Galerkin methods for a singularly perturbed hyperbolic problem with nonlocal nonlinearity

Abstract

We consider the numerical approximation of hyperbolic and parabolic problems with nonlocal nonlinearity by Galerkin methods, and provide optimal order L 2 error estimates in the continuous-time case. Three-level discrete schemes are then introduced and used to investigate the relationship between a singularly perturbed hyperbolic problem with small parameter ϵ 2 multiplying the highest time derivative and the reduced problem of parabolic type. For small ϵ 2, the problem of stiffness in the hyperbolic problem can be avoided by utilizing the solution of the reduced problem in accordance with a recent asymptotic result of Esham and Weinacht. The advantage of using a two-term asymptotic expansion is also briefly considered.