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

摘要

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(Nx~-(m+1)ln~m+1Nx+Nyu~-(m+1)ln~m+1Ny in L~2 norm is obtained for a reaction-diffusion model by using the mth order (m≥ 2) tensor--product element, thus answering some open problems posed by Roos in [H.-G. Roos, Layer-adapted grids for singular perturbation problems, Z. Angew. Math. Mech. 78(5) (l998) 29l --309] and [H.-G. Ross, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations (Springerwerlag, Berlin, l996) 2781. Here, N. and N are the number of partitions in the x-- and y-directions, respectively. Numerical results are provided supporting our theoretical analysis.