Issue 
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 5, October 2023



Page(s)  411  420  
DOI  https://doi.org/10.1051/wujns/2023285411  
Published online  10 November 2023 
Mathematics
CLC number: O241
Uniform Convergence Analysis of the Discontinuous Galerkin Method on LayerAdapted Meshes for Singularly Perturbed Problem
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China
^{†} To whom correspondence should be addressed. Email: ycheng@usts.edu.cn
Received:
2
March
2023
This paper concerns a discontinuous Galerkin (DG) method for a onedimensional singularly perturbed problem which possesses essential characteristic of second order convectiondiffusion problem after some simple transformations. We derive an optimal convergence of the DG method for eight layeradapted meshes in a general framework. The convergence rate is valid independent of the small parameter. Furthermore, we establish a sharper L^{2}error estimate if the true solution has a special regular component. Numerical experiments are also given.
Key words: layeradapted meshes / singularly perturbed problem / uniform convergence / discontinuous Galerkin method
Biography: SHI Jiamin, female, Undergraduate, research direction: computational method. Email: 21200210103@qq.com
Fundation item: Supported by the National Natural Science Foundation of China (11801396) and National College Students Innovation and Entrepreneurship Training Project (202210332019Z)
© Wuhan University 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
Singularly perturbed problems attract much attention in such applications as optimal control, chemical reactions, fluid dynamics and electrical networks^{[1]}. The exact solution generally displays boundary layers which causes many numerical difficulties to the traditional finite element method on the quasiuniform mesh. To obtain satisfactory numerical approximations, several numerical strategies were developed, such as the layeradapted meshes, the fitted operator methods and the stabilised numerical method, see Refs. [13] for a survey.
Discontinuous Galerkin (DG) method is a finite element method whose test function and trail function have possible discontinuous points at the element edges^{[4]}. As such, the discontinuous finite element space provides much flexibility in solving those problems exhibiting large gradients, boundary layers or even discontinuous interfaces, see Ref. [5] for a survey.
For the singularly perturbed problem whose solution exhibits boundary layers, numerical investigations were performed in Ref. [6] for the DG method. Local behavior was explored on the uniform mesh. Uniform convergence and superconvergence were observed on Shishkin mesh. Along this direction, research on the DG method for singularly perturbed problems was developed in a series of papers, see Refs. [710].
However, the above error estimates are often performed on Shishkin mesh, which has a simple structure. Owing to the influence of a logarithmic factor, the convergence rate will be deteriorated as the degree of piecewise polynomials goes larger. It is therefore of much interest to derive optimal convergence rate on general graded layeradapted meshes. Recently, we studied the local DG method on several layeradapted meshes. Some uniformly optimal error estimates were established for second order convectiondiffusion problem^{[11,12]}.
This paper concerns a uniform convergence of the DG method under a much larger range of the layeradapted meshes. These meshes contain five common Shishkintype meshes as well as three common Bakhvalovtype meshes^{[13,14]}. Based on a general analysis framework, we establish optimal convergence for the DG method independent of small perturbation parameter. In particular, we establish a sharper norm convergence rate under the situation that the smooth component of the true solution is a piecewise polynomial. Some numerical results are given to confirm our prediction.
We organize this paper as follows. First, layeradapted meshes and the DG method are introduced in Section 1. Then we present a local projector as well as its approximation error. Our main result follows in Section 2. Finally, we supplement some numerical results to validate our error estimate.
1 LayerAdapted Meshes and the DG Method
Consider a onedimensional model problem
which possesses some essential characteristics of the following secondorder problem
In fact, one can transform (2) into a problem with b=0 if a and b are bothconstants. Then introduce the following transformation
one obtains the model problem (1).
Assume that problem (1) exhibits a boundary layer at x =1. Assume that the exact solution of (1) can be expressed as^{[15]}
which implies a decomposition Here is the regular component and is the layer component satisfying
1.1 LayerAdapted Meshes
Let be a monotonically increasing, continuous and piecewisely differentiable function. Let . Introduce a mesh transition parameter
where is a constant. Assume that for a small . Otherwise, the problem is nonsingularly perturbed and one can carry out the error analysis in a classical framework. Let be an even integer. Denote , where is the rough region and is the refined region; they have both equalelements. The mesh points are given by
