The purpose of this workshop is to bring together a group of researchers at the forefront of development of new numerical approximation schemes and fast solution methods for the resulting discrete equations, based on the approach of structure preserving discretizations, with the goals of fostering communication and laying out promising future directions for research. Occuring ten years after a successful "Hot Topics" workshop at the IMA on a related topic, and building on related conferences held later at the Centre of Mathematics for Applications at the University of Oslo, this workshop will focus on further research progress that has occurred since those meetings.
Progress in structure preserving discretizations has included the development of the finite element exterior calculus and its applications (e.g., to the equations of linear elasticity, the stationary Stokes equations, the Hodge Laplacian, problems on hypersurfaces, and a posteriori error estimation). Advances have also occurred in the development of other approaches to numerical discretization that also preserve some of the properties of a PDE, (e.g.,the discrete exterior calculus and mimetic finite difference schemes). Tools from topology and geometry, such as de Rham cohomology and Hodge theory, play an important role in understanding the well-posedness of the continuous problem, and it is natural that their discrete versions should aid in the development of stable numerical discretization schemes.
Part of the purpose of this workshop is to communicate how these and other tools from what has been traditionally considered pure mathematics can play an important role in numerical analysis. This workshop will also be an occasion to celebrate the 60th birthday of Douglas N. Arnold, whose work has made profound contributions to the field of numerical analysis of partial differential equations.
A meeting of the Finite Element Circus, devoted to the theory and applications of the finite element method and related areas of numerical analysis and partial differential equations, will immediately follow the IMA Special Workshop, taking place on October 24-25, 2014 at the same location.