# Weakly-normal basis vector fields in RKHS with an application to shape Newton methods

## TL;DR

This paper develops a novel RKHS-based basis for vector fields normal to curves, enabling discretization of shape Newton methods and analysis of their convergence properties.

## Contribution

It introduces a new RKHS-based basis for normal vector fields and applies it to discretize shape Newton methods, analyzing their convergence.

## Key findings

- Basis functions are effectively approximated.
- Discretized shape Newton methods converge reliably.
- Impact of discretization on convergence rates is characterized.

## Abstract

We construct a space of vector fields that are normal to differentiable curves in the plane. Its basis functions are defined via saddle point variational problems in reproducing kernel Hilbert spaces (RKHSs). First, we study the properties of these basis vector fields and show how to approximate them. Then, we employ this basis to discretise shape Newton methods and investigate the impact of this discretisation on convergence rates.