Global uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems:higher-order elements

Jichun Li and I.M. Navon

In this paper, we develop a general higher-order finite element method for solving singularly perturbed elliptic linear and quasilinear problems in two space dimensions. We prove that a quasioptimal global uniform convergence rate of O(N_x^{-(m+1)} ln^{m+1} N_x +N_y^{-(m+1)} ln^{m+1} N_y) in L^2 norm is obtained for a reaction-diffusion model by using the m-th order (m \ge 2) tensor-product element, thus answering some open problems posed by Roos in [1] and [2, p.278]. Here N_x and N_y are the number of partitions in the x- and y-directions,respectively. Numerical results are provided supporting our theoretical analysis.