The MINI mixed virtual element for the Stokes equation

TL;DR

The paper develops a MINI-inspired mixed virtual element method for the 2D Stokes problem on polygonal meshes, combining equal-order velocity and pressure spaces with bubble enrichment and a pressure stabilization to ensure stability. It introduces computable bilinear forms a_h, b_h and c_h, proves k-consistency, stability, and optimal error estimates in energy and L2 norms, and confirms the theory with extensive numerical tests on diverse polygonal meshes. A duality argument yields L2 velocity error estimates on convex domains, and static condensation provides an equal-order virtual element formulation by eliminating the bubble DOFs. Overall, the MINI-VEM enables high-order accurate, robust Stokes discretizations on general meshes with practical and efficient implementation features.

Abstract

We present and discuss a generalization of the popular MINI mixed finite element for the 2D Stokes equation by means of conforming virtual elements on polygonal meshes. We prove optimal error estimates for both velocity and pressure. Theoretical results are confirmed by several numerical tests performed with different choices of polynomial accuracy and meshes.

Figures (8)

• Figure 1: Graphical representation of the MINI mixed finite element. Degrees of freedom of the linear part are represented by red circles, whereas the bubble degree of freedom is indicated by a blue square.
• Figure 2: Graphical representation of the MINI mixed virtual element for $k=1,2$. The degrees of freedom of the degree $k$ contribution are denoted by red circles (vertices), green triangles (internal points on edges) and yellow diamonds (internal moments). The degrees of freedom of the additional bubbles are denoted by blue squares.
• Figure 3: Meshes used for studying the condition number. Mesh features are collected in Table \ref{['tab:mesh_info']}. From left to right: hexagons (Level 3), Voronoi (Level 2), random polygons (Level 1), diamond with zoom (Level 5).
• Figure 4: Condition number of MINI--VEM with respect to pressure stabilization parameter $\alpha$. Effect of polynomial basis: scaled monomials $\mathcal{M}_{k}$ (left column) compared with $L^2$ orthonormal basis $\mathcal{Q}_{k}$ (right column).
• Figure 5: Exact solutions of Test 1.

Theorems & Definitions (31)

• Theorem 3.3
• Lemma 5.1
• Remark 5.2
• Proposition 5.3
• Proposition 5.4
• Proposition 6.1
• proof
• Proposition 6.2
• Corollary 6.3
• Proposition 7.1