[Paper Review] A Nonconforming Finite Element Method for Fourth Order Curl Equations in R^3

Bin Zheng, Qiya Hu+1 more|arXiv (Cornell University)|Jan 31, 2010

Advanced Numerical Methods in Computational Mathematics15 references3 citations

TL;DR

This paper presents a nonconforming finite element method for solving fourth-order curl equations in three dimensions, specifically arising in magnetohydrodynamics (MHD) models. The method uses a novel element with minimal degrees of freedom that enforces tangential continuity of the curl, achieving optimal convergence rates and providing explicit basis functions for implementation.

ABSTRACT

In this paper we present a nonconforming finite element method for solving fourth order curl equations in three dimensions arising from magnetohydrodynamics models. We show that the method has an optimal error estimate for a model problem involving both curl^2 and curl^4 operators. The element has a very small number of degrees of freedom and it imposes the inter-element continuity along the tangential direction which is appropriate for the approximation of magnetic fields. We also provide explicit formulae of basis functions for this element.

Motivation & Objective

• To develop a stable and efficient finite element method for fourth-order curl operators in three-dimensional magnetohydrodynamics (MHD) problems.
• To address the challenge of discretizing the $(\nabla\times)^4$ operator in MHD models, which is difficult with standard conforming elements due to high degrees of freedom.
• To construct a nonconforming finite element that ensures inter-element tangential continuity of the curl, suitable for magnetic field approximation.
• To provide explicit basis functions for the new element to enable practical implementation.
• To establish optimal error estimates for the method under standard Sobolev regularity assumptions.

Proposed method

• The method employs a nonconforming finite element space based on piecewise quadratic polynomials enriched with curl-related degrees of freedom.
• The finite element space enforces inter-element continuity of the tangential component of the curl, ensuring consistency with magnetic field physics.
• A bilinear form $ a_h(\cdot, \cdot) $ is constructed using volume and facet integrals involving $ \nabla\times\mathbf{v}_h $, incorporating stabilization terms via $ \alpha $ and $ \beta $.
• The method is analyzed using a nonconforming Galerkin formulation, with consistency and approximation error estimates derived via integration by parts and trace inequalities.
• The analysis relies on the second Strang lemma to establish optimal convergence in $ H^1 $-like norms of the curl and its gradient.
• Explicit basis functions for the element are derived and provided, enabling direct implementation in finite element codes.

Experimental results

Research questions

• RQ1 Can a nonconforming finite element method be constructed that efficiently discretizes fourth-order curl operators in three dimensions with minimal degrees of freedom?
• RQ2 Does enforcing tangential continuity of the curl across element boundaries lead to optimal convergence for fourth-order curl problems?
• RQ3 How can the $(\nabla\times)^4$ operator in MHD models be approximated robustly using a nonconforming approach?
• RQ4 What is the optimal error estimate achievable by such a method under standard regularity assumptions?
• RQ5 Can explicit basis functions be derived for the new element to support practical implementation?

Key findings

• The proposed nonconforming finite element method achieves optimal convergence rates in the energy norm, with error bounded by $ \lesssim h\|\mathbf{u}\|_{4,\Omega} $ for $ \mathbf{u} \in (H^4(\Omega))^3 $.
• The method exhibits optimal error estimates for the solution, its curl, and the gradient of its curl, ensuring high-order accuracy.
• The element has only 30 degrees of freedom per tetrahedral element, significantly reducing computational cost compared to conforming methods requiring 220 degrees of freedom.
• The method preserves the physical structure of magnetic fields by enforcing tangential continuity of the curl across element interfaces.
• Explicit formulae for the basis functions are derived, enabling direct implementation in finite element software.
• The consistency error is bounded by $ \lesssim h(\|\nabla\times\Delta(\nabla\times\mathbf{u})\| + \|\nabla(\nabla\times\mathbf{u})\|_{1} + \|\nabla\times\mathbf{u}\| + \|\nabla\times\nabla\times\mathbf{u}\|) $, confirming optimal approximation properties.