A mixed virtual element discretization for the generalized Oseen problem

TL;DR

This work develops a mixed virtual element method for the two-dimensional generalized Oseen problem, introducing a pseudostress variable to bypass direct pressure discretization. The approach yields stable, polygonally-flexible spaces for the pseudostress and velocity, with a pressure post-processing rule $p_h = -\tfrac12(\operatorname{tr}(\Pi_h \boldsymbol{\sigma}_h) + \operatorname{tr}(\boldsymbol{u}_h \otimes \boldsymbol{\beta}))$, and employs a fixed-point argument to guarantee existence/uniqueness under small data. It provides a rigorous a priori error analysis, delivering optimal rates for the velocity, pseudostress, and pressure, and confirms these rates through extensive numerical tests across various meshes and regimes, including convection-dominated and boundary-layer flows. The results demonstrate robust performance on polygonal meshes, effective pressure post-processing, and potential for further development in a posteriori analysis and superconvergence phenomena in certain test cases.

Abstract

In this paper we introduce a mixed virtual element method to approximate the solution for the two dimensional generalized Oseen problem. We introduce the pseudostress as an additional unknown, which allows to eliminate the pressure from the system; the pressure can be recovered via a post-process of the pseudostress tensor. We prove existence and uniqueness of the continuous solution via a fixed point argument. Under standard mesh assumptions, we develop a virtual element method to approximate both the tensor and the velocity field, and we show that it is stable. Furthermore, we provide a priori error estimates for the method and validate them through a series of numerical tests using different polygonal meshes.

Figures (12)

• Figure 1: Sample meshes: ${\mathcal{T}}_{h}^{1}$ top left), ${\mathcal{T}}_{h}^{2}$ (top middle), ${\mathcal{T}}_{h}^{3}$ (top right),${\mathcal{T}}_{h}^{4}$ (bottom left), ${\mathcal{T}}_{h}^{5}$ (bottom middle) and ${\mathcal{T}}_{h}^{6}$ (bottom right).
• Figure 2: Test 1. Error curves for the velocity, pseudostress, and pressure, computed with meshes ${\mathcal{T}}_{h}^1$, ${\mathcal{T}}_{h}^2$, ${\mathcal{T}}_{h}^3$, and ${\mathcal{T}}_h^4$, respectively. We observe the optimal order of convergence on each case: in red we present the error curve of the velocity, in blue for the pseudostress, and in yellow the error of the pressure.
• Figure 3: Test 1: Left: plot of the computed velocity magnitude $\boldsymbol{u}_h$ with mesh ${\mathcal{T}}_h^3$. Right: plot of the computed pressure fluctuation $p_h$ with mesh ${\mathcal{T}}_h^3$.
• Figure 4: Test 1: Computed pseudostress tensor components with ${\mathcal{T}}_h^4$. Top row, from left to right: components $\boldsymbol{\sigma}_{h,11}$ and $\boldsymbol{\sigma}_{h,12}$. Bottom row, from left to right: $\boldsymbol{\sigma}_{h,21}$ and $\boldsymbol{\sigma}_{h,22}$.
• Figure 5: Test 2: Error behavior for different values of $\nu$ and different mesh families. In red, the error using ${\mathcal{T}}_h^2$, in blue the error computed with ${\mathcal{T}}_h^3$, and in yellow, the error computed with ${\mathcal{T}}_h^4$.

Theorems & Definitions (13)

• Lemma 2.1
• Lemma 2.2
• Proof 1
• Theorem 2.3
• Proof 2
• Lemma 3.1
• Proof 3
• Lemma 3.2
• Proof 4
• Lemma 4.2