Local Discontinuous Galerkin Method for a Third-Order Singularly Perturbed Problem of Convection-Diffusion Type

Li Yan; Zhoufeng Wang; Yao Cheng

doi:10.1515/cmam-2022-0176

Article

Local Discontinuous Galerkin Method for a Third-Order Singularly Perturbed Problem of Convection-Diffusion Type

Li Yan , Zhoufeng Wang and Yao Cheng

Published/Copyright: December 6, 2022

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Computational Methods in Applied Mathematics Volume 23 Issue 3

Abstract

The local discontinuous Galerkin (LDG) method is studied for a third-order singularly perturbed problem of convection-diffusion type. Based on a regularity assumption for the exact solution, we prove almost O ⁢ ( N - ( k + 1 2 ) ) (up to a logarithmic factor) energy-norm convergence uniformly in the perturbation parameter. Here, k ≥ 0 is the maximum degree of piecewise polynomials used in discrete space, and N is the number of mesh elements. The results are valid for the three types of layer-adapted meshes: Shishkin-type, Bakhvalov–Shishkin-type, and Bakhvalov-type. Numerical experiments are conducted to test the theoretical results.

Keywords: Local Discontinuous Galerkin Method; Third-Order Singularly Perturbed Problem; Convection-Diffusion; Shishkin-Type Mesh; Bakhvalov-Type Mesh; Uniform Convergence

MSC 2010: 65N15; 65N30; 65N50

Appendix

In this part, we provide a technical lemma used in the proof of our main result.

Lemma 4.

Suppose ( X , Y ) ∈ V N 2 satisfies

(A.1) 〈 Y , s 〉 I j + ε ⁢ ( 〈 X , s ′ 〉 I j - X ^ j ⁢ s j - + X ^ j - 1 ⁢ s j - 1 + ) = F j ⁢ ( s )

in each element I j ∈ Ω N and for any test function s ∈ V N , where F j ⁢ ( s ) : V N → R is a linear functional, and

X ^ j = { X j + , j = 0 , 1 , … , N - 1 , 0 , j = N .

Then the local estimate holds

(A.2) ∥ Y ∥ I j ≤ C ⁢ ε ⁢ ( h j - 1 ⁢ ∥ X ∥ I j + h j - 1 2 ⁢ | [ [ X ] ] j | ) + | F j ⁢ ( Y ) | ∥ Y ∥ I j

for each element I j ∈ Ω N , where C > 0 is independent of ε and h j .

Proof.

Take s = Y into (A.1), use integration by parts, an inverse inequality and the Cauchy–Schwarz inequality to get

∥ Y ∥ I j 2 = ε ⁢ ( - 〈 X , Y ′ 〉 I j + X j + ⁢ Y j - - X j - 1 + ⁢ Y j - 1 + ) + F j ⁢ ( Y )

= ε ⁢ ( 〈 X ′ , Y 〉 I j + Y j - ⁢ [ [ X ] ] j ) + F j ⁢ ( Y )

≤ C ⁢ ε ⁢ ( h j - 1 ⁢ ∥ X ∥ I j ⁢ ∥ Y ∥ I j + h j - 1 2 ⁢ | [ [ X ] ] j | ⁢ ∥ Y ∥ I j ) + | F j ⁢ ( Y ) |

for j = 1 , 2 , … , N - 1 . Hence,

∥ Y ∥ I j ≤ C ⁢ ε ⁢ ( h j - 1 ⁢ ∥ X ∥ I j + h j - 1 2 ⁢ | [ [ X ] ] j | ) + | F j ⁢ ( Y ) | ∥ Y ∥ I j

hold for j = 1 , 2 , … , N - 1 .

