Hanne Hardering

Philipp Grohs, Hanne Hardering, Oliver Sander et al.

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.

Hanne Hardering

We study the approximation of functions that map a Euclidean domain $Ω\subset \mathbb{R}^{d}$ into an $n$-dimensional Riemannian manifold $(M,g)$ minimizing an elliptic, semilinear energy in a function set $H\subset W^{1,2}(Ω,M)$. The approximation is given by a restriction of the energy minimization problem to a family of conforming finite-dimensional approximations $S_{h}\subset H$. We provide a set of conditions on $S_{h}$ such that we can prove a priori $W^{1,2}$- and $L^{2}$-approximation error estimates comparable to standard Euclidean finite elements. This is done in an intrinsic framework, independently of embeddings of the manifold or the choice of coordinates. A special construction of approximations ---geodesic finite elements--- is shown to fulfill the conditions, and in the process extended to maps into the tangential bundle.

Hanne Hardering

The Aubin--Nitsche trick is a common tool to show $L^2$-error estimates for discretizations of $H^1$-elliptic linear partial differential equations arising for example as Euler--Lagrange equations of a quadratic energy functional. The technique itself is linear: for quasilinear problems it is not applicable. We generalize the Aubin--Nitsche trick to a class of minimization problems closely related to semi-linear partial differential equations.