Core Concepts

The authors examine the dimensions of various inf-sup stable mixed finite element spaces on tetrahedral meshes in 3D with exact divergence constraints, comparing the standard Scott-Vogelius elements of higher polynomial degree and low order methods on split meshes.

Abstract

The main focus of this work is to analyze the dimensions of mixed finite element spaces with exact divergence constraints in 3D. The authors consider the following key points:
They review various split mesh methods, including the Alfeld split and the Worsey-Farin split, that allow for exact divergence constraints to be satisfied for low-order finite elements.
The authors develop a counting strategy to express the degrees of freedom for a given polynomial degree and mesh split in terms of a few mesh quantities, such as the number of vertices, edges, faces, and tetrahedra. They investigate bounds and asymptotic behavior of these mesh quantities under uniform mesh refinement.
Applying the counting method, the authors compare the dimensions of the standard Scott-Vogelius elements of higher polynomial degree (k ≥ 4) and the low-order split mesh methods. They find that the Scott-Vogelius elements, especially for k = 4, are competitive or even more efficient in terms of the number of degrees of freedom compared to most of the low-order split methods.
The authors also provide insights into the Stokes complex for the Worsey-Farin split, which yields further understanding of the structure of these exactly divergence-free finite element spaces.
The main conclusion is that the Scott-Vogelius elements, particularly for k = 4, deserve further investigation as they appear to be an efficient choice for exactly divergence-free finite element approximations in 3D.

Stats

The number of tetrahedra T in the mesh is 8 times the number of tetrahedra in the previous mesh after uniform refinement.
The number of vertices V in the refined mesh is the sum of the number of vertices V and the number of edges E in the previous mesh.
The number of boundary edges Eb in the refined mesh is 2 times the number of edges E plus 3 times the number of faces F in the previous mesh.
The number of boundary faces Fb in the refined mesh is 4 times the number of faces F in the previous mesh.

Quotes

"Even for k = 6, this pair has only at most 50% higher dimension than most of the lowest-order split methods considered here."
"The only exception to this is the lowest-order case on the so-called Worsey-Farin split mesh. By comparing the actual number of degrees of freedom rather than the polynomial degree, we find that the Scott-Vogelius method is competitive or at least not much more costly than standard low-order methods."
"As soon as one considers methods of higher order than linear, the Scott-Vogelius element for k = 4 is the most efficient (by a factor 2)."

Key Insights Distilled From

by L. Ridgway S... at **arxiv.org** 04-22-2024

Deeper Inquiries

To further understand the behavior of average mesh quantities and the dimensions of exactly divergence-free finite element spaces, exploring different mesh refinement schemes beyond uniform refinement can provide valuable insights. One such scheme is adaptive mesh refinement (AMR), where the mesh is refined based on certain criteria such as error indicators, solution gradients, or geometric features. By adaptively refining the mesh in regions of interest, AMR can lead to more efficient discretizations with fewer degrees of freedom while maintaining accuracy. Additionally, hierarchical refinement schemes like quadtree or octree structures can be explored, allowing for non-uniform refinement based on the local features of the domain. These hierarchical structures can capture complex geometries more efficiently and may impact the distribution of average mesh quantities in a different way compared to uniform refinement. Investigating these alternative mesh refinement schemes can provide a comprehensive understanding of how different refinement strategies influence the dimensions of finite element spaces and the behavior of average mesh quantities.

When comparing the computational costs between the Scott-Vogelius elements and low-order split mesh methods, it is essential to consider factors beyond just the dimensions of the discrete spaces. The conditioning of the linear systems plays a crucial role in the efficiency and accuracy of the solution techniques. The Scott-Vogelius elements, being higher-order methods, may lead to better-conditioned systems compared to low-order split mesh methods. This improved conditioning can result in faster convergence of iterative solvers and more stable solutions. Additionally, the effectiveness of solution techniques like static condensation should be taken into account. Static condensation can reduce the computational cost by eliminating the pressure variable from the system, leading to smaller systems that are cheaper to solve. By comparing the conditioning of the linear systems, convergence rates of solution techniques, and the overall computational efficiency, a comprehensive assessment of the computational costs between the two methods can be made.

Extending the analysis of the Stokes complex to other split mesh methods beyond the Worsey-Farin split can provide valuable insights into the suitability of different finite element spaces for pressure-robust approximations of the Stokes and Navier-Stokes equations. By exploring alternative split mesh methods such as the Alfeld split, Powell-Sabin split, or other custom splits, one can evaluate the inf-sup stability, accuracy, and computational efficiency of the resulting discrete spaces. Understanding how different split mesh methods handle the exact divergence constraints and maintain stability can guide the selection of appropriate finite element spaces for pressure-robust approximations. Additionally, investigating the performance of these methods on challenging geometries or under varying flow conditions can provide a comprehensive understanding of their robustness and applicability in practical fluid dynamics simulations. By extending the analysis to a wider range of split mesh methods, researchers can make informed decisions on the most suitable finite element spaces for pressure-robust approximations in complex fluid flow problems.

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