Abstract
we present a first supercloseness analysis for higher order FEM/LDG coupled method for solving singularly perturbed convection-diffusion problem. Based on piecewise polynomial approximations of degree , a supercloseness property of in DG norm is established on S-type mesh. Numerical experiments complement the theoretical results.
1. Introduction
In this paper we are interested in the construction and validation of high-order finite element approximations to problems of type where is a small positive parameter and , , and are sufficiently smooth functions with the following properties: for some constants and . This assumption guarantees that (1) has a unique solution in for all [1]. Typically the solution of (1) has an exponential boundary layer at .
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 , where is the -weighted energy norm, is the exact solution, and is the computed solution. On the same method using a B-S mesh [4] improved this result to . 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 S-meshes for solving singularly perturbed problems (SPPs) were considered in [7–15]. Xie and her collaborators [8–10] 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 reaction-diffusion 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 S-type 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.
2. 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 for some integer . Notice that ; here and below we take . Moreover, we suppose that which is realistic for this type of problems.
Let be a partition of the domain . Let be a partition of the domain and . We choose where is a monotonically increasing mesh-generating function satisfying and . Given an arbitrary function fulfilling these conditions, a S-type mesh is defined.
We define a mesh-characterizing function that is closely related to by which is monotonically decreasing with and . Table 1 gives some examples of S-type meshes introduced in [6].
Denote the length of any subinterval by . Some properties of S-type mesh are given in the following lemma.
Lemma 1 (see [21]). Assume that the piecewise differentiable mesh-generating function satisfies the conditions Let , be the points for a S-type mesh. Then, the estimates, hold true.
Remark 2. Taking , the proof of this lemma is similar to the statements given on page 142 of [21]. From (6) we can also get a simpler bound
Set and . We denote by and the values of at , from the right cell and the left cell of , respectively.
2.2. The Weak Formulation of the FEM/LDG Coupled Method
The S-type mesh defined in Section 2.1 is fine on and coarse on . We discretize problem (1) by using the FEM on where the mesh is fine enough and strong stable LDG method is used on coarse mesh part . 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 , , and in . Rewrite problem (1) as the following equivalent transmission problem: with boundary conditions
Let us now denote by the space of polynomials of degree at most on and define the finite element space and as follows: The space is a standard conforming finite element space, whereas the functions in are completely discontinuous across interelement boundaries.
We will search for approximate solutions of (10) and (11) in the finite element space 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 such that for all test function , and for all test function and for all , where , , and are the numerical fluxes, which approximate the traces of and on the boundary of the elements of . 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 at the interior node are given by respectively. We now define the numerical fluxes and 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,
3. 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 . Under this condition, we obtain the higher order uniform convergence of the coupled method.
Primal Formulation. Let us introduce the space where For , we define as the unique element in satisfying for all .
As a result, from (14) we get Similar to the definition of , for , we define as the unique element in satisfying for all .
Following [19], using the lifting operators and , the flux form FEM/LDG coupled method (13)–(15) with numerical fluxes (17)–(19) can be rewritten as the primal form: find such that with Here, and have been defined in by a trivial extension.
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 , we define the following norms and seminorms for : where is the usual Sobolev norm defined on region .
According to [25] (page 422), when the coefficient , there exists a constant , such that
Lemma 4. If , there exists a constant , such that
Proof. The proof is similar to Lemma 3.2 in [19].
From Lemma 4, we easily get which implies the uniqueness of the solution to (25). Further, since (25) is a linear problem over the finite-dimensional space , 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).
Remark 5. In fact, following [25] or [8], for any , we can prove the existence and uniqueness of the solution to the problem (13)–(15) with numerical fluxes (17)–(19). In this paper, we are only interested in the special case .
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 , we define nodal functionals by Now a local interpolation is defined by which can be extended to a continuous global interpolation via set Obviously, if , 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 and is the length of element .
Lemma 8. Assume that the piecewise differential mesh-generating function satisfies (6). Let the exact solution of the problem (1) be decomposed into a smooth and layered part, respectively, and are the interpolants of and on a -type mesh respectively. Then, one has and the estimates
Proof. Our proof is based on arguments given by Tobiska [26]. The linearity of the interpolation operator implies .
