A Globally Uniformly Convergent Finite Element Method for a Singularly Perturbed Elliptic Problem in Two Dimensions

Uniformly Convergent Finite Element Methods for Singularly Perturbed Elliptic Boundary Value Problems I: Reaction-diffusion Type

{ We consider the bilinear nite element method on a Shishkin mesh for the singularly perturbed elliptic boundary value problem ?" 2 (@ 2 u @x 2 + @ 2 u @y 2) + a(x; y)u = f(x; y) in two space dimensions. By using a very sophisticated asymptotic expansion of Han et al. 11] and the technique we used in 17], we prove that our method achieves almost second-order uniform convergence rate in L 2-norm...

A Global Uniformly Convergent Finite Element Method for a Quasilinear Singularly Perturbed Elliptic Problem

Uniformly Convergent Nite Element Methods for Singularly Perturbed Elliptic Boundary Value Problems: Convection-diiusion Type

In this paper we consider the standard bilinear nite element method (FEM) and the corresponding streamline diiusion FEM for the singularly perturbed elliptic boundary value problem ?" (@ 2 u @x 2 + @ 2 u @y 2) ? b(x; y) ru + a (x; y)u = f (x; y) in the two space dimensions. By using the asymptotic expansion method of Vishik and Lyusternik 42] and the technique we used in 25, 26], we prove that ...

Analysis of a Streamline-Diffusion Finite Element Method on Bakhvalov-Shishkin Mesh for Singularly Perturbed Problem

Abstract. In this paper, a bilinear Streamline-Diffusion finite element method on Bakhvalov-Shishkin mesh for singularly perturbed convection – diffusion problem is analyzed. The method is shown to be convergent uniformly in the perturbation parameter ǫ provided only that ǫ ≤ N. An O(N(lnN)) convergent rate in a discrete streamline-diffusion norm is established under certain regularity assumpti...

Global Uniformly Convergent Nite Element Methods for Singularly Perturbed Elliptic Boundary Value Problems: Higher-order Elements

{ In this paper, we develop a general higher-order nite 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 ?(m+1) x ln m+1 N x + N ?(m+1) y ln m+1 N y) in L 2 norm is obtained for a reaction-diiusion model by using the m-th order (m 2) tensor-product element, thus a...