A New Meshless Method for Solving 3D Inverse Conductivity Issues of Highly Nonlinear Elliptic Equations

Abstract

In this research, the 3D inverse conductivity issues of highly nonlinear elliptic partial differential equations (PDEs) are investigated numerically. Even some researchers have utilized several schemes to overcome these multi-dimensional forward issues of those PDEs; nevertheless, an effective numerical algorithm to solve these 3D inverse conductivity issues of highly nonlinear elliptic PDEs is still not available. We apply two sets of single-parameter homogenization functions as the foundations for the answer and conductivity function to cope with the 3D inverse conductivity issue of highly nonlinear PDEs. The unknown conductivity function can be retrieved by working out another linear system produced from the governing equation by collocation skill, while the answer is acquired by dealing with a linear system to gratify over-specified Neumann boundary condition on a fractional border. As this new computational approach is based on a concrete theoretical foundation, it can result in a deeper understanding of 3D inverse conductivity issues with symmetry and asymmetry geometries. The homogenization functions method is rather stable, effective, and accurate in revealing the conductivity function when the over-specified Neumann data with a large level of noise of 28%.