A uniform finite element method for a conservative singularly perturbed problem

摘要

We examine the problem ϵ(p(x)u′) + (q(x)u)′ − r(x)u = f(x) for 0 < x < 1, p>0, q>0, r ⩾ 0; p, q, r and f in C2[0, 1], ϵ in (0, 1], u(0) and u(1) given. Existence of a unique solution u and bounds on u and its derivatives are obtained. Using finite elements on an equidistant mesh of width h we generate a tridiagonal difference scheme which is shown to be uniformly second order accurate for this problem (i.e., the nodal errors are bounded by Ch2, where C is independent of h and ϵ). With a natural choice of trial functions, uniform first order accuracy is obtained in the L∞(0, 1) norm. Using trial functions which interpolate linearly between the nodal values generated by the difference scheme gives uniform first order accuracy in the L1(0, 1) norm.