Designed basis functions for speed and stability

Putting physical information into a numerical solution method can be used to produce fast solution methods. These methods can be extended to improve the stability of the solution procedure and answer. Iterative solution that requires a stable (well conditioned) formulation. However, truly large problems are also sensitive to stability issues. This paper shows a range of methods that may be used to make truly large problems manageable. Some methods improve how local interactions are handled and improve stability for dense meshes. Others improve how distant interactions are treated and improve stability when the problem size measured in wavelengths increases.