[Paper Review] Amiable mixed schemes for fourth order curl equations

Shuo Zhang|arXiv (Cornell University)|Jul 17, 2016

Advanced Numerical Methods in Computational Mathematics13 references3 citations

TL;DR

This paper introduces stable, low-order mixed finite element schemes for fourth-order curl equations by introducing auxiliary variables to reformulate the problems into well-posed mixed formulations. The approach ensures optimal convergence rates using $H^1$, $H({ m curl})$, and $L^2$ finite elements, with regularity results confirmed on convex polyhedral domains and convergence proven via standard variational analysis.

ABSTRACT

In this paper, amiable mixed schemes are presented for two variants of fourth order curl equations. Specifically, mixed formulations for the problems are constructed, which are well-posed in Babuska-Brezzi's sense and admit stable discretizations by finite element spaces of low smoothness and of low degree. The regularities of the mixed formulations and thus equivalently the primal problems are established, and some finite elements examples are given which can exploit the regularity of the solutions to an optimal extent.

Motivation & Objective

• To develop stable, well-posed mixed formulations for two variants of fourth-order curl equations with minimal smoothness requirements.
• To establish regularity results for the mixed formulations and their equivalence to the primal problems on convex polyhedral domains.
• To design finite element discretizations that exploit solution regularity optimally using low-degree, low-smoothness finite element spaces.
• To enable efficient, scalable solution strategies by ensuring algebraic and topological nesting on nested grids.
• To provide a foundation for future work on parameter-robust discretizations and eigenvalue problems.

Proposed method

• Introduce auxiliary variables to reduce the fourth-order curl problem into a system of second-order equations, forming a mixed formulation.
• Construct mixed variational formulations that are well-posed in the Babuška-Brezzi sense using $L^2$, $H({ m curl})$, and $H^1$-type finite element spaces.
• Prove regularity of the mixed formulation by establishing $H^2$-regularity for the vector potential and its curl on convex polyhedrons.
• Use standard finite elements such as Nédélec elements and piecewise polynomials to achieve stable and convergent discretizations.
• Apply duality arguments and a priori error estimates to derive convergence rates in energy and $L^2$ norms.
• Ensure algebraic and topological nesting of finite element spaces on nested grids for multilevel method compatibility.

Experimental results

Research questions

• RQ1 Can stable mixed formulations be constructed for fourth-order curl equations using low-smoothness, low-degree finite elements?
• RQ2 What is the regularity of the solution to the mixed formulation, and does it confirm the $H^2$-regularity of the primal solution on convex polyhedral domains?
• RQ3 Can the mixed formulation be discretized in a way that allows optimal convergence and efficient solution via existing preconditioners?
• RQ4 How can the boundary conditions on the curl of the solution be properly enforced in the variational setting?
• RQ5 Can the resulting scheme be extended to parameter-robust or eigenvalue problems in future work?

Key findings

• The mixed formulations for both problem variants (A) and (B) are well-posed and stable under Babuška-Brezzi conditions with $L^2$, $H({ m curl})$, and $H^1$ spaces.
• The solution $\undertilde{u}$ and its curl $\nabla\times\undertilde{u}$ achieve $H^2$-regularity on convex polyhedral domains, confirming assumptions from prior works.
• Optimal convergence rates are achieved: $\|\undertilde{\varphi} - \undertilde{\varphi}_h\|_{0,\Omega} \leq Ch^2\|\undertilde{f}\|_{0,\Omega}$ for the mixed variable $\undertilde{\varphi}$.
• For problem (B), the convergence rate in the $H({\rm curl})$-norm is $O(h)$, improving to $O(h^2)$ if $\undertilde{f} \in \undertilde{H}^1(\Omega)$.
• The finite element spaces used—such as $N^2_{h0}$ and $\mathcal{L}^2_{h0}$—allow algebraic and topological nesting on nested grids, enabling efficient multilevel solvers.
• The scheme is implementable with standard finite element packages and compatible with existing optimal preconditioners.