A finite element method with a large mesh-width for a stiff two-point boundary value problem

Abstract

We describe a finite element method for computation of numerical approximations of the solution of the second order singularly perturbed two-point boundary value problem on [−1, 1] ϵ u″ + pu′ = f, u(−1) = u(1) = 0, 0 < ϵ ∠ 1, (′ = d dx ) On a quasi-uniform mesh we construct exponentially fitted trial spaces which consist of piece-wise polynomials and of exponentials which fit locally to the singular solution of the equation or its adjoint. We discretise the Galerkin form for the boundary problem using such exponentially fitted trial spaces. We derive rigorous bounds for the error of discretisation with respect to the energy norm and we obtain superconvergence at the mesh-points, the error depending on ϵ, the mesh-width and the degree of the piece-wise polynomials.