In recent years computational geometry and numerical analysis have been put in close contacts in fields such as computer aided design (CAD) and scientific computing. We will investigate the different approaches to discretization in differential respectively computational geometry and in finite element analysis. Although some attempts have been made to put concepts on a common ground the focus in geometry has traditionally been on the discretization of shapes and exact integrability while the view in finite element analysis was directed more on the discretization of function spaces and approximation issues. The research of this project will concentrate on an improved linking of both aspects of discretization techniques to gain better insight and improve integration of technologies.

The project is based on combined efforts in discrete differential geometry and finite element methods for geometric partial differential equations. Especially, discretizations using polyhedral surfaces and piecewise linear functions on them proved to be very successful in both theory and applications. This development led to the creation of counterparts of geometric and metric properties of smooth surfaces on polyhedral surfaces and to insights on convergence properties of them. Some of these questions, e.g. the convergence of surface area, are already more 100 years old (cf the Lantern of Schwarz). A geometric view onto finite element spaces of piecewise linear functions helped to develop a consistent theory of discrete differential forms on polyhedral surfaces, where discrete analogs to important theorems, like the Hodge decomposition hold exactly (and not only in the limit of refinement). For applications in engineering, computer aided design, and computer graphics the discretization of the Laplace-Beltrami operator of surface to polyhedral surfaces is a prominent examples. It has been used in various applications including physical simulation, parametization, geometric modeling, shape analysis, and surface optimization.

The long term aim of this project is to extend the geometric view onto finite element constructions to higher order elements, constructed from subdivision processes as well as from NURBS surface representations. As one of the starting points we will consider the geometric problems of constructing minimal and cmc surfaces in this setting.

A related recent development in the computer aided design community is isogeometric analysis, where also NURBS and subdivision schemes are used to build discrete function spaces, e.g. finite element spaces. Since the focus in isogeometric analysis is less on differential geometric problems we see interesting contact points.