An improved pointwise error estimate of the LDG method for 1-d singularly perturbed reaction-diffusion problem

The local discontinuous Galerkin (LDG) method on a Shishkin mesh is studied for a onedimensional singularly perturbed reaction-diffusion problem. Based on the discrete Green's function, we derive some improved pointwise error estimates in the regular and layer regions. For the LDG approximation to the solution itself, the convergence rate of the pointwise error in the regular region is sharp. For the LDG approximation to the derivative of the solution, the convergence rate of the pointwise error in the whole domain is optimal. We establish also optimal pointwise error estimates in the case that the regular component of the exact solution belongs to the finite element space. Numerical experiments are presented to validate the theoretical results.(c) 2023 IMACS. Published by Elsevier B.V. All rights reserved.

Automated summary

The local discontinuous Galerkin (LDG) method on a Shishkin mesh is investigated for a one-dimensional singularly perturbed reaction-diffusion problem. Improved pointwise error estimates are derived based on the discrete Green's function in the regular and layer regions. The convergence rates of the pointwise error for both the LDG approximation to the solution and its derivative are analyzed, showing optimal rates in different domains. Moreover, optimal pointwise error estimates are established when the regular component of the exact solution belongs to the finite element space. Numerical experiments are conducted to validate the theoretical findings.