Open Source or free electric field mapping software
QuickField Student edition - 2D finite element package - Windows

Trace Analyzer - Transmission line parameter calculator for PCB's - Win/Mac/Linux

SATE - Static Field Analysis Toolkit - 2D E-Field - Windows

Radia - 3D Static magnetic fields - Mathematica plugin

PDN Mesh - 2D field problems (Poisson and Helmholtz Equations). - Windows

OpenEMS - open electromagnetic field solver using the FDTD method. Matlab or Octave are used as an easy and flexible scripting interface. - Win/Linux

NEC2 - (Method of Moments) simulates the electromagnetic response of antennas and metal structures. - Win?linux

MMP - The Multiple Multipole Program (MMP) - Win/Unix

MEEP - Meep (or MEEP) is a free finite-difference time-domain (FDTD) simulation software package developed at MIT to model electromagnetic systems. - Win/Linux

LC - LC is simulation tool for the analysis of the electromagnetic properties of electrical interconnects. - Linux/Unix compile

EM Explorer - EM Explorer is a 3D electromagnetic (EM) solver for scattering problems of periodic structures illuminated by arbitrary incident fields including planewaves, Gaussian beams and focused Hi-NA beams - Windows

Atribution: Blurbs copied from respective web sites.