Tailored finite point method for the interface problem

In this paper, we propose a tailored-finite-point method for a numerical simulation of the second order elliptic equation with discontinuous coefficients. Our finite point method has been tailored to some particular properties of the problem, then we can get the approximate solution with the same behaviors as that of the exact solution very naturally. Especially, in one-dimensional case, when the coefficients are piecewise linear functions, we can get the exact solution with only one point in each subdomain. Furthermore, the stability analysis and the uniform convergence analysis in the energy norm are proved. On the other hand, our computational complexity is only $\O(N)$ for $N$ discrete points. We also extend our method to two-dimensional problems.