Study of the numerical solutions for the Electrical Impedance Equation in the plane: A pseudoanalytic approach of the forward Dirichlet boundary value problem

TL;DR

This paper introduces a pseudoanalytic function theory-based numerical method to solve the Electrical Impedance Equation's Dirichlet problem in the plane, effectively handling arbitrary conductivities and non-smooth domains.

Contribution

It develops a novel pseudoanalytic approach for numerically solving the forward Dirichlet problem with arbitrary conductivity functions, including non-smooth domains, without extra regularization.

Findings

Method accurately approximates solutions in smooth domains.

Converges in non-smooth domains under certain conditions.

Applicable to arbitrary non-vanishing conductivity functions.

Abstract

Employing a limiting case of a conjecture for constructing piecewise separable-variables functions, the elements of the Pseudoanalytic Function Theory are used for numerically approaching solutions of the forward Dirichlet boundary value problem, corresponding to the Electrical Impedance Equation in the plane, when the electrical conductivity is an arbitrary non-vanishing function, fully defined within a bounded domain. The new method is studied considering a variety of examples when the bounded domain coincides with the unit circle, and it is also included a description of its behaviour in non-smooth domains, selecting special cases that do not require additional regularization techniques, for warranting the convergence of the approach at the non-smooth regions, when certain requirements are fulfilled.