Analysis of an LDG-FEM for a two-dimensional singularly perturbed convection-reaction-diffusion problem with interior and boundary layers

Abstract

This article is concerned with a two-dimensional singularly perturbed convection-reaction-diffusion interface problem. In the considered problem, the convection and reaction coefficients and the source term have discontinuities along interface lines, which are parallel to the x- and y-axes. The coefficient of the highest-order term is a small positive parameter denoted by ε. Due to discontinuities in the coefficients and the source term, interior and boundary layers appear in the solution when ε approaches zero. A Local Discontinuous Galerkin method is constructed on an appropriate Shishkin mesh. The test functions in the Local Discontinuous Galerkin method are piecewise polynomials that lie in the space Qr of piecewise polynomials of degree at most r in each variable, where r is a positive integer. We established that the error in the computed solution converges at the rate of O((N−1ln⁡N)r+12) in an energy-norm. Numerical results are given to support the theoretical results.