A mathematical model of steady state heat conduction consisting of an elliptic nature in a composite medium is the object of interest in the present article which arises in various streams of science and engineering. By steady we mean that temperatures are constant with time, as the result, the heat flow is also constant with time and by one dimensional we mean that temperature is a function of a single dimension or spatial coordinate. In this article, we have proposed a numerical approach based on extended finite element method on uniform mesh for this interface problem with second order accuracy in L²-norm and first order accuracy in H1-norm. To show the superiority of our approach we have solved two interface problems with interface at different positions and discuss the behavior of our solutions.

elliptic interface problems, extended finite element method, steady state heat conduction, immersed interface method.