(i) The proof of (38): by Lemma 1, Lemma 7, and (33), we have
for any . Hence, we obtain
From (33), we get
Consider the element . The local nodal functional can be estimated by
thus, we have from the local representation,
the estimate
where we used , are basis functions on element . And then, we obtain . Combining this with (46), we get
Collecting (43), (45), and (50), we conclude
(ii) The proof of (39): by (38), we easily get
(iii) The proof of (40): by Lemma 1, Lemma 7, and (33), we easily obtain
(iv) The proof of (41): let . Then on the fine part of the mesh, we have
where we have used Lemma 7 and (7). From (9) we get , and therefore for . The representation
yields
since , and by (9). We sum the overall subintervals in the layer region to get
where and are used.
(v) The proof of (42): by triangle inequality
we estimate and , respectively. From (33), we obtain
An inverse inequality yields
and it remains to bound . Consider the element . From local representation of , it follows
where we used . Here is the reference element of and are the basis functions on . Summing up we get
Recall that the mesh size on the coarse mesh has been denoted by and satisfies . Integrating the inequality
over and summing up for , we obtain
Therefore, we get
And then, we have
Combining this with (59), we conclude (42).
The following statement is the direct consequence of Lemma 8.
Theorem 9. Under the conditions of Lemma 8, one has satisfies for S-type mesh.
Proof. Since is continuous in , we have and . Then, . From (40), (41), and (42) of Lemma 8, we obtain Using this together with (39), we easily conclude the result of Theorem 9.
Now we turn to estimate .
Lemma 10. Under the conditions of Lemma 8, one has where .
Proof. (i) The proof of (69): by Lemma 8 and integrating by parts, we get
Recalling that is continuous in , we have , . And then,
Combining this with (71), we can easily get (69).
(ii) The proof of (70): by Cauchy-Schwarz inequality,
where by Lemma 7,
and from (33), (65),
Thus, we obtain
where .
Remark 11. The factor is bounded by a constant for all meshes listed in Table 1. But there also exists counterexample that increases with the increasing of ; see [21] for details.
Theorem 12. Under the conditions of Lemma 8. Assuming , then satisfies
Proof. By Lemma 3 and Lemma 4, we first obtain
By the definition of , we have , . Consequently,
Firstly, we consider the term . By Lemma 7, Lemma 8, and (29), we have
Now consider the term . By Lemma 10, we get
By (39), the second term in the right hand side of (79) can be easily estimated with
This, combined with (81), yields
Collecting (78), (80), and (83), we conclude Theorem 12.
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: on or on and on . Here is a parameter in the definition of numerical fluxes, and , are unions of some edges of elements. In our paper, a parameter in (18), which plays the same role as , takes value as . 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). Taking , then
Corollary 15. Let be the solution obtained by the coupled method (13)–(15) with numerical fluxes (17)–(19). Under the assumption of Theorem 14, one has where is a problem-related norm defined by
Proof. From (23), we have . Since , we obtain . In terms of (29), we conclude , which implies the conclusion.
4. Numerical Experiments
In this section, we numerically verify the sharpness of our theoretical findings. In our numerical experiments, we take , in (17) and (18).
Example 16. We solve the model problem (1) with , , and taking such that the exact solution is which exhibits a boundary layer with the width at the outflow boundary .
In the following, “ln-ord” denotes the exponent in a convergence order of the form , while “ord” denotes the exponent in a convergence order of the form .
The errors and for the FEM/LDG coupled method with higher order th elements are shown in Table 2. We have chosen in our calculations on S-meshes and B-S meshes. From Table 2, we observe that the numerical results for agree with those predicted in Theorem 14. Note that the closeness errors have a supercloseness property of order if th polynomial is used. This phenomenon indicates that the supercloseness result of order proved in Theorem 12 is not optimal.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by Zhejiang Provincial Natural Science Foundation of China (Grant no. LQ12A01014) and Zhejiang Provincial Department of Education (Grant no. Y201330020).