Finite difference, finite element and finite volume methods applied to two-point boundary value problems

This paper considers the finite difference, finite element and finite volume methods applied to the two-point boundary value problem -d/dx(p(x) du/dx) = f(x), a < x < b, u(a) = u(b) = 0. By using an inversion formula of a nonsingular tridiagonal matrix, explicit expressions of approximate solutions by three methods are given, which lead to a unified understanding of these methods as well as their unified error estimates. Numerical examples are also given.