In Table 1, we list eight common layeradapted meshes, which are simplified as S, BS, mBS,VS, pS, B, mVB and RSmeshes. Here is mesh characterizing function. See Ref. [13] for more details.
Set , where each element has the mesh size . Assume that , then one has for each mesh in Table 1. This property will be frequently used in the following analysis.
Lemma 1^{[12]} Define
Then one has
Here does not depend on and N.
Layeradapted meshes
1.2 The DG Method
Define a finite element space as
where is a space of polynomial with degree no larger than k. The functions in the above discontinuous finite element space have possible discontinuous points at the cell ends. Define and the jumps as
The DG method reads: Find such that
holds for any and , where
Denote Rewrite the scheme (5) into a compact form: Find such that
where
One obtains an energy norm
Let in (6) then one has and which implies the uniquely existence of the computed solution determined by the DG method (6).
2 Convergence Analysis
Divide the error as follows
where is the local GaussRadau projecter such that for any function and each element ,
From Ref.[16], one can verify the wellposedness of the above projection. Furthermore,
Lemma 2 Assume in the definition of the layeradapted mesh (4) . Let with each component and satisfying (3), then one has
Furthermore, if , one has
Proof Denote for . From (3) and (10), one has
For the monotonic increasing function using the stability (8), one has
On the refined domain, if , one obtains from Lemma 1 that
Consequently, (11) follows from (14)(16). Note that if , (13) follows from (15),(16).
By the approximation property (10), one has
By Lemma 1 and one obtains
which leads to (12).
Theorem 1 Let q and are respectively the solutions of (1) and (5) . Then one has
Furthermore, if , one has
Proof For any, one has Galerkin orthogonality which leads to
Then one gets from (11) that
Hence,
If , by (13), one has
Remark 1 Theorem 1 presents a convergence rate for an equal number of mesh elements in the rough and refined domains. However, it is possible to explore different number of mesh elements in rough and refined domains (denote and respectively). Following the similar line, one derives a convergence rate . Then it is possible to use larger to balance the influence of on the convergence rate and arrive at a convergence .
3 Numerical Experiments
We perform the DG method (5) on the eight meshes given by Table 1. Set For the pSmesh, take For the RSmesh, take Compute the convergence rate by the formulae
The quantities and are used to reflect the convergence rates from the error bounds of the forms and , respectively.
Example 1 Consider the problem (1) with and the true solution is set as
As mentioned before, for a large , one can use arbitrary mesh with maximum mesh size h and expect a convergence rate . In Table 2, we list the numerical results on four meshes. One observes that the convergence rates on uniform mesh and Bmesh are both , the convergence rate on Smesh is , while the convergence rate on BSmesh is because its maximum mesh size is bounded by
Table 3 and Table 4 list the error of the DG method on the eight meshes for and , respectively. One observes for the Smesh a convergence rate , while for the BS, mBS, VS, B and mVB meshes a general convergence rate . For the pSmesh, the convergence rate is a little smaller than k+1 because of the influence of the logarithmic factor (here ). For the RSmesh, the convergence rate behaviors as which agrees with theoretical prediction (17) in view of l=3 and Thus, the numerical convergence rate is generally for every meshes, which confirms our theoretical convergence rate in Theorem 1.
Furthermore, Figure 1 and Table 5 show that the error is uniform regarding the singular perturbation parameter in the case that and N=64 except the Smesh and RSmesh which are slightly influenced by the factor as is suitably large.
Fig. 1 error on the singular perturbation parameter for Example 1 
Example 2 We continue example 1 but employ a different true solution
Now the regular component of q is in the finite element space. Let ,. Tables 67 show a general convergence rate for each mesh. Furthermore, Figure 2 and Table 8 demonstrate the influence of the factor on the upper bound of these errors, which confirms our prediction (18).
Fig. 2 error on the singular perturbation parameter for Example 2 
error on 4 meshes for Example 1, where
error on 8 layeradapted meshes for Example 1, where
error on 8 layeradapted meshes for Example 1, where
error on 8 layeradapted meshes for Example 1, here k=2 and N=64
error on 8 layeradapted meshes for Example 2, where
error on 8 layeradapted meshes for Example 2, where
error on 8 layeradapted meshes for Example 2, here k=2 and N=64
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All Tables
All Figures
Fig. 1 error on the singular perturbation parameter for Example 1 

In the text 
Fig. 2 error on the singular perturbation parameter for Example 2 

In the text 
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