arXiv subjects

Hanne Hardering

Publications and source records attributed to Hanne Hardering.

9 recordsLinked to original sources

arXiv2026

Odd behaviour of even geometries: an explanation for superconvergent geometric consistency errors

Piecewise polynomial surface approximations used in surface finite element methods often seem to behave better than their standard approximation properties suggest if their polynomial order is even. We explain this superconvergence through cancellation of leading interpolation errors on suitably structured meshes that naturally arise in some refinement processes. This cancellation improves weighted integral estimates for functions, derivatives, and geometric quantities. Applications include estimates for surface normals, the Weingarten map, and Gaussian curvature. Numerical experiments reproduce the predicted parity-dependent behaviour and support the proposed explanation of superconvergent geometric consistency errors, while the corresponding pointwise errors retain their standard orders.

math.NA↗

arXiv2025

Surface Minkowski tensors to characterize shapes on curved surfaces

We introduce surface Minkowski tensors to characterize rotational symmetries of shapes embedded in curved surfaces. The definition is based on a modified vector transport of the shapes boundary co-normal into a reference point which accounts for the angular defect that a classical parallel transport would introduce. This modified transport can be easily implemented for general surfaces and differently defined embedded shapes, and the associated irreducible surface Minkowski tensors give rise to the classification of shapes by their normalized eigenvalues, which are introduced as shape measures following the flat-space analog. We analyze different approximations of the embedded shapes, their influence on the surface Minkowski tensors, and the stability to perturbations of the shape and the surface. The work concludes with a series of numerical experiments showing the applicability of the approach on various surfaces and shape representations and an application in biology in which the characterization of cells in a curved monolayer of cells is considered.

math.NA↗

arXiv2023

Parametric Finite Element Discretization of the Surface Stokes Equations

We study a higher-order surface finite element (SFEM) penalty-based discretization of the tangential surface Stokes problem. Several discrete formulations are investigated which are equivalent in the continuous setting. The impact of the choice of discretization of the diffusion term and of the divergence term on numerical accuracy and convergence, as well as on implementation advantages, is discussed. We analyze the inf-sup stability of the discrete scheme in a generic approach by lifting stable finite element pairs known from the literature. A discretization error analysis in tangential norms then shows optimal order convergence of an isogeometric setting that requires only geometric knowledge of the discrete surface.

math.NA↗

arXiv2021

Tangential Errors of Tensor Surface Finite Elements

We discretize a tangential tensor field equation using a surface-finite element approach with a penalization term to ensure almost tangentiality. It is natural to measure the quality of such a discretization intrinsically, i.e., to examine the tangential convergence behavior in contrast to the normal behavior. We show optimal order convergence with respect to the tangential quantities in particular for an isogeometric penalization term that is based only on the geometric information of the discrete surface.

math.NA↗

arXiv2020

Quartic $L^p$-convergence of cubic Riemannian splines

We prove quartic convergence of cubic spline interpolation for curves into Riemannian manifolds as the grid size of the interpolation grid tends to zero. In contrast to cubic spline interpolation in Euclidean space, where this result is classical, the interpolation operator is no longer linear. Still, concepts from the linear setting may be generalized to the Riemannian case, where we try to use intrinsic Riemannian formulations and avoid charts as much as possible.

math.NA↗

arXiv2018

Sparips

Persistent homology of the Rips filtration allows to track topological features of a point cloud over scales, and is a foundational tool of topological data analysis. Unfortunately, the Rips-filtration is exponentially sized, when considered as a filtered simplicial complex. Hence, the computation of full persistence modules is impossible for all but the tiniest of datasets; when truncating the dimension of topological features, the situation becomes slightly less intractable, but still daunting for medium-sized datasets. It is theoretically possible to approximate the Rips-filtration by a much smaller and sparser, linear-sized simplicial complexs, however, possibly due to the complexity of existing approaches, we are not aware of any existing implementation. We propose a different sparsification scheme, based on cover-trees, that is easy to implement, while giving similar guarantees on the computational scaling. We further propose a visualization that is adapted to approximate persistence diagrams, by incorporating a variant of error bars and keeping track of all approximation guarantees, explicitly.

cs.CG↗

arXiv2018

Projection-Based Finite Elements for Nonlinear Function Spaces

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.

math.NA↗

arXiv2017

The Aubin--Nitsche Trick for Semilinear Problems

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.

math.NA↗

arXiv2016

$L^{2}$-Discretization Error Bounds for Maps into Riemannian Manifolds

We study the approximation of functions that map a Euclidean domain $\Omega\subset \mathbb{R}^{d}$ into an $n$-dimensional Riemannian manifold $(M,g)$ minimizing an elliptic, semilinear energy in a function set $H\subset W^{1,2}(\Omega,M)$. 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.

math.NA↗