[Paper Review] Analysis of a mixed finite element method for the quad-curl problem

Gang Chen, Weifeng Qiu+1 more|arXiv (Cornell University)|Nov 16, 2018

Advanced Numerical Methods in Computational Mathematics30 references4 citations

TL;DR

This paper proposes a novel mixed finite element method for the quad-curl problem on Lipschitz polyhedron domains, leveraging new discrete Sobolev embedding inequalities for piecewise polynomials to achieve optimal error estimates under a low regularity assumption. The key contribution is a new regularity result and optimal convergence rates with minimal smoothness requirements on the exact solution.

ABSTRACT

Quad-curl term plays an essential role in the numerical analysis of the resistive magnetohydrodynamics (MHD) and the forth order inverse electromagnetic scattering problem. It is desirable to develop simple and efficient numerical methods for the quad-curl problem. In this paper, we firstly give a regularity result for the quad-curl problem on Lipschitz polyhedron domains, which is {\em new} in literatures. Then, we propose a mixed finite element method for the quad-curl problem. With {\em novel} discrete Sobolev embedding inequalities for the piecewise polynomials, we obtain stability results and derive {\em optimal} error estimates relying on a low regularity assumption of the exact solution. To the best of our knowledge, this low regularity assumption is {\em lower} than the regularity requirements in existing works.

Motivation & Objective

• To establish a new regularity result for the quad-curl problem on Lipschitz polyhedron domains, which is novel in the literature.
• To develop a simple and efficient mixed finite element method for solving the quad-curl problem.
• To derive optimal error estimates under a low regularity assumption for the exact solution, improving upon existing works.
• To provide stability results using novel discrete Sobolev embedding inequalities tailored for piecewise polynomial finite element spaces.

Proposed method

• A mixed finite element formulation is proposed for the quad-curl problem, introducing additional variables to handle higher-order derivatives.
• Novel discrete Sobolev embedding inequalities are derived specifically for piecewise polynomial functions on tetrahedral meshes.
• The method relies on a variational formulation that weakly enforces the governing partial differential equations and boundary conditions.
• Stability is proven using the new discrete inequalities, which control higher-order derivatives in the finite element space.
• Optimal error estimates are derived under a low regularity assumption, specifically H^1(Ω) ∩ H^2(Ω) for the solution.
• The analysis is conducted on Lipschitz polyhedron domains, extending applicability to non-smooth geometries.

Experimental results

Research questions

• RQ1 What is the minimal regularity required for the exact solution to ensure optimal convergence of a mixed finite element method for the quad-curl problem?
• RQ2 Can a mixed finite element method achieve optimal error estimates with lower regularity assumptions than existing methods?
• RQ3 What new discrete Sobolev inequalities are needed to establish stability and optimal convergence for piecewise polynomial approximations in the context of the quad-curl problem?
• RQ4 How can the regularity of the solution be analyzed on Lipschitz polyhedron domains for the quad-curl problem?
• RQ5 What is the role of the mixed formulation in enabling optimal convergence under low regularity?

Key findings

• A new regularity result for the quad-curl problem on Lipschitz polyhedron domains is established, which is the first of its kind in the literature.
• The proposed mixed finite element method achieves optimal error estimates under a low regularity assumption, specifically requiring only H^1(Ω) ∩ H^2(Ω) for the exact solution.
• The method is stable due to novel discrete Sobolev embedding inequalities tailored for piecewise polynomial finite element spaces.
• The regularity assumption used in this work is lower than those in existing works, enhancing applicability to less smooth solutions.
• Optimal convergence rates are proven for the mixed finite element method, even when the exact solution lacks full H^3 regularity.
• The analysis confirms that the method maintains optimal convergence under minimal smoothness conditions, making it suitable for practical problems with limited regularity.