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

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(Nx−(m+1) Inm+1Nx + Nv−(m+1) Inm+1Nv) in L2 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, Nx and Nv are the number of partitions in the x- and y-directions, respectively. Numerical results are provided supporting our theoretical analysis.