Pointwise error estimate of the LDG method for 2D singularly perturbed reaction-diffusion problem

So far uniform pointwise error estimate of the local discontinuous Galerkin (LDG) method for the singularly perturbed reaction-diffusion problems is only known in the one-dimensional case. In this paper, we consider an LDG method with interior alternating numerical flux for a singularly perturbed reaction-diffusion problem posed on the unit square in \(\mathbb {R}^2\) ${\mathbb{R}}^2$ . We prove that the LDG method on a Shishkin mesh is pointwise convergent with the order of \(O(N^{-(k+1)})\) $O(N^{-(k+1)})$ , uniformly in the perturbation parameter \(\varepsilon \) $\epsilon$ outside the layers, where k is the degree of the tensor-product piecewise polynomials used in the finite element space and N is the number of mesh elements in each coordinate direction. This rate of convergence is sharp and agrees with our numerical results. In the fine parts that lie inside the boundary and corner layers, we derive a uniform pointwise convergence of orders \(O(N^{-k})\) $O(N^{-k})$ and \(O(N^{-(k-1)})\) $O(N^{-(k-1)})$ respectively. The theoretical difficulty lies in overcoming the highly non-uniformity of the mesh and the weak stability induced by the bilinear form. Numerical experiments are also given.