Analogously, one can obtain the conclusion for j = N by noticing that [ [ X ] ] N = - X N - . ∎

References

[1] Y. Cheng, On the local discontinuous Galerkin method for singularly perturbed problem with two parameters, J. Comput. Appl. Math. 392 (2021), Paper No. 113485. 10.1016/j.cam.2021.113485Search in Google Scholar

[2] Y. Cheng, Y. Mei and H.-G. Roos, The local discontinuous Galerkin method on layer-adapted meshes for time-dependent singularly perturbed convection-diffusion problems, Comput. Math. Appl. 117 (2022), 245–256. 10.1016/j.camwa.2022.05.004Search in Google Scholar

[3] Y. Cheng, L. Yan, X. Wang and Y. Liu, Optimal maximum-norm estimate of the LDG method for singularly perturbed convection-diffusion problem, Appl. Math. Lett. 128 (2022), Paper No. 107947. 10.1016/j.aml.2022.107947Search in Google Scholar

[4] Y. Cheng and Q. Zhang, Local analysis of the local discontinuous Galerkin method with generalized alternating numerical flux for one-dimensional singularly perturbed problem, J. Sci. Comput. 72 (2017), no. 2, 792–819. 10.1007/s10915-017-0378-ySearch in Google Scholar

[5] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4, North-Holland, Amsterdam, 1978. 10.1115/1.3424474Search in Google Scholar

[6] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2440–2463. 10.1137/S0036142997316712Search in Google Scholar

[7] F. A. Howes, Nonlinear dispersive systems: theory and examples, Stud. Appl. Math. 69 (1983), no. 1, 75–97. 10.1002/sapm198369175Search in Google Scholar

[8] F. A. Howes, The asymptotic solution of a class of third-order boundary value problems arising in the theory of thin film flows, SIAM J. Appl. Math. 43 (1983), no. 5, 993–1004. 10.1137/0143065Search in Google Scholar

[9] T. Linß, The necessity of Shishkin decompositions, Appl. Math. Lett. 14 (2001), no. 7, 891–896. 10.1016/S0893-9659(01)00061-1Search in Google Scholar

[10] T. Linß, Layer-adapted Meshes for Reaction-Convection-Diffusion Problems, Lecture Notes in Math. 1985, Springer, Berlin, 2010. 10.1007/978-3-642-05134-0Search in Google Scholar

[11] R. E. O’Malley, Jr., Introduction to Singular Perturbations, Appl. Math. Mech. (English Ed.) 14, Academic Press, New York, 1974. Search in Google Scholar

[12] H.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, 2nd ed., SSpringer Ser. Comput. Math. 24, Springer, Berlin, 2008. Search in Google Scholar

[13] H.-G. Roos, L. Teofanov and Z. Uzelac, Uniformly convergent difference schemes for a singularly perturbed third order boundary value problem, Appl. Numer. Math. 96 (2015), 108–117. 10.1016/j.apnum.2015.06.002Search in Google Scholar

[14] R. S. Temsah, Spectral methods for some singularly perturbed third order ordinary differential equations, Numer. Algorithms 47 (2008), no. 1, 63–80. 10.1007/s11075-007-9147-6Search in Google Scholar

[15] S. Valarmathi and N. Ramanujam, An asymptotic numerical method for singularly perturbed third-order ordinary differential equations of convection-diffusion type, Comput. Math. Appl. 44 (2002), no. 5–6, 693–710. 10.1016/S0898-1221(02)00183-9Search in Google Scholar

[16] Z. Xie, Z. Zhang and Z. Zhang, A numerical study of uniform superconvergence of LDG method for solving singularly perturbed problems, J. Comput. Math. 27 (2009), no. 2–3, 280–298. Search in Google Scholar

[17] H. Zarin, H.-G. Roos and L. Teofanov, A continuous interior penalty finite element method for a third-order singularly perturbed boundary value problem, Comput. Appl. Math. 37 (2018), no. 1, 175–190. 10.1007/s40314-016-0339-3Search in Google Scholar

[18] H. Zhu and Z. Zhang, Convergence analysis of the LDG method applied to singularly perturbed problems, Numer. Methods Partial Differential Equations 29 (2013), no. 2, 396–421. 10.1002/num.21711Search in Google Scholar

[19] H. Zhu and Z. Zhang, Uniform convergence of the LDG method for a singularly perturbed problem with the exponential boundary layer, Math. Comp. 83 (2014), no. 286, 635–663. 10.1090/S0025-5718-2013-02736-6Search in Google Scholar

Received: 2022-09-03

Accepted: 2022-10-26

Published Online: 2022-12-06

Published in Print: 2023-07-01

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/cmam-2022-0176

Keywords for this article

Local Discontinuous Galerkin Method; Third-Order Singularly Perturbed Problem; Convection-Diffusion; Shishkin-Type Mesh; Bakhvalov-Type Mesh; Uniform Convergence