Abstract

Partial differential equations with a nonlinear pointwise constraint defined by a manifold occur in a variety of applications: the magnetization of a ferromagnet can be described by a unit length vector field and the orientation of the rod-like molecules that constitute a liquid crystal is often modeled by a restricted vector field. Other applications arise in geometric modeling, nonlinear bending of solids, and quantum mechanics. Nodal finite element methods have to appropriately relax the pointwise constraint leading to a variational crime. Since exact solutions are typically nonunique and do not admit higher regularity properties, the correctness of discretizations has to be established by weaker means avoiding unrealistic conditions. The iterative solution of the nonlinear systems of equations can be based on linearizations of the constraint or by using appropriate constraint-preserving reformulations. The talk focusses on the approximation of harmonic maps and wave maps. The latter arise as a model problem in general relativity.