Error estimates for a vorticity-based velocity-stress formulation of the Stokes eigenvalue problem

TL;DR

This paper introduces a new mixed finite element method for the 2D Stokes eigenvalue problem using stress and velocity, providing error estimates, spectral correctness, and adaptive algorithms validated through numerical tests.

Contribution

It develops a vorticity-based mixed formulation with Nédélec elements, derives convergence and spectral correctness, and proposes an adaptive scheme with an error estimator.

Findings

The method converges with optimal order.

The spectral problem is solved accurately.

The adaptive scheme improves computational efficiency.

Abstract

The aim of this paper is to analyze a mixed formulation for the two dimensional Stokes eigenvalue problem where the unknowns are the stress and the velocity, whereas the pressure can be recovered with a simple postprocess of the stress. The stress tensor is written in terms of the vorticity of the fluid, leading to an alternative mixed formulation that incorporates this physical feature. We propose a mixed numerical method where the stress is approximated with suitable N\'edelec finite elements, whereas the velocity is approximated with piecewise polynomials of degree $k\geq 0$. With the aid of the compact operators theory we derive convergence of the method and spectral correctness. Moreover, we propose a reliable and efficient a posteriori error estimator for our spectral problem. We report numerical tests in different domains, computing the spectrum and convergence orders, together with a computational analysis for the proposed estimator. In addition, we use the corresponding error estimator to drive an adaptive scheme, and we report the results of a numerical test, that allow us to assess the performance of this approach.