A Hybrid Asymptotic-Finite Element Method for Stiff Two-Point Boundary Value Problems

Abstract

An accurate and efficient numerical method has been developed for a nonlinear stiff second order two-point boundary value problem. The scheme combines asymptotic methods with the usual solution techniques for two-point boundary value problems. A new modification of Newton's method or quasilinearization is used to reduce the nonlinear problem to a sequence of linear problems. The resultant linear problem is solved by patching local solutions at the knots or equivalently by projecting onto an affine subset constructed from asymptotic expansions. In this way, boundary layers are naturally incorporated into the approximation. An adaptive mesh is employed to achieve an error of $O({1 / {N^2 }}) + O(\sqrt \varepsilon )$. Here, N is the number of intervals and $\varepsilon \ll 1$ is the singular perturbation parameter. Numerical computations are presented.