# Edge and Nodal finite element methods in MATLAB for Magnetic induction tomography

What is the difference between edge finite elements and nodal finite elements?

This for use in modeling the eddy current problem in classical electromagnetism. I am attempting to convert MATLAB code written for Electrical Impedance Tomography (EIT) into a code for Magnetic Induction Tomography (MIT). EIT involves electrodes with applied alternating currents and requires nodal FEM. Whereas MIT involves coils and induction of eddy currents and requires edge FEM.

The code for both EIT and MIT involves solving the forward and inverse problems in a simulation for tomographic image reconstruction of conductivity maps.

The MATLAB code for conversion is found at EIDORS:
http://eidors3d.sourceforge.net/index.shtml

• Have you checked chapter 3 from the FEnICs book? – nicoguaro Feb 6 '17 at 22:53
• No I haven't I will check it out. – Brendan Darrer Feb 7 '17 at 9:50

## 1 Answer

The paper: I. Anjam, J. Valdman, "Fast MATLAB assembly of FEM matrices in 2D and 3D: Edge elements", Applied Mathematics and Computation, 267, (2015), 252–263; states the following. http://www.sciencedirect.com/science/article/pii/S0096300315004191

"A finite element discretization is done in terms of edge elements, typically Raviart–Thomas elements  for H(div) problems and Nédélec elements  for H(curl) problems. Edge element basis functions are not defined on the nodes of 2D triangular or 3D tetrahedral meshes, but on edges and faces. Edge elements provide only partial continuity over element boundaries: continuity of normal vector component for H(div) problems and continuity of tangential vector component for H(curl) problems."

They give a figure for this representation. Edge elements are derived from nodal shape functions:

"In Edge FEM on a tetrahedral mesh, a vector field is represented using a basis vector function" (1):

$N_{ij} = L_i\bigtriangledown L_j - L_j\bigtriangledown L_i$ ____________ (1)

The linear shape function of the node $i$ has the form,

$L_i (x,y,z) = a_i + b_i x + c_i y + d_i z$ 