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

TL;DR

This paper addresses elliptic interface problems with discontinuous coefficients and, in general, discontinuous solutions or fluxes. It provides a rigorous proof of second-order $L^{\infty}$ convergence for the classical harmonic average method in 1D under homogeneous jump conditions and introduces an improved derivative-free harmonic average finite difference method that retains second-order accuracy even when jumps occur in the solution or flux. The proof leverages a second-kind Green's function and a carefully constructed correction at irregular grid points, demonstrating error cancellation mechanisms that yield $O(h^{2})$ global accuracy. The authors extend the improved method to 2D with axis-parallel interfaces and validate the theory through numerical experiments, showing robust, second-order convergence in both 1D and 2D settings, with the method producing an $M$-matrix system and remaining derivative-free throughout.

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.

Figures (2)

• Figure 2: (a), computed (little 'o's) and exact solution (solid line) when $k_1=30, k_2=5$ and $n=256$. (b), the error plot.
• Figure 3: (a), the computed solution when $\beta^-=1.5$, $\beta^+=3$, $k_1=5, k_2=3$ and $n=64 \times 64$. (b), the corresponding error plot.

Theorems & Definitions (12)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Lemma 1
• Lemma 2
• Theorem 1
• Remark 5
• Example 5.1
• Example 5.2