# New error estimates of linear triangle finite elements for the Steklov eigenvalue problem

## TL;DR

This paper develops new, optimal error estimates for linear finite element methods applied to the Steklov eigenvalue problem on concave polygons, improving existing theoretical results with supporting numerical experiments.

## Contribution

It introduces a novel proof approach leveraging regularity estimates and edge average interpolation, providing sharper error bounds for both conforming and nonconforming finite elements.

## Key findings

- Optimal error estimates in boundary norm for eigenfunctions
- Improved theoretical bounds over previous results
- Numerical experiments confirm theoretical predictions

## Abstract

In this paper we make a further discussion on the finite elements approximation for the Steklov eigenvalue problem on concave polygonal domain. We make full use of the regularity estimate and the characteristic of edge average interpolation operator of nonconforming Crouzeix-Raviart element, which is different from the existing proof argument, and prove a new and optimal error estimate in $\|\cdot\|_{0,\partial\Omega}$ for the eigenfunction of linear conforming finite element and the nonconforming Crouzeix-Raviart element, which is an improvement of the current results. Finally, we present some numerical experiments to support the theoretical analysis.