Many problems in science and mathematics may be transformed via a linear or nonlinear change of coordinates into related problems. A tranformation which carries a problem or equation into itself is called a *symmetry* of the equation. Usually the symmetry is derived from underlying geometric symmetries of the domain or body on which the problem is considered. The thoughtful incorporation of symmetry in the analysis and solution of such problems may result in enormous gains in insight and efficiency. The focus of this research is to study systematic techniques for exploiting symmetry in the application of numerical analysis to the construction of solutions for equations having non-trivial symmetry.
The numerical treatment of problems such as partial differential equations and integral equations generally involves discretizations. In this setting a general principle may be formulated: if a domain enjoys a symmetry group, and a differential or operator equation which expresses a coordinate-free physical law is to be solved on that domain, then (if the discretization is chosen to incorporate or respect the symmetries), the resulting numerical problem (which is usually a matrix equation) is amenable to a block decomposition which greatly reduces the computational cost of determining the solution.
This research is being conducted jointly with Profs. Gene Allgower and Kurt Georg of the Department of Mathematics here at Colorado State University.