A finite element method for a singularly perturbed boundary value problem
We examine the problem: \(\epsilon u''+a(x)u'-b(x)u=f(x)\) for \(0<x<1\), \(a(x)\geq \alpha >0\), b(x)\(\geq \beta\), \(\alpha^ 2+4\epsilon \beta >0\), a, b and f in \(C^ 2[0,1]\), \(\epsilon\) in (0,1], u(0) and u(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of width h is uniformly second order accurate for this problem (i.e., the nodal errors are bounded by \(Ch^ 2\), where C is independent of h and \(\epsilon)\). With a natural choice of trial functions, uniform first order accuracy is obtained in the \(L^{\infty}(0,1)\) norm. On choosing piecewise linear trial functions (''hat'' functions), uniform first order accuracy is obtained in the \(L^ 1(0,1)\) norm.
scientific article; zbMATH DE number 3934339
scientific article