Open Access
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

© Wuhan University 2023

Licence Creative CommonsThis 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 quasi-uniform mesh. To obtain satisfactory numerical approximations, several numerical strategies were developed, such as the layer-adapted meshes, the fitted operator methods and the stabilised numerical method, see Refs. [1-3] 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. [7-10].

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 layer-adapted meshes. Recently, we studied the local DG method on several layer-adapted meshes. Some uniformly optimal error estimates were established for second order convection-diffusion 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 Shishkin-type meshes as well as three common Bakhvalov-type 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, layer-adapted 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 Layer-Adapted Meshes and the DG Method

Consider a one-dimensional model problem

(1)

which possesses some essential characteristics of the following second-order problem

(2)

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

(3)

1.1 Layer-Adapted 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 non-singularly 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

(4)

In Table 1, we list eight common layer-adapted meshes, which are simplified as S-, BS-, mBS-,VS-, pS-, B-, mVB- and RS-meshes. 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.

Table 1

Layer-adapted 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

(5)

holds for any and , where

Denote Rewrite the scheme (5) into a compact form: Find such that

(6)

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 Gauss-Radau projecter such that for any function and each element ,

(7)

From Ref.[16], one can verify the well-posedness of the above projection. Furthermore,

(8)

(9)

(10)

Lemma 2   Assume in the definition of the layer-adapted mesh (4) . Let with each component and satisfying (3), then one has

(11)

(12)

Furthermore, if , one has

(13)

Proof   Denote for . From (3) and (10), one has

(14)

For the monotonic increasing function using the stability (8), one has

(15)

On the refined domain, if , one obtains from Lemma 1 that

(16)

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

(17)

Furthermore, if , one has

(18)

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 pS-mesh, take For the RS-mesh, 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 B-mesh are both , the convergence rate on S-mesh is , while the convergence rate on BS-mesh 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 S-mesh a convergence rate , while for the BS-, mBS-, VS-, B- and mVB- meshes a general convergence rate . For the pS-mesh, the convergence rate is a little smaller than k+1 because of the influence of the logarithmic factor (here ). For the RS-mesh, 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 S-mesh and RS-mesh which are slightly influenced by the factor as is suitably large.

thumbnail 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 6-7 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).

thumbnail Fig. 2

-error on the singular perturbation parameter for Example 2

Table 2

-error on 4 meshes for Example 1, where

Table 3

-error on 8 layer-adapted meshes for Example 1, where

Table 4

-error on 8 layer-adapted meshes for Example 1, where

Table 5

-error on 8 layer-adapted meshes for Example 1, here k=2 and N=64

Table 6

-error on 8 layer-adapted meshes for Example 2, where

Table 7

-error on 8 layer-adapted meshes for Example 2, where

Table 8

-error on 8 layer-adapted meshes for Example 2, here k=2 and N=64

References

  1. Roos H G, Stynes M, Tobiska L. Numerical Methods for Singularly Perturbed Differential Equations[M]. Berlin: Springer-Verlag, 1996. [CrossRef] [Google Scholar]
  2. Stynes M, Stynes D. Convection-diffusion Problems: An Introduction to Their Analysis and Numerical Solution[M]. Halifax: American Mathematical Society, 2018. [Google Scholar]
  3. Miller J J H, Riordan E O, Shishkin G I. Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions[M]. Singapore: World Scientific, 1996. [CrossRef] [Google Scholar]
  4. Reed W H, Hill T R. Triangular Mesh Methods for the Neutron Transport Equation[M]. Los Alamos: Los Alamos Scientific Laboratory, 1973. [Google Scholar]
  5. Cockburn B. Discontinuous Galerkin methods for convection-dominated problems, in high-order methods for computational physics[C]//Lecture Notes in Computational Science and Engineering. Berlin: Springer-Verlag, 1999, 9: 69-224. [Google Scholar]
  6. Xie Z Q, Zhang Z Z, Zhang Z M. A numerical study of uniform superconvergence of LDG method for solving singularly perturbed problems[J]. J Comput Math, 2009, 27(2-3): 280-298. [MathSciNet] [Google Scholar]
  7. Zhu H, Zhang Z. Convergence analysis of the LDG method applied to singularly perturbed problems[J]. Numerical Methods Partial Differential Equation, 2013, 29(2): 396-421. [CrossRef] [Google Scholar]
  8. Zhu H, Zhang Z. Uniform convergence of the LDG method for a singularly perturbed problem with the exponential boundary layer[J]. Math Comp, 2014, 286(83): 635-663. [Google Scholar]
  9. Cheng Y, Song C J, Mei Y J. Local discontinuous Galerkin method for time-dependent singularly perturbed semilinear reaction-diffusion problems[J]. Comput Methods Appl Math, 2021, 21(1): 31-52. [CrossRef] [MathSciNet] [Google Scholar]
  10. Cheng Y, Zhang Q, Wang H J. Local analysis of the local discontinuous Galerkin method with the generalized alternating numerical flux for two-dimensional singularly perturbed problem[J]. Int Numer Anal Mod, 2018, 15(6): 785-810. [Google Scholar]
  11. Cheng Y. On the local discontinuous Galerkin method for singularly perturbed problem with two parameters[J]. J Comput Appl Math, 2021, 392: 113485. [CrossRef] [Google Scholar]
  12. Cheng Y, Mei Y J, Roos H G. The local discontinuous Galerkin method on layer-adapted meshes for time-dependent singularly perturbed convection-diffusion problems[J]. Comput Math Appl, 2022, 117: 245-256. [MathSciNet] [Google Scholar]
  13. Linß T. Layer-adapted meshes for convection-diffusion problems[J]. Comput Methods Appl Mech Engrg, 2003, 192: 9-10. [Google Scholar]
  14. Roos H G. Layer-adapted grids for singular perturbation problems[J]. Z Angew Math Mech, 1998, 78: 291-309. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  15. Xie Z Q, Zhang Z M. Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problems in 1-D[J]. Math Comp, 2010, 79(269): 35-45. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  16. Ciarlet P. The Finite Element Method for Elliptic Problem[M]. Amsterdam: North-Holland, 1975. [Google Scholar]

All Tables

Table 1

Layer-adapted meshes

Table 2

-error on 4 meshes for Example 1, where

Table 3

-error on 8 layer-adapted meshes for Example 1, where

Table 4

-error on 8 layer-adapted meshes for Example 1, where

Table 5

-error on 8 layer-adapted meshes for Example 1, here k=2 and N=64

Table 6

-error on 8 layer-adapted meshes for Example 2, where

Table 7

-error on 8 layer-adapted meshes for Example 2, where

Table 8

-error on 8 layer-adapted meshes for Example 2, here k=2 and N=64

All Figures

thumbnail Fig. 1

-error on the singular perturbation parameter for Example 1

In the text
thumbnail Fig. 2

-error on the singular perturbation parameter for Example 2

In the text

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