Parameter uniform fitted mesh finite difference scheme for elliptical singularly perturbed problems with mixed shifts in two dimensions

Abstract

In this work, we embarked on the research of linear two-dimensional elliptic singularly perturbed problems having positive and negative shifts in both directions of the reaction terms, whose solution exhibits characteristic boundary layers. The study of this class of problems was started in the mid-eighties, but all the studies were restricted to 1D. The Taylor series expansion is used to estimate decelerated terms of the problem and the emerging problem is discretized using the fitted mesh finite difference method to establish parameter uniform error estimates. The effect of positive and negative shifts on the behaviour of the solution is explained by executing numerical experiments on two test examples.

Keywords:

• Singularly perturbed problem
• characteristic boundary layer
• positive shift
• negative shift
• Shishkin mesh
• fitted mesh scheme
• ϵ-uniform convergence

AMS Subject Classifications:

• 65N22
• 35J25
• 39A06
• 39A14

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author acknowledges the financial support received from the Council of Scientific and Industrial Research (File No. 09/1112(0006)/2018-EMR-I) in the form of Senior Research Fellowship.