[Paper Review] A locally mass conserving quadratic velocity, linear pressure element

R. W. Thatcher, David J. Silvester|arXiv (Cornell University)|Jan 31, 2020

Computational Fluid Dynamics and Aerodynamics9 references4 citations

TL;DR

This paper introduces a locally mass-conserving (LC) finite element for Stokes flow, combining quadratic velocity and linear pressure with added element-wise constant pressure enrichment. It proves optimal convergence and stability on a wide range of triangular grids via a novel patch-based stability analysis, outperforming the Taylor–Hood element in resolving complex flow features like multiple recirculation zones.

ABSTRACT

By supplementing the pressure space for the Taylor-Hood element a triangular element that satisfies continuity over each element is produced. Making a novel extension of the patch argument to prove stability, this element is shown to be globally stable and give optimal rates of convergence on a wide range of triangular grids. This theoretical result is extended in the discussion given in the appendix, showing how optimal convergence rates can be obtained on all grids. Two examples are presented, one illustrating the convergence rates and the other illustrating difficulties with the Taylor-Hood element which are overcome by the element presented here.

Motivation & Objective

• To address the lack of local mass conservation in the Taylor–Hood element, which only enforces continuity globally, not element-wise.
• To develop a finite element that ensures local incompressibility (local continuity) while maintaining optimal convergence rates.
• To establish a new stability proof for the enriched pressure space using an extended patch argument, applicable to a broad class of unstructured triangular meshes.
• To demonstrate through numerical examples that the new element resolves complex flow structures—such as multiple recirculation zones—more accurately than the Taylor–Hood element, even on coarse grids.

Proposed method

• The method uses quadratic velocity shape functions ($P_2$) and enriches the continuous linear pressure ($P_1$) space with element-wise constant functions ($P_0$), forming a locally conservative pressure space.
• The discrete inf-sup condition is proven using a novel patch-based stability argument, where patches are enclosed in larger, overlapping extended patches to handle degeneracy in constant function representations.
• The analysis extends the patch techniques of Stenberg and Boland–Nicolaides, adapting them to handle the non-unique representation of constants in the enriched pressure space.
• The method is applied to both Stokes and Navier–Stokes problems, with the discrete equations solved in finite element subspaces satisfying velocity and pressure continuity constraints.
• A pressure stabilization strategy is introduced for non-triangulated corners: fixing centroid and vertex pressures ensures solvability without compromising accuracy.
• Theoretical stability is validated numerically on two benchmark problems, including a swirling flow with multiple recirculation zones.

Experimental results

Research questions

• RQ1 Can a finite element be constructed that ensures local mass conservation (i.e., divergence-free velocity per element) while maintaining optimal convergence rates?
• RQ2 Is the enriched pressure space—linear plus element-wise constant—stable on unstructured triangular grids, including highly distorted or stretched elements?
• RQ3 Does the proposed element outperform the Taylor–Hood element in resolving complex flow features such as multiple recirculation zones?
• RQ4 Can a new patch-based stability argument be developed to handle the non-unique representation of constant functions in the pressure space?

Key findings

• The locally mass-conserving (LC) element achieves optimal convergence rates for velocity and pressure on a wide range of triangular grids, including those violating the standard regularity condition $h_{ riangle} \leq \sigma \rho_{\triangle}$.
• The LC element resolves three distinct recirculation zones in a swirling flow problem on a $16 \times 4$ grid, while the Taylor–Hood element fails to capture more than one complete recirculation even on a $64 \times 16$ grid.
• Numerical results show that the LC element produces consistent solutions across grid refinements, whereas the Taylor–Hood element exhibits erratic and inconsistent behavior for the same problem.
• The stability proof relies on enclosing patches in overlapping extended patches, a technique that overcomes the challenge of multiple representations of constant functions in the pressure space.
• The method remains stable and accurate even when grids do not triangulate into all corners, provided centroid and vertex pressures are fixed in boundary-adjacent elements.