Workshop Singularly Perturbed Problems
Talks & Slides
Analytic regularity and finite element approximation of coupled systems of singularly perturbed reaction-diffusion equations
We consider a coupled system of two singularly perturbed reaction-diffusion equations with two small parameters 0 < ε ≤ μ ≤ 1, each multiplying the highest derivative in the equations. The presence of these parameters causes the solution to have boundary layers that overlap and interact, based on the relative size of ε and μ.
We present full asymptotic expansions together with error bounds that cover the complete range 0 < ε ≤ μ ≤ 1. Under the assumption of analytic input data, we obtain derivative growth estimates for the terms in the asymptotic expansion that are explicit in the perturbation parameters and the expansion order.
Next, we propose and analyze an h p finite element scheme which includes elements of size O(p ε) and O(p μ) near the boundary, where p is the degree of the approximating polynomials. We show that as p → ∞, the method converges at an exponential rate, independently of the singular perturbation parameters, when the error is measured in the energy norm.
This is joint work with J. M. Melenk and L. Oberbroeckling.