The mixed discontinuous Galerkin method for the Oseen eigenvalue problem

TL;DR

This paper develops and analyzes a mixed discontinuous Galerkin method for solving the non-self-adjoint Oseen eigenvalue problem, providing optimal error estimates and effective adaptive algorithms validated by numerical experiments.

Contribution

It introduces an adjoint-consistent DG formulation with optimal error estimates and residual-based a posteriori error estimators for the Oseen eigenvalue problem.

Findings

The scheme achieves high-accuracy eigenvalue approximations.

Error estimators are reliable and effective.

Numerical results confirm computational efficiency.

Abstract

The Oseen eigenvalue problem plays a important role in the stability analysis of fluids. The problem is non-self-adjoint due to the presence of convection field. In this paper, we present a comprehensive investigation of the mixed discontinuous Galerkin (DG) method, employing Pk-Pk-1(k>=1) elements to solve the Oseen eigenvalue problem in Rd(d=2,3). We first develop an adjoint-consistent DG formulation for the problem. We then derive optimal a priori error estimates for the approximate eigenpairs, and propose residual type a posteriori error estimators. Furthermore, we prove the reliability and effectiveness of these estimators for approximate eigenfunctions, as well as the reliability of the estimator for approximate eigenvalues. To validate our approach, we conduct numerical computations on both uniform and adaptively refined meshes. The numerical results demonstrate that our scheme is computationally efficient and capable of yielding high-accuracy approximate eigenvalues.