Construction of Multi-Resolution Wavelet Based Mesh Free Method in Solving Poisson and Imaginary Helmholtz Problem

In this paper, we propose a new multi-resolution wavelet based mesh free method for numerical analysis of electromagnetic field problems. In problems with variable object geometries or mechanical movements, the mesh free methods yield more accurate simulation results compared to the finite element approach in solving the inverse problem, because they are based on a set of nodes without using the connectivity of the elements. The wavelet based mesh free method requires effectively no local integration in the vicinity of nodes in numerical implementations. Moreover, wavelets give a more efficient approximation using multi-resolution analysis. On the other hand, boundary condition constraints are difficult to be applied on the wavelet based mesh free method. In order to apply boundary and interface conditions, we utilize a new form of jump functions in the set of basic functions. The boundary and interface conditions are applied effectively using the suggested slope jump functions. The simulation results of the proposed method using two different jump functions in solving some simple boundary problems are compared. The results are validated by analytical solutions. The results of this study can be used in future for inverse problem of Magnetic resonance electrical impedance tomography (MREIT) studies as an imaging technique for reconstructing the cross-sectional conductivity distribution of a human brain or body using EIT technique integrated with the MRI.