A uniform finite element method for a conservative singularly perturbed problem

Uniform error estimates in the finite element method for a singularly perturbed reaction-diffusion problem

Consider the problem− 2∆u+u = f with homogeneous Neumann boundary condition in a bounded smooth domain in RN . The whole range 0 < ≤ 1 is treated. The Galerkin finite element method is used on a globally quasi-uniform mesh of size h; the mesh is fixed and independent of . A precise analysis of how the error at each point depends on h and is presented. As an application, first order error estima...

Finite element methods for a singularly perturbed transmission problem

We consider a one-dimensional singularly perturbed transmission problem with two different diffusion coefficients. The solution will contain boundary layers only in the part of the domain where the diffusion coefficient is high. We derive and analyze various finite element approaches for the approximation of the solution and conduct numerical computations that show the robustness (or lack there...

A Parameter–Uniform Finite Difference Method for a Singularly Perturbed Initial Value Problem: a Special Case

A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are not necessarily equal. The components of the solution exhibit overlapping layers. A Shishkin piecewise– uniform mesh is constructed, which is used, in conjunction with a cl...

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

We analyze a new Galerkin finite element method for numerically solving a linear convection-dominated convection-diffusion problem in two dimensions. The method is shown to be convergent, uniformly in the perturbation i ii parameter, of order h ' in a global energy norm which is stronger than the L norm. This order is optimal in this norm for our choice of trial functions.

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