Published by De Gruyter December 6, 2022

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

Li Yan , Zhoufeng Wang and Yao Cheng

From the journal Computational Methods in Applied Mathematics

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

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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 - . ∎

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Received: 2022-09-03

Accepted: 2022-10-26

Published Online: 2022-12-06

Published in Print: 2023-07-01