A Tailored Finite Point Method for a Singular Perturbation Problem on an Unbounded Domain

• Published in: Journal of scientific computing Vol. 36; no. 2; pp. 243 - 261
• Main Authors: Han, Houde, Huang, Zhongyi, Kellogg, R. Bruce
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.08.2008; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Applications of mathematics; Boundaries; Computation; Computational Mathematics and Numerical Analysis; Construction; Finite element method; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Singular perturbation; Theoretical; Tailored finite point method; Unbounded domain; Singular perturbation problem; Artificial boundary condition
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-008-9187-7

In this paper, we propose a tailored-finite-point method for a kind of singular perturbation problems in unbounded domains. First, we use the artificial boundary method (Han in Frontiers and Prospects of Contemporary Applied Mathematics, [ 2005 ]) to reduce the original problem to a problem on bounded computational domain. Then we propose a new approach to construct a discrete scheme for the reduced problem, where our finite point method has been tailored to some particular properties or solutions of the problem. From the numerical results, we find that our new methods can achieve very high accuracy with very coarse mesh even for very small ε . In the contrast, the traditional finite element method does not get satisfactory numerical results with the same mesh.