Supercloseness of weak Galerkin methods in a weighted and balanced norm for singularly perturbed reaction–diffusion problems

TOPRAKSEVEN Ş., Zhu P.

Mathematics and Computers in Simulation, cilt.246, ss.491-508, 2026 (SCI-Expanded, Scopus) identifier

  • 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.