A robust weak Galerkin method for singularly perturbed fourth-order convection–diffusion problems

Suayip Toprakseven; Seza Dinibutun; Suayip Toprakseven; Seza Dinibutun

doi:10.3934/era.2026153

Electronic Research Archive

2026, Volume 34, Issue 5: 3410-3446. doi: 10.3934/era.2026153

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A robust weak Galerkin method for singularly perturbed fourth-order convection–diffusion problems

Suayip Toprakseven ^{1
,
,},
Seza Dinibutun ²

1.
Department of Mathematics, Faculty of Art and Science, Yozgat Bozok University, Yozgat 66100, Türkiye
2.
Department of Mathematics and Natural Sciences, International University of Science and Technology in Kuwait, Ardiya 92400, Kuwait

Received: 27 February 2026 Revised: 13 April 2026 Accepted: 17 April 2026 Published: 22 April 2026

We propose a uniform weak Galerkin finite element method (WG-FEM) to solve singularly perturbed fourth-order convection–diffusion problems exhibiting boundary layers. The method is designed to handle small perturbation parameters $ \varepsilon $, ensuring accurate resolution of sharp layers without spurious oscillations. We construct appropriate layer-adapted meshes and use uniform estimates to analyze the stability and convergence of the weak Galerkin scheme. Numerical experiments confirm the theoretical results and demonstrate that the method achieves uniform convergence with respect to the perturbation parameter, effectively capturing the boundary layer behavior on layer-adapted meshes. The proposed approach provides a reliable and efficient tool for high-order singularly perturbed problems with mixed derivative boundary conditions.
- weak Galerkin method,
- singularly perturbed problems,
- fourth-order convection-diffusion,
- boundary layers,
- uniform convergence,
- layer-adapted meshes
Citation: Suayip Toprakseven, Seza Dinibutun. A robust weak Galerkin method for singularly perturbed fourth-order convection–diffusion problems[J]. Electronic Research Archive, 2026, 34(5): 3410-3446. doi: 10.3934/era.2026153

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Abstract

We propose a uniform weak Galerkin finite element method (WG-FEM) to solve singularly perturbed fourth-order convection–diffusion problems exhibiting boundary layers. The method is designed to handle small perturbation parameters $ \varepsilon $, ensuring accurate resolution of sharp layers without spurious oscillations. We construct appropriate layer-adapted meshes and use uniform estimates to analyze the stability and convergence of the weak Galerkin scheme. Numerical experiments confirm the theoretical results and demonstrate that the method achieves uniform convergence with respect to the perturbation parameter, effectively capturing the boundary layer behavior on layer-adapted meshes. The proposed approach provides a reliable and efficient tool for high-order singularly perturbed problems with mixed derivative boundary conditions.

References

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