Kaibo Hu

Kaibo Hu, Yicong Ma, Jinchao Xu

This paper is devoted to the design and analysis of some structure-preserving finite element schemes for the magnetohydrodynamics (MHD) system. The main feature of the method is that it naturally preserves the important Gauss law, namely $\nabla\cdot\boldsymbol{B}=0$. In contrast to most existing approaches that eliminate the electrical field variable $\boldsymbol{E}$ and give a direct discretization of the magnetic field, our new approach discretizes the electric field $\boldsymbol{E}$ by Nédélec type edge elements for $H(\mathrm{curl})$, while the magnetic field $\boldsymbol{B}$ by Raviart-Thomas type face elements for $H(\mathrm{div})$. As a result, the divergence-free condition on the magnetic field holds exactly on the discrete level. For this new finite element method, an energy stability estimate can be naturally established in an analogous way as in the continuous case. Furthermore, well-posedness is rigorously established in the paper for both the Picard and Newton linearization of the fully nonlinear systems by using the Brezzi theory for both the continuous and discrete cases. This well-posedness naturally leads to robust (and optimal) preconditioners for the linearized systems.

Yicong Ma, Kaibo Hu, Xiaozhe Hu et al.

In this paper, we develop two classes of robust preconditioners for the structure-preserving discretization of the incompressible magnetohydrodynamics (MHD) system. By studying the well-posedness of the discrete system, we design block preconditioners for them and carry out rigorous analysis on their performance. We prove that such preconditioners are robust with respect to most physical and discretization parameters. In our proof, we improve the existing estimates of the block triangular preconditioners for saddle point problems by removing the scaling parameters, which are usually difficult to choose in practice. This new technique is not only applicable to the MHD system, but also to other problems. Moreover, we prove that Krylov iterative methods with our preconditioners preserve the divergence-free condition exactly, which complements the structure-preserving discretization. Another feature is that we can directly generalize this technique to other discretizations of the MHD system. We also present preliminary numerical results to support the theoretical results and demonstrate the robustness of the proposed preconditioners.

Kaibo Hu, Jinchao Xu

In this paper, we develop a class of mixed finite element scheme for stationary magnetohydrodynamics (MHD) models, using magnetic field $\bm B$ and current density $\bm j$ as the discretization variables. We show that the Gauss's law for the magnetic field, namely $\nabla\cdot\bm{B}=0$, and the energy law for the entire system are exactly preserved in the finite element schemes. Based on some new basic estimates for $H^{h}(\mathrm{div})$, we show that the new finite element scheme is well-posed. Furthermore, we show the existence of solutions to the nonlinear problems and the convergence of Picard iterations and finite element methods under some conditions.

Juncai He, Kaibo Hu, Jinchao Xu

We prove generalized Gaffney inequalities and the discrete compactness for finite element differential forms on $s$-regular domains, including general Lipschitz domains. In computational electromagnetism, special cases of these results have been established for edge elements with weakly imposed divergence-free conditions and used in the analysis of nonlinear and eigenvalue problems. In this paper, we generalize these results to discrete differential forms, not necessarily with strongly or weakly imposed constraints. The analysis relies on a new Hodge mapping and its approximation property. As an application, we show $L^{p}$ estimates for several finite element approximations of the scalar and vector Laplacian problems.

Snorre H. Christiansen, Jun Hu, Kaibo Hu

We construct 2D and 3D finite element de Rham sequences of arbitrary polynomial degrees with extra smoothness. Some of these elements have nodal degrees of freedom (DoFs) and can be considered as generalisations of scalar Hermite and Lagrange elements. Using the nodal values, the number of global degrees of freedom is reduced compared with the classical Nédélec and Brezzi-Douglas-Marini (BDM) finite elements, and the basis functions are more canonical and easier to construct. Our finite elements for ${H}(\mathrm{div})$ with regularity $r=2$ coincide with the nonstandard elements given by Stenberg (Numer Math 115(1): 131-139, 2010). We show how regularity decreases in the finite element complexes, so that they branch into known complexes. The standard de Rham complexes of Whitney forms and their higher order version can be regarded as the family with the lowest regularity. The construction of the new families is motivated by the finite element systems.%, and we also establish local exact sequences (geometric decomposition) for the new elements.

Snorre Harald Christiansen, Kaibo Hu

We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously introduced notion of Finite Element Systems, and the examples include conforming mixed finite elements for Stokes' equation. In dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes. In dimension 3 we define one low order complex, which may be branched into Whitney forms at a chosen index. Stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension. The finite element spaces all consist of composite polynomials. The framework guarantees some nice properties of the spaces, in particular the existence of commuting interpolators. It also shows that some of the examples are minimal spaces.

Andrew Gillette, Kaibo Hu, Shuo Zhang

We propose two general operations on finite element differential complexes on cubical meshes that can be used to construct and analyze sequences of "nonstandard" convergent methods. The first operation, called DoF-transfer, moves edge degrees of freedom to vertices in a way that reduces global degrees of freedom while increasing continuity order at vertices. The second operation, called serendipity, eliminates interior bubble functions and degrees of freedom locally on each element without affecting edge degrees of freedom. These operations can be used independently or in tandem to create nonstandard complexes that incorporate Hermite, Adini and "trimmed-Adini" elements. We show that the resulting elements provide convergent, non-conforming methods for problems requiring stronger regularity and satisfy a discrete Korn inequality. We discuss potential benefits of applying these elements to Stokes, biharmonic and elasticity problems.

Chengbin Zhu, Snorre H. Christiansen, Kaibo Hu et al.

We prove convergence and stability of the discrete exterior calculus (DEC) solutions for the Hodge-Laplace problems in two dimensions for families of meshes that are non-degenerate Delaunay and shape regular. We do this by relating the DEC solutions to the lowest order finite element exterior calculus (FEEC) solutions. A Poincaré inequality and a discrete inf-sup condition for DEC are part of this proof. We also prove that under appropriate geometric conditions on the mesh the DEC and FEEC norms are equivalent. Only one side of the norm equivalence is needed for proving stability and convergence and this allows us to relax the conditions on the meshes.

Kaibo Hu, Ting Lin, Bowen Shi

We construct a family of finite element sub-complexes of the conformal complex on tetrahedral meshes and show their exactness on contractible domains. This complex includes vector fields and symmetric and traceless tensor fields, connected through the conformal Killing operator, the linearized Cotton-York operator, and the divergence operator, respectively. This leads to discrete versions of transverse traceless (TT) tensors, i.e., symmetric, traceless and divergence-free matrix fields, in continuum mechanics and general relativity. We also show the inf-sup stability of the $H(\operatorname{div})$-conforming finite element symmetric and traceless tensors paired with discontinuous vectors.

Patrick E. Farrell, Mingdong He, Kaibo Hu et al.

Magnetic relaxation drives plasma toward lower-energy equilibria under helicity constraints. In ideal magnetohydrodynamics (MHD), helicity is locally conserved, while resistive theories such as Taylor relaxation preserve only global helicity. This distinction has important implications for structure-preserving numerical methods. We compare three finite element formulations: an unconstrained scheme that does not conserve helicity, a mixed method based on finite element exterior calculus that preserves all local helicities, and a Lagrange multiplier approach that enforces only global helicity conservation. Numerical experiments on braided and knotted magnetic fields show that local helicity preservation prevents spurious reconnection and maintains nontrivial topology in ideal MHD or magneto-friction, whereas enforcing only global helicity allows further relaxation through local reconnection. Numerical results on magnetic knots and braids are provided. These results clarify how different levels of discrete helicity constraints influence magnetic relaxation and equilibrium structure in numerical computation.

Kaibo Hu, Weifeng Qiu, Ke Shi

We discuss a class of magnetic-electric fields based finite element schemes for stationary magnetohydrodynamics (MHD) systems with two types of boundary conditions. We establish a key $L^{3}$ estimate for divergence-free finite element functions for a new type of boundary conditions. With this estimate and a similar one in [Hu&Xu,2018], we rigorously prove the convergence of Picard iterations and the finite element schemes with weak regularity assumptions. These results demonstrate the convergence of the finite element methods for singular solutions.

Snorre H. Christiansen, Kaibo Hu, Espen Sande

We propose a general strategy to derive null-homotopy operators for differential complexes based on the Bernstein-Gelfand-Gelfand (BGG) construction and properties of the de Rham complex. Focusing on the elasticity complex, we derive path integral operators $\mathscr{P}$ for elasticity satisfying $\mathscr{D}\mathscr{P}+\mathscr{P}\mathscr{D}=\mathrm{id}$ and $\mathscr{P}^{2}=0$, where the differential operators $\mathscr{D}$ correspond to the linearized strain, the linearized curvature and the divergence, respectively. In general we derive path integral formulas in the presence of defects. As a special case, this gives the classical Cesàro-Volterra path integral for strain tensors satisfying the Saint-Venant compatibility condition.