We consider a coupled system of two singularly perturbed reaction-diffusion equations in one dimension. Associated with the two singular perturbation parameters 0 < ε μ 1 are boundary layers of length scales O(ε) and O(μ). We propose and analyse an hp finite element scheme which includes elements of size O(εp) and O(μp) near the boundary, where p is the degree of the approximating polynomials. We show that under the assumption of analytic input data, the method yields exponential rates of convergence, independently of ε and μ and independently of the relative size of ε to μ. In particular, the full range 0 < ε μ 1 is covered by our analysis. Numerical computations supporting the theory are also presented.
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(s) to have boundary layers which overlap and interact, based on the relative size of ε and µ. We construct full asymptotic expansions together with error bounds that cover the complete range 0 < ε ≤ µ ≤ 1. For the present case of analytic input data, we derive derivative growth estimates for the terms of the asymptotic expansion that are explicit in the perturbation parameters and the expansion order.
We consider the approximation of a coupled system of two singularly perturbed reaction-diffusion equations by the finite element method. The solution to such problems contains boundary layers which overlap and interact, and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters. We present results on a high order hp finite element scheme which includes elements of size O(εp) and O(µp) near the boundary, where ε, µ are the singular perturbation parameters and p is the degree of the approximating polynomials. Under the assumption of analytic input data, the method yields exponential rates of convergence as p → ∞, independently of ε and µ.
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