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

In this paper, we develop a general higher-order finite element method for solving singularly perturbed elliptic linear and quasilinear problems in tw o space dimensions. We prove that a quasioptimal global uniform convergence rate of O(N-x(-(m + 1)) Inm + 1Nx + N-y(-(m + 1)) Inm + 1Ny) in L-2 norm i s obtained for a reaction-diffusion model by using the mth order (m greater than or equal to 2) tenser-product element, thus answering some open probl ems posed by Roos in [H.-G. Roos, Layer-adapted grids for singular perturba tion problems, Z. Angew. Math. Mech. 78(5) (1998) 291-309] and [H.-G. Ross, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Diffe rential Equations (Springer-Verlag, Berlin, 1996) 278]. Here, N-x and N-y a re the number of partitions in the x- and y-directions, respectively. Numer ical results are provided supporting our theoretical analysis. (C) 1999 Els evier Science S.A. All rights reserved.