Obtaining Potential Field Solution with Spherical Harmonics and Finite   Differences
  Potential magnetic field solutions can be obtained based on the synoptic magnetograms of the Sun. Traditionally, a spherical harmonics decomposition of the magnetogram is used to construct the current and divergence free magnetic field solution. This method works reasonably well when the order of spherical harmonics is limited to be small relative to the resolution of the magnetogram, although some artifacts, such as ringing, can arise around sharp features. When the number of spherical harmonics is increased, however, using the raw magnetogram data given on a grid that is uniform in the sine of the latitude coordinate can result in inaccurate and unreliable results, especially in the polar regions close to the Sun.   We discuss here two approaches that can mitigate or completely avoid these problems: i) Remeshing the magnetogram onto a grid with uniform resolution in latitude, and limiting the highest order of the spherical harmonics to the anti-alias limit; ii) Using an iterative finite difference algorithm to solve for the potential field. The naive and the improved numerical solutions are compared for actual magnetograms, and the differences are found to be rather dramatic.   We made our new Finite Difference Iterative Potential-field Solver (FDIPS) a publically available code, so that other researchers can also use it as an alternative to the spherical harmonics approach. 
Gabor Toth, Bart van der Holst, and Zhenguang Huang
astro-ph.SR