Projection-Based Finite Elements for Nonlinear Function Spaces

TL;DR

This paper introduces a new projection-based finite element method for approximating functions into nonlinear manifolds, providing optimal error bounds and numerical verification for harmonic maps.

Contribution

It develops a novel approximation space using polynomial interpolation and projection for nonlinear manifold-valued functions, with proven optimal error bounds.

Findings

Optimal interpolation error bounds in Lebesgue and Sobolev norms.

Error bounds for variations of discrete functions.

Numerical verification of theoretical error bounds.

Abstract

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 C\'ea 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.