A High‐Order Numerical Method for Semilinear Convection–Diffusion Equation on Bakhvalov‐type Mesh

ABSTRACT This study analyzes a high‐order weak Galerkin finite element method (WG‐FEM) to solve the one‐dimensional unsteady convection–diffusion equation with a nonlinear reaction term on a non‐uniform mesh. The proposed method employs piecewise polynomials of degree in the interior, combined with a constant polynomial approximation at each element boundary. We apply WG‐FEM on a Bakhvalov mesh for spatial discretization, coupled with the Crank–Nicolson scheme for time discretization on a uniform mesh. The standard Lagrange interpolation is commonly used in error analysis for the finite element method when solving convection‐dominated problems. However, when applied to Bakhvalov‐type meshes, this interpolation can lead to instability, particularly within specific section of the mesh known as problematic region. To address this issue and accurately capture the solution's behavior in this region, a special interpolation, represented as , is introduced. The primary outcome of this study demonstrates that the weak Galerkin solution on the Bakhvalov mesh achieves a parameter‐free error bound of order in the spatial direction, along with second‐order convergence in the temporal direction. Numerous numerical experiments have been conducted to validate these theoretical findings.