Supercloseness of weak Galerkin methods in a weighted and balanced norm for singularly perturbed reaction–diffusion problems
Mathematics and Computers in Simulation, cilt.246, ss.491-508, 2026 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 246
- Basım Tarihi: 2026
- Doi Numarası: 10.1016/j.matcom.2026.02.019
- Dergi Adı: Mathematics and Computers in Simulation
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Compendex, INSPEC, MathSciNet, Public Affairs Index, zbMATH
- Sayfa Sayıları: ss.491-508
- Anahtar Kelimeler: Balanced norm, Layer-adapted meshes, Reaction–diffusion, Singularly perturbed, Supercloseness, Weak Galerkin finite element method
- Yozgat Bozok Üniversitesi Adresli: Evet
Özet
In this paper, we analyze the error of the weak Galerkin finite element method (WG-FEM) for singularly perturbed reaction–diffusion equations using piecewise discontinuous bilinear polynomials on a 2D Shishkin mesh. To accurately capture boundary layer effects, we introduce a weighted and balanced norm that is stronger than the standard energy norm. Unlike traditional approaches, which rely on global or local L2-projections and face challenges in 2D settings, our balanced norm is defined using a weight function together with the bilinear interpolation operator. By employing integral identities, we establish supercloseness results on Shishkin meshes. Numerical experiments confirm the sharpness of the theoretical analysis.