Tailored finite point method for the interface problem

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