The Singularity-Separated Method for the Singular Perturbation Problems in 1-D

Abstract

The singularity-separated method (SSM) for the singular perturbation problem $-\epsilon u''+bu' + cu = f(x), u(0) = u(1) = 0$, is proposed for the first time. The solution is expressed as $u = w-ν$, where $w$ is the solution of corresponding third boundary value problem and $ν$ is an exact singular function. We have proved a global regularity, $||w||_2 ≤ C$, where the constant $C$ is independent of $\epsilon$, and discussed three kinds of finite element (FE) methods with SSM. Numerical results show that these FE-solutions have the high accuracy when only one element in boundary layer is taken.