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Materials such as ferromagnets, liquid crystals, and granular media involve orientation degrees of freedom. Mathematical descriptions of such materials involve fields of nonlinear objects such as unit vectors, rotations matrices, or unitary matrices.  Classical numerical methods like the finite element method cannot be applied in such situations, because linear and polynomial interpolation is not defined for such nonlinear objects.  Instead, a variety of heuristic approaches is used in the literature, which are difficult to analyze rigorously.  We present nonlinear generalizations of the finite element method that allow to treat problems with orientation degrees of freedom in a mathematically sound way.  This allows
to show solvability of the discrete problems, makes the construction of efficient solvers easier, and allows to obtain reliable bounds for the finite element approximation error.  We use the technique to calculate stable configurations of chiral magnetic skyrmions, and wrinkling patterns of a thin elastic polyimide film.