The uniform convergence of a weak Galerkin finite element method in the balanced norm for reaction–diffusion equation

Abstract

In this paper, a weak Galerkin finite element method is implemented to solve the one-dimensional singularly perturbed reaction–diffusion equation. This weak Galerkin finite element scheme uses piecewise polynomial with degree k≥1 in the interior part of each element and piecewise constant function at the nodes of each element. The existence and uniqueness of the weak Galerkin finite element solution are proved. Based on the interpolation operator and the corresponding approximation properties, an ɛ−uniform error bound of O(N−(k+1)+ɛ(N−1lnN)k) in the energy-like norm is investigated rigorously. Furthermore, an ɛ−uniform error bound of O(N−(k+1/2)+(N−1lnN)k) in the balanced norm is established by the weighted local L2 projection and its corresponding approximation properties. Finally, numerical experiments validate the theoretical results. Moreover, numerical results show that this weak Galerkin finite element solution achieves the convergence rate of O((N−1lnN)k+1) in the L2 norm and the discrete L∞ norm uniformly with respect to the singular perturbation parameter ɛ.