Projection-Based Finite Elements for Nonlinear Function Spaces
Provides a rigorous theoretical foundation for finite element methods on nonlinear manifolds, benefiting computational geometry and PDEs on manifolds.
The paper introduces projection-based finite elements for functions valued in nonlinear manifolds, proving optimal interpolation and discretization error bounds for harmonic maps, with numerical verification.
We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embedding space, and then projecting pointwise onto the manifold. We show optimal interpolation error bounds with respect to Lebesgue and Sobolev norms. Additionally, we show similar bounds for the test functions, i.e., variations of discrete functions. Combining these results with a nonlinear Céa lemma, we prove optimal $L^2$ and $H^1$ discretization error bounds for harmonic maps from a planar domain into a smooth manifold. All these error bounds are also verified numerically.