Analysis of harmonic average method for interface problems with discontinuous solutions and fluxes

TL;DR

This paper rigorously proves the second order convergence of the harmonic average method for 1D interface problems with discontinuous coefficients and introduces an improved method that handles discontinuous solutions and fluxes, confirmed by numerical experiments.

Contribution

The paper provides the first rigorous proof of second order convergence for the harmonic average method with discontinuous coefficients and proposes an improved method for more general discontinuities.

Findings

Proved second order convergence for harmonic averaging with discontinuous coefficients and continuous solutions.

Developed an improved harmonic average method for discontinuous solutions and fluxes.

Numerical experiments in 1D and 2D validate the theoretical results.

Abstract

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.