Geometric Finite Elements

Geometric finite elements (GFE) generalize the idea of Galerkin methods to variational problems for unknowns that map into nonlinear spaces. In particular, GFE methods introduce proper discrete function spaces that are conforming in the sense that values of geometric finite element functions are in the codomain manifold ℳ $$\mathcal {M}$$ at any point. Several types of such spaces have been constructed, and some are even completely intrinsic, i.e., they can be defined without any surrounding space. GFE spaces enable the elegant numerical treatment of variational problems posed in Sobolev spaces with nonlinear codomain space. Indeed, as GFE spaces are geometrically conforming, such variational problems have natural formulations in GFE spaces. These correspond to the discrete formulations of classical finite element methods. Also, the canonical projection onto the discrete maps commutes with the differential for a suitable notion of the tangent bundle as a manifold, and we therefore also obtain natural weak formulations. Rigorous results exist that show the optimal behavior of the a priori L 2 and H 1 errors under reasonable smoothness assumptions. Although the discrete function spaces are no vector spaces, their elements can nevertheless be described by sets of coefficients, which live in the codomain manifold. Variational discrete problems can then be reformulated as algebraic minimization problems on the set of coefficients. These algebraic problems can be solved by established methods of manifold optimization. This text will explain the construction of several types of GFE spaces, discuss the corresponding test function spaces, and sketch the a priori error theory. It will also show computations of the harmonic maps problem, and of two example problems from nanomagnetics and plate mechanics.