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1. Finite element de Rham and Stokes complexes in three dimensions

   https://doi.org/10.1090/mcom/3859  

   Chen, Long; Huang, Xuehai (January 2024, Mathematics of Computation)

   Finite element de Rham complexes and finite element Stokes complexes with varying degrees of smoothness in three dimensions are systematically constructed in this paper. Smooth scalar finite elements in three dimensions are derived through a non-overlapping decomposition of the simplicial lattice. H(div)-conforming finite elements and H(curl)-conforming finite elements with varying degrees of smoothness are devised based on these smooth scalar finite elements. The finite element de Rham complexes with corresponding smoothness and commutative diagrams are induced by these elements. The div stability of the H(div)-conforming finite elements is established, and the exactness of these finite element complexes is proven. 

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2. A finite element elasticity complex in three dimensions

   https://doi.org/10.1090/mcom/3739  

   Chen, Long; Huang, Xuehai (September 2022, Mathematics of Computation)

   A finite element elasticity complex on tetrahedral meshes and the corresponding commutative diagram are devised. The H 1 H^1 conforming finite element is the finite element developed by Neilan for the velocity field in a discrete Stokes complex. The symmetric div-conforming finite element is the Hu-Zhang element for stress tensors. The construction of an H ( inc ) H(\operatorname {inc}) -conforming finite element of minimum polynomial degree 6 6 for symmetric tensors is the focus of this paper. Our construction appears to be the first H ( inc ) H(\operatorname {inc}) -conforming finite elements on tetrahedral meshes without further splitting. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace of the inc \operatorname {inc} operator. The polynomial elasticity complex and Koszul elasticity complex are created to derive the decomposition. The trace of the inc \operatorname {inc} operator is induced from a Green’s identity. Trace complexes and bubble complexes are also derived to facilitate the construction. Two-dimensional smooth finite element Hessian complex and div ⁡ div \operatorname {div}\operatorname {div} complex are constructed. 

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3. H(div)-conforming finite element tensors with constraints

   https://doi.org/10.1016/j.rinam.2024.100494  

   Chen, Long; Huang, Xuehai (August 2024, Results in Applied Mathematics)

   A unified construction of H(div)-conforming finite element tensors, including vector element, symmetric matrix element, traceless matrix element, and, in general, tensors with linear constraints, is developed in this work. It is based on the geometric decomposition of Lagrange elements into bubble functions on each sub-simplex. Each tensor at a sub-simplex is further decomposed into tangential and normal components. The tangential component forms the bubble function space, while the normal component characterizes the trace. Some degrees of freedom can be redistributed to (n-1)-dimensional faces. The developed finite element spaces are H(div)-conforming and satisfy the discrete inf-sup condition. Intrinsic bases of the constraint tensor space are also established. 

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4. Hybridizable symmetric stress elements on the barycentric refinement in arbitrary dimensions

   https://doi.org/10.1090/mcom/4180  

   Chen, Long; Huang, Xuehai (December 2025, Mathematics of Computation)

   Hybridizable $$H(\operatorname{div})$$-conforming finite elements for symmetric tensors on simplices with barycentric refinement are developed in this work for arbitrary dimensions and any polynomial order. By employing barycentric refinement and an intrinsic tangential-normal decomposition, novel basis functions are constructed to redistribute degrees of freedom while preserving $$H(\operatorname{div})$$-conformity and symmetry, and ensuring inf-sup stability. These hybridizable elements enhance computational flexibility and efficiency, with applications to mixed finite element methods for linear elasticity. 

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5. A mass conserving mixed stress formulation for the Stokes equations

   https://doi.org/10.1093/imanum/drz022  

   Gopalakrishnan, Jay; Lederer, Philip L; Schöberl, Joachim (May 2019, IMA Journal of Numerical Analysis)

   Abstract We propose stress formulation of the Stokes equations. The velocity $$u$$ is approximated with $$H(\operatorname{div})$$-conforming finite elements providing exact mass conservation. While many standard methods use $H^1$-conforming spaces for the discrete velocity $$H(\operatorname{div})$$-conformity fits the considered variational formulation in this work. A new stress-like variable $$\sigma $$ equalling the gradient of the velocity is set within a new function space $$H(\operatorname{curl} \operatorname{div})$$. New matrix-valued finite elements having continuous ‘normal-tangential’ components are constructed to approximate functions in $$H(\operatorname{curl} \operatorname{div})$$. An error analysis concludes with optimal rates of convergence for errors in $$u$$ (measured in a discrete $H^1$-norm), errors in $$\sigma $$ (measured in $L^2$) and the pressure $$p$$ (also measured in $L^2$). The exact mass conservation property is directly related to another structure-preservation property called pressure robustness, as shown by pressure-independent velocity error estimates. The computational cost measured in terms of interface degrees of freedom is comparable to old and new Stokes discretizations. 

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