem (1) is usually solved by strong stability numerical methods on the layer-adapted mesh, such as Shishkin-mesh (S-mesh) [2] or Bakhvalov-Shishkin mesh (B-S mesh) [3,4].In [5], a bilinear Galerkin finite element method was applied to (1) using a S-mesh, and it was shown that ‖ −    Gal ‖ 1, = O( −1 ln ), where ‖ ⋅ ‖ 1, is the weighted energy norm,  is the exact solution, and   Gal is the computed solution.On the same method using a B-S mesh [4] improved this result to O( −1 ).Roos and Linß [6] provided the so-called S-type mesh which was a class of generalized Shishkin-mesh including S-mesh and B-S mesh.
A popular stabilization technique is the discontinuous Galerkin (DG) methods which were introduced in the early 1970s for the numerical solution of first order hyperbolic problems.Simultaneously, but quite independently, they were proposed as nonstandard schemes for the approximation of elliptic and parabolic problems.The DG methods on Smeshes for solving singularly perturbed problems (SPPs) were considered in [7][8][9][10][11][12][13][14][15].Xie and her collaborators [8][9][10] 2 Abstract and Applied Analysis investigated the superconvergence and uniform superconvergence properties of the local discontinuous Galerkin (LDG) method on S-mesh for 1D and/or 2D convection-diffusion type SPP.Zhu et al. [13] proved the uniformly convergence properties of the LDG methods with higher order elements on S-mesh for general 1D convection-diffusion and reactiondiffusion type SPPs.And recently Zhu and Zhang [14,15] analyzed the uniform convergence properties of the LDG methods with bilinear and higher order elements on S-mesh for 2D SPP, respectively.On the other hand, the uniformly convergence of the NIPG method with bilinear elements on S-mesh was analyzed by Zarin and Roos [11] for 2D convection-diffusion type SPP with parabolic layers.In order to reduce the degrees of freedom of NIPG method, Roos and Zarin [7] and Zarin [12] analyzed the uniformly convergence of FEM/NIPG coupled method with bilinear element on S-mesh for 2D convection-diffusion type SPP with exponentially layers or characteristic layers.LDG method has much more advantages than the others in the DG methods family [16], but it also has more degrees of freedom than the others.By this motivation, Zhu and his collaborators [17,18] analyzed the uniformly convergence property of FEM/LDG coupled method with linear/bilinear element on S-mesh for 1D/2D convection-diffusion type SPP with boundary layer.Recently, Zhu and his collaborator [19] analyzed the uniformly convergence property of higher order FEM/LDG coupled method on S-mesh for 1D convection-diffusion type SPP with boundary layer.
A supercloseness property is a useful tool to prove superconvergence by postprocessing.Recently, Franz [20] numerically studies the supercloseness properties for higher order finite element methods and the streamline diffusion finite element methods on 2D Bakhvalov-Shishkin meshes.By the authors' knowledge, there is a few works about uniform supercloseness result of higher order DG method for solving SPP on S-type mesh.In this paper, we are interested in uniformly convergence properties and supercloseness properties of higher order FEM/LDG coupled method for 1D SPP of convection-diffusion type on S-type mesh.The paper is organized as follows.In Section 2, we introduce the Stype mesh and the FEM/LDG coupled method.The stability and error analysis of the FEM/LDG coupled method with higher order elements on S-type mesh is given in Section 3. A numerical example is presented in Section 4. It aims to validate our theoretical result.
In the sequel with  we will denote a generic positive constant independent of the perturbation parameter  and mesh size.

The S-Type Mesh and the FEM/LDG Coupled Method
2.1.The S-Type Mesh.Let  be an even integer.Denote by  the transition parameter which indicates where the mesh changes from fine to coarse.This parameter is given by where our trial space, which is defined below, comprises functions that are piecewise in P  for some integer  ≥ 1.
Notice that  ≪ 1; here and below we take  = (( + 1.5)/) ln .Moreover, we suppose that  ≤  −1 which is realistic for this type of problems.

The Weak Formulation of the FEM/LDG Coupled Method.
The S-type mesh defined in Section 2.1 is fine on Ω 1 = [0, ] and coarse on Ω 2 = [, 1].We discretize problem (1) by using the FEM on Ω 1 where the mesh is fine enough and strong stable LDG method is used on coarse mesh part Ω 2 .The derived method is the so-called FEM/LDG coupled method.The motivation to this coupled approach is to construct a numerical scheme with strong stability property but has less degrees of freedom than LDG method.
Rewrite problem (1) as the following equivalent transmission problem: with boundary conditions Let us now denote by P  () the space of polynomials of degree at most  on  and define the finite element space V 1  and V 2  as follows: The space V 1  is a standard conforming finite element space, whereas the functions in V 2  are completely discontinuous across interelement boundaries.
We will search for approximate solutions ( 1  ,  2  ,   ) of ( 10) and (11) that satisfy (10) and (11) in a weak sense.The FEM/LDG coupled method (see more details in [17,22]) for problems (10) and ( 11) is defined as follows: find for all test function V 1 ∈ V 1  , and for all test function and for all   ∈ T 2  , where Û2  , Ũ2  , and Q are the numerical fluxes, which approximate the traces of  2  and   on the boundary of the elements of T 2  .To complete the specification of the method, it only remains to define the numerical fluxes.
The Numerical Fluxes.We use the following notation to describe the numerical fluxes at the interior nodes.The average and jump of the trace of smooth function V ∈  2 (Ω 2 ) at the interior node   are given by respectively.We now define the numerical fluxes Û2  and Q by Here the scalars  = () and  = () are auxiliary parameters.Their purpose is to enhance the stability and accuracy properties of the LDG method (see [23,24]).
The numerical flux associated with the convection is the classical upwinding flux; namely,

