Harmonic average method has been widely utilized to deal with heterogeneous coefficients in solving differential equations. One remarkable advantage of the harmonic averaging method is that no derivative of the coefficient is needed. Furthermore, the coefficient matrix of the finite difference equations is an M-matrix which guarantees the stability of the algorithm. It has been numerically observed but not theoretically proved that the method produces second order pointwise accuracy when the solution and flux are continuous even if the coefficient has finite discontinuities for which the method is inconsistent ($O(1)$ in the local truncation errors). It has been believed that there are some fortunate error cancellations. The harmonic average method does not converge when the solution or the flux has finite discontinuities. In this paper, not only we rigorously prove the second order convergence of the harmonic averaging method for one-dimensional interface problem when the coefficient has a finite discontinuities and the solution and the flux are continuous, but also proposed an {\em improved harmonic average method} that is also second order accurate (in the $L^{\infty}$ norm), which allows discontinuous solutions and fluxes along with the discontinuous coefficients. The key in the convergence proof is the construction of the Green's function. The proof shows how the error cancellations occur in a subtle way. Numerical experiments in both 1D and 2D confirmed the theoretical proof of the improved harmonic average method.

翻译：调和平均法已被广泛用于处理微分方程求解中的异质系数问题。该方法的一个显著优点是不需要系数的导数。此外，有限差分方程的系数矩阵为M矩阵，这保证了算法的稳定性。数值实验观察到（但未得到理论证明）：即使系数存在有限间断点（此时方法不一致，局部截断误差为$O(1)$），当解和通量连续时，该方法仍能产生二阶逐点精度。学界普遍认为存在某种误差的幸运抵消。当解或通量存在有限间断时，调和平均法不收敛。本文不仅严格证明了一维界面问题中，当系数存在有限间断而解与通量连续时，调和平均法具有二阶收敛性；还提出了一种改进的调和平均方法，该方法同样具有二阶精度（在$L^{\infty}$范数意义下），且允许解和通量随系数同时出现间断。收敛性证明的关键在于格林函数的构造。该证明揭示了误差如何以精妙的方式实现抵消。一维与二维数值实验均验证了改进调和平均法的理论证明。