Finite element method on Shishkin mesh for a singularly perturbed problem with an interior layer. (November 2021) - Zhang, Jin; Ma, Xiaoqi; Lv, Yanhui
Abstract: In this paper, we discuss a singularly perturbed convection diffusion problem, whose solution contains an interior layer caused by the discontinuity of the convection coefficient. For this kind of problem, uniform convergence of the finite element method has not been established. For this aim, after proving the minimum principle and stability result, we derive the asymptotic expansion of the solution and a priori estimates of its smooth and layer components. Then we construct a finite element method on Shishkin mesh, and prove uniform convergence of almost order k in the energy norm, where k is the order of piecewise polynomials.