Stability and Error Analysis of the FEM/LDG Coupled Method
This section is devoted to the existence and uniqueness of the solution of the coupled method ( 13)-( 15) with numerical fluxes ( 17)-( 19) and its corresponding error analysis.Firstly, we rewrite our method in the primal form by eliminating   following Arnold et al. [16].And then we get stability of the FEM/LDG coupled method, if the stabilization parameter  is taken of order O(1/).Under this condition, we obtain the higher order uniform convergence of the coupled method.
Primal Formulation.Let us introduce the space where For V ∈ V(), we define L 1 (V) as the unique element in for all  ∈ V 2  .As a result, from (14) we get Similar to the definition of L 1 (V), for V ∈ V(), we define for all  ∈ V 2  .
From the following lemma, the primal formulation is consistent.

Lemma 3.
Let  be the exact solution of the problems (10) and (11).Then the primal form (25) has the Galerkin orthogonality property Proof.The proof is similar to Lemma 3.1 in [19].
Stability Analysis.To consider the stability of the primal form A  , we define the following norms and seminorms for V ∈ V(): where ‖ ⋅ ‖ 0, is the usual Sobolev norm defined on region .
Abstract and Applied Analysis 5 According to [25] (page 422), when the coefficient  = O( −1 ), there exists a constant  > 0, such that Lemma 4. If  = O(1/), there exists a constant  > 0, such that Proof.The proof is similar to Lemma 3.2 in [19]. From which implies the uniqueness of the solution to (25).Further, since ( 25) is a linear problem over the finite-dimensional space V  , the existence of the solution follows from its uniqueness.Consequently, by (23), we get the existence and uniqueness of the solution to the problem ( 13)-( 15) with numerical fluxes ( 17)- (19).
Error Analysis.We are now going to provide a -uniform estimate for the error  −   in the norm (28).The error analysis presented in this paper relies on a priori estimate of the exact solution of (1) and a special interpolation which was firstly introduced in [26].
Lemma 6 (see [27,Lemma 1.9]).Let  be some positive integer.Consider the boundary value problem (1) with the assumption of (2).Its exact solution  can be composed as  = +, where the smooth part S and the layer part  satisfy Next we introduce a special interpolant in [26] that will be useful later.On each element  = [ −1 ,   ], we define  + 1 nodal functionals N  by Now a local interpolation I   |∈ P  () is defined by which can be extended to a continuous global interpolation I ∈ V  via set Obviously, if  = 1, this special interpolation is just the Lagrange linear interpolation.
The following error estimate is adapted from Lemma 7 of [26].

Lemma 7.
The special interpolant has the following properties: where  is any element of partition T N and ℎ  is the length of element .
Lemma 8. Assume that the piecewise differential meshgenerating function  satisfies (6).Let the exact solution  = +  of the problem (1) be decomposed into a smooth and layered part, respectively, I and I are the interpolants of  and  on a -type mesh respectively.Then, one has I = I + I and the estimates Proof.Our proof is based on arguments given by Tobiska [26].The linearity of the interpolation operator implies I = I( + ) = I + I.
The following statement is the direct consequence of Lemma 8.

Theorem 9. Under the conditions of Lemma 8, one has
for S-type mesh.
Proof.Since  − I is continuous in Ω, we have || * = 0 and of Lemma 8, we obtain Using this together with (39), we easily conclude the result of Theorem 9.
Remark 13.Uniform convergence of higher order LDG method on 2D Shishkin-mesh was considered in [15].From Theorem 3.1 of [15], a similar result can be obtained as our Theorem 12. From Theorem 3.1 and Remark 3.3 of [15], we can find the following conditions must hold: 0 ≤  11 ≤ O(1) on E or 0 ≤  11 ≤ O() on E  + and  11 = 0 on E \ E  + .Here  11 is a parameter in the definition of numerical fluxes, and E, E  + are unions of some edges of elements.In our paper, a parameter  in (18), which plays the same role as  11 , takes value as  = O(1/).This does not fulfill the condition of Theorem 3.1 of [15].
The combination of Theorem 9 and Theorem 12 leads to our main results directly, that is, the following.Theorem 14.Let  and   be the solutions of the continuous problem (10) and the discrete problem (25), respectively.Assume that the piecewise differential mesh-generating function  satisfies (6)

Numerical Experiments
In this section, we numerically verify the sharpness of our theoretical findings.In our numerical experiments, we take  = 1/,  = 1/2 in ( 17) and (18).In the following, "ln-ord" denotes the exponent  in a convergence order of the form O(( −1 ln )  ), while "ord" denotes the exponent  in a convergence order of the form O( − ).
The errors |‖ −   ‖|  and |‖  −   ‖|  for the FEM/LDG coupled method with higher order th elements are shown in Table 2.We have chosen  = 10 −6 in our calculations on