Isogeometric Analysis for singularly perturbed problems in 1-D: a numerical study

TL;DR

This paper investigates the use of isogeometric analysis with B-splines for 1-D singularly perturbed reaction-convection-diffusion problems, providing guidelines for knot placement to ensure uniform exponential convergence.

Contribution

It offers practical guidelines for knot selection in isogeometric analysis to achieve optimal convergence in singularly perturbed problems.

Findings

Guidelines for knot placement to ensure uniform convergence

Demonstration of exponential convergence in numerical experiments

Application to three different singularly perturbed problems

Abstract

We perform numerical experiments on one-dimensional singularly perturbed problems of reaction-convection-diffusion type, using isogeometric analysis. In particular, we use a Galerkin formulation with B-splines as basis functions. The question we address is: how should the knots be chosen in order to get uniform, exponential convergence in the maximum norm? We provide specific guidelines on how to achieve precisely this, for three different singularly perturbed problems.