[Paper Review] Curl-curl conforming elements on tetrahedra

Qian Zhang, Zhimin Zhang|arXiv (Cornell University)|Jul 20, 2020

Advanced Numerical Methods in Computational Mathematics26 references4 citations

TL;DR

This paper constructs H(curl²)-conforming finite elements on tetrahedral meshes for solving 3D quad-curl problems, a fourth-order curl system arising in electromagnetics and magnetohydrodynamics. The proposed elements achieve optimal interpolation error estimates and enable conforming finite element discretization in general Lipschitz domains, with numerical validation confirming their accuracy and convergence.

ABSTRACT

In [24], we proposed H(curl^2)-conforming elements on both a triangle and a rectangle. This family of elements provides a brand new method to solve the quad-curl problem in 2 dimensions. In this paper, we turn our focus to 3 dimensions and construct an H(curl^2)-conforming tetrahedral finite element. The newly proposed element has been proved to have the optimal interpolation error estimate. Having tetrahedral elements, we can solve the quad-curl problem in any Lipschitz domain by conforming finite element method. We also provide several numerical examples of using our element to solve the quad-curl problem. The results of the numerical experiments show the effectiveness and correctness of our element.

Motivation & Objective

• To develop H(curl²)-conforming finite elements on tetrahedral meshes for solving 3D quad-curl problems.
• To overcome the challenge of constructing curl-curl-conforming elements in 3D with weaker regularity requirements than H¹(curl) elements.
• To ensure optimal interpolation error estimates despite the large number of degrees of freedom (315 for the lowest-order element).
• To provide a conforming finite element method that converges to the exact solution in H(curl²) spaces, avoiding projection errors inherent in H²-conforming methods.
• To enable accurate numerical solution of quad-curl problems in arbitrary Lipschitz domains using tetrahedral meshes.

Proposed method

• Construct H(curl²)-conforming finite elements on tetrahedra by defining degrees of freedom (DOFs) involving point evaluations of the vector field, its curl, and normal derivatives on edges and faces.
• Use a reference element and a transformation matrix D to map DOFs from the reference to a physical tetrahedron, ensuring unisolvence and conformity.
• Introduce intermediate elements with DOFs related to the reference element to facilitate proving optimal interpolation error estimates.
• Employ a method from [8] to compute basis functions for high-DOF elements (315 DOFs) due to the infeasibility of traditional Lagrange-type basis computation.
• Represent all DOFs on the physical element as linear combinations of functional values on the reference element via transformation matrices involving the Jacobian and edge/face normal vectors.
• Prove that the interpolation error converges optimally in the H(curl²) norm by expressing the interpolation operator in terms of the DOFs and verifying consistency and stability.

Experimental results

Research questions

• RQ1 Can H(curl²)-conforming finite elements be constructed on tetrahedral meshes in 3D to solve the quad-curl problem with optimal convergence rates?
• RQ2 How can one ensure unisolvence and conformity of curl-curl-conforming elements when the DOFs involve normal derivatives on edges and faces, which do not naturally transfer between reference and physical elements?
• RQ3 What is the optimal interpolation error estimate for the proposed H(curl²)-conforming finite element space on tetrahedra?
• RQ4 Can the proposed method achieve convergence to the exact solution in H(curl²) space, avoiding the projection error that arises with H²-conforming elements?
• RQ5 How can basis functions be efficiently computed for high-order H(curl²)-conforming elements with 315 degrees of freedom?

Key findings

• The proposed H(curl²)-conforming finite elements on tetrahedra achieve optimal interpolation error estimates in the H(curl²) norm, confirming their theoretical reliability.
• The construction of the elements is mathematically rigorous, with unisolvence and conformity verified through detailed analysis of DOFs and their transformation.
• The lowest-order element has 315 degrees of freedom, significantly higher than standard Nédélec elements, reflecting the need to capture curl and second-order derivatives.
• Numerical experiments confirm the correctness and convergence of the method, showing optimal convergence rates for the quad-curl problem in 3D.
• The method enables conforming finite element discretization of the quad-curl problem in any Lipschitz domain, overcoming limitations of H²-conforming and nonconforming approaches.
• A computational framework with code available at https://github.com/QianZhangMath/3D-curl-curl-conforming-FE allows efficient basis function computation despite the high DOF count.