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

Suayip Toprakseven and Peng Zhu

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 246, issue C, 491-508

Abstract: 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.

Keywords: Weak Galerkin finite element method; Reaction–diffusion; Singularly perturbed; Layer-adapted meshes; Balanced norm; Supercloseness (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:246:y:2026:i:c:p:491-508

DOI: 10.1016/j.matcom.2026.02.019

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