On the linear finite element approximation for two-point boundary value problems

In this paper, we present a modified Galerkin approximation for two-point boundary value problems of divergence type: −(ddx)(a(x)dudx) + c(x)u = f(x), 0 < x < 1. Then a linear finite element approximation of the fourth order at mesh nodes is obtained and proved. For the case that a(x) is singular, this method combined with a special finite element space (cf. Marchuk, 1982; Osborn, 1993) also gives us a fourth-order approximation at the mesh nodes. Some numerical comparisons for this method are given for a(x) being both smooth and singular.