9403392 Allgower Many problems in science display symmetries with respect to a group of transformations. The equations describing such problems are then said to be `equivariant' with respect to this symmetry group. The focus of this project is to continue to study systematic techniques for exploiting symmetry in the numerical solution of such equivariant equations. This exploitation leads to more stable solution methods and to a greatly increased numerical efficiency. One of the methods studied by the investigators (the symmetry reduction method) can be viewed as a generalization of Fourier Transform methods (which exploit symmetry under a cyclic group). Among other projects, the investigators further develop the symmetry reduction method and study hidden symmetries, optimal equivariant preconditioners for problems that are only `nearly'equivariant, domain reduction via symmetries, the numerical solution of nonlinear equivariant equations and bifurcation phenomena. Preliminary numerical codes are further developed into a collection of codes for a range of important applications. Geometric symmetries have always played an important role in theoretical physics and some branches of applied mathematics. However, exploiting these symmetries has only lately been developed in the areas of numerical analysis and computational mathematics. The aim of this project is to use geometric and algebraic techniques to give efficient algorithms for solving large numerical problems that enjoy such symmetries. Very large problems are broken down (via symmetry considerations) into smaller problems that can be handled independently and more efficiently on modern architectures of high-performance computing. Problems that are only `nearly' equivariant (i.e., where certain symmetry structures have imperfections) occur quite frequently in engineering (e.g., in elasticity or flow problems, or in structural mechanics). An efficient exploitation of these imperfect sym metries would reduce the computational effort for solving such problems. This has two direct benefits: firstly, one can solve moderately sized problems much faster, possibly in real time; secondly, very large problems that are inaccessible via standard methods can be solved with this symmetry reduction method.
