Abstract
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 0( N x −( m+1) In m+1 N x + N v −(m+1) In m+1 N v) 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) (1998) 291–309] and [H.-G. Ross, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations (Springer-Verlag, Berlin, 1996) 278]. Here, N x and N v are the number of partitions in the x- and y-directions, respectively. Numerical results are provided supporting our theoretical analysis.
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