A Mixed finite element method for the velocity-pseudostress formulation of the Oseen eigenvalue problem

Abstract

In this paper, we introduce and analyze a mixed formulation for the Oseen eigenvalue problem by introducing the pseudostress tensor as a new unknown, allowing us to eliminate the fluid pressure. The well-posedness of the solution operator is established using a fixed-point argument. For the numerical analysis, we use the tensorial versions of Raviart-Thomas and Brezzi-Douglas-Marini elements to approximate the pseudostress, and piecewise polynomials for the velocity. Convergence and a priori error estimates are derived based on compact operator theory. We present a series of numerical tests in two and three dimensions to confirm the theoretical findings.

Figures (9)

• Figure 1: Test \ref{['subsec:square-domain2D']}. Velocity streamlines and pressures surface plot for the first and fourth computed eigenvalue with $\boldsymbol{\beta}_1$.
• Figure 2: Test \ref{['subsec:square-domain2D']}. Computed eigenvalues distribution on the square domain with different choices of $\boldsymbol{\beta}$ and $N=100$.
• Figure 3: Test \ref{['subsec:lshape-2D']}. Velocity streamlines and pressures surface plot for the first and fourth computed eigenvalue with $\boldsymbol{\beta}_1$.
• Figure 4: Test \ref{['subsec:lshape-2D']}. Computed eigenvalues distribution on the lshape domain with different choices of $\boldsymbol{\beta}$ and $N=64$.
• Figure 5: Test \ref{['subsec:cube-domain3D']}. Velocity streamlines for the first and fourth eigenmode on the unit cube domain with convective velocity $\boldsymbol{\beta}=(1,0,0)^\texttt{t}$.

Theorems & Definitions (18)

• Remark 2.1
• Lemma 2.2
• Lemma 2.3
• Proof 1
• Theorem 2.4
• Proof 2
• Remark 2.5
• Lemma 2.6
• Lemma 3.1
• Proof 3