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1. Discrete elasticity exact sequences on Worsey–Farin splits

   https://doi.org/10.1051/m2an/2023084  

   Gong, Sining; Gopalakrishnan, Jay; Guzmán, Johnny; Neilan, Michael (November 2023, ESAIM: Mathematical Modelling and Numerical Analysis)

   We construct conforming finite element elasticity complexes on Worsey–Farin splits in three dimensions. Spaces for displacement, strain, stress, and the load are connected in the elasticity complex through the differential operators representing deformation, incompatibility, and divergence. For each of these component spaces, a corresponding finite element space on Worsey–Farin meshes is exhibited. Unisolvent degrees of freedom are developed for these finite elements, which also yields commuting (cochain) projections on smooth functions. A distinctive feature of the spaces in these complexes is the lack of extrinsic supersmoothness at subsimplices of the mesh. Notably, the complex yields the first (strongly) symmetric stress finite element with no vertex or edge degrees of freedom in three dimensions. Moreover, the lowest order stress space uses only piecewise linear functions which is the lowest feasible polynomial degree for the stress space. 

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2. A Conservative Eulerian Finite Element Method for Transport and Diffusion in Moving Domains

   https://doi.org/10.1515/cmam-2024-0055  

   Olshanskii, Maxim; von_Wahl, Henry (June 2025, Computational Methods in Applied Mathematics)

   Abstract The paper introduces a finite element method for an Eulerian formulation of partial differential equations governing the transport and diffusion of a scalar quantity in a time-dependent domain. The method follows the idea from[C. Lehrenfeld and M. Olshanskii,An Eulerian finite element method for PDEs in time-dependent domains,ESAIM Math. Model. Numer. Anal. 53 2019, 2, 585–614]of a solution extension to realise the Eulerian time-stepping scheme. However, a reformulation of the partial differential equation is suggested to derive a scheme which conserves the quantity under consideration exactly on the discrete level. For the spatial discretisation, the paper considers an unfitted finite element method. Ghost-penalty stabilisation is used to realise the discrete solution extension and gives a scheme robust against arbitrary intersections between the mesh and geometry interface. The stability is analysed for both first- and second-order backward differentiation formula versions of the scheme. Several numerical examples in two and three spatial dimensions are included to illustrate the potential of this method. 

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3. Quasi-projectivity of images of mixed period maps

   https://doi.org/10.1515/crelle-2023-0070  

   Bakker, Benjamin; Brunebarbe, Yohan; Tsimerman, Jacob (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove a mixed version of a conjecture of Griffiths: that the closure of the image of any admissible mixed period map is quasi-projective, with a natural ample bundle. Specifically, we consider the map from the image of the mixed period map to the image of the period map of the associated graded. On the one hand, we show in a precise manner that the parts of this map parametrizing extension data of non-adjacent-weight pure Hodge structures are quasi-affine. On the other hand, extensions of adjacent-weight pure polarized Hodge structures are parametrized by a compact complex torus (the intermediate Jacobian) equipped with a natural theta bundle which is ample in Griffiths transverse directions.Our proof makes heavy use of o-minimality, and recent work with B. Klingler associating an ${ℝ}_{\mathrm{an},\exp }${\mathbb{R}_{\mathrm{an},\exp}}-definable structure to mixed period domains and admissible mixed period maps. 

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4. A mixed elasticity formulation for fluid–poroelastic structure interaction

   https://doi.org/10.1051/m2an/2021083  

   Li, Tongtong; Yotov, Ivan (January 2022, ESAIM: Mathematical Modelling and Numerical Analysis)

   We develop a mixed finite element method for the coupled problem arising in the interaction between a free fluid governed by the Stokes equations and flow in deformable porous medium modeled by the Biot system of poroelasticity. Mass conservation, balance of stress, and the Beavers–Joseph–Saffman condition are imposed on the interface. We consider a fully mixed Biot formulation based on a weakly symmetric stress-displacement-rotation elasticity system and Darcy velocity-pressure flow formulation. A velocity-pressure formulation is used for the Stokes equations. