Abstract
A singularly perturbed reaction-diffusion equation in two dimensions is considered. We assume analyticity of the input data, i.e., the boundary of the domain is an analytic curve and the right-hand side is analytic. We show that the hp version of the finite element method leads to robust exponential convergence provided that one layer of needle elements of width $O(p \eps)$ is inserted near the domain boundary, that is, the rate of convergence is $O(\mbox{\rm exp} (-b p))$ and independent of the perturbation parameter $\eps$. Additionally, we show that the use of numerical quadrature for the evaluation of the stiffness matrix and the load vector retains the exponential rate of convergence. In particular, the spectral element method based on the use of a Gauss--Lobatto quadrature rule with (p+1)x (p+1) points yields robust exponential convergence.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
