Asymptotics and Computation of the Solution to the Conductivity Equation in the Presence of Adjacent Inclusions with Extreme Conductivities

TL;DR

This paper explicitly characterizes the singular behavior of solutions to the conductivity equation with adjacent extreme-inclusion conductivities and demonstrates efficient numerical computation of the gradient.

Contribution

It provides an explicit characterization of the singular term for solutions with adjacent extreme conductivities and applies this for improved numerical gradient computation.

Findings

Explicit singular term characterization for adjacent extreme conductivities

Numerical methods leveraging the singular term for efficient gradient computation

Demonstration of the approach's effectiveness through numerical experiments

Abstract

When inclusions with extreme conductivity (insulator or perfect conductor) are closely located, the gradient of the solution to the conductivity equation can be arbitrarily large. And computation of the gradient is extremely challenging due to its nature of blow-up in a narrow region in between inclusions. In this paper we characterize explicitly the singular term of the solution when two circular inclusions with extreme conductivities are adjacent. Moreover, we show through numerical computations that the characterization of the singular term can be used efficiently for computation of the gradient in the presence adjacent inclusions.