# A least-squares collocation method for nonlinear higher-index differential-algebraic equations

## TL;DR

This paper presents a novel least-squares collocation method for solving nonlinear higher-index differential-algebraic equations, demonstrating promising numerical results and establishing initial convergence theory for nonlinear cases.

## Contribution

It introduces a direct numerical approach using overdetermined polynomial least-squares collocation for nonlinear higher-index DAEs, with the first convergence proof for nonlinear problems.

## Key findings

- Numerical experiments show impressive results.
- The method is not much more expensive than standard collocation.
- First convergence proof for nonlinear problems.

## Abstract

We introduce a direct numerical treatment of nonlinear higher-index differential-algebraic equations by means of overdetermined polynomial least-squares collocation. The procedure is not much more computationally expensive than standard collocation methods for regular ordinary differential equations. The numerical experiments show impressive results. In contrast, the theoretical basic concept turns out to be considerably challenging. So far, quite recently convergence proofs for linear problems have been published. In the present paper we come up to a first convergence result for nonlinear problems.