The mixed discontinuous Galerkin method for the Oseen eigenvalue problem

TL;DR

This work develops a high-order mixed discontinuous Galerkin method for the non-self-adjoint Oseen eigenvalue problem, including an adjoint-consistent formulation, a priori error estimates, and residual-based a posteriori estimators. An adaptive algorithm based on Dörfler marking is designed to efficiently resolve eigenpairs in 2D and 3D domains. The authors prove key stability and convergence properties for the DG scheme and demonstrate, through extensive numerical experiments, that the adaptive method yields high-accuracy eigenvalues with superior efficiency compared to Mini-element approaches. The results highlight the method's potential for stable, accurate eigenvalue computations in fluid stability analyses and Navier–Stokes solvers.

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.

Figures (6)

• Figure 1: The error curves on adaptive refinement meshes of the first four eigenvalues using Algorithm 1 on L-shaped domain for $\beta(x,y)=\beta_{1}(x,y)$ and $\mu=1$ (left: DG $\mathbb{P}_{2}-\mathbb{P}_{1}$ with $\gamma=40$; right: DG $\mathbb{P}_{3}-\mathbb{P}_{2}$ with $\gamma=90$).
• Figure 2: The error curves on adaptive refinement meshes of the first four eigenvalues using Algorithm 1 on L-shaped domain for $\beta(x,y)=\beta_{2}(x,y)$ and $\mu=1$ (left: DG $\mathbb{P}_{2}-\mathbb{P}_{1}$ with $\gamma=40$; right: DG $\mathbb{P}_{3}-\mathbb{P}_{2}$ with $\gamma=90$).
• Figure 3: The error curves on adaptive refinement meshes of the first four eigenvalues using Algorithm 1 on L-shaped domain for $\beta(x,y)=\beta_{3}(x,y)$ and $\mu=1$ (left: DG $\mathbb{P}_{2}-\mathbb{P}_{1}$ with $\gamma=40$; right: DG $\mathbb{P}_{3}-\mathbb{P}_{2}$ with $\gamma=90$).
• Figure 4: The error curves on adaptive refinement meshes of the first four eigenvalues using Algorithm 1 on L-shaped domain for $\beta(x,y)=\beta_{4}(x,y)$ and $\mu=1$ (left: DG $\mathbb{P}_{2}-\mathbb{P}_{1}$ with $\gamma=40$; right: DG $\mathbb{P}_{3}-\mathbb{P}_{2}$ with $\gamma=90$.
• Figure 5: The error curves on adaptive refinement meshes for the first four eigenvalues on the unit cube domain for $\beta=(0,0,1)^{t}$ and $\mu=1$ (left: DG $\mathbb{P}_{2}-\mathbb{P}_{1}$ with $\gamma=50$ ; right: DG $\mathbb{P}_{3}-\mathbb{P}_{2}$ with $\gamma=150$).

Theorems & Definitions (16)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.5
• Theorem 2.6
• Lemma 2.7
• Theorem 2.8
• Theorem 3.1
• Lemma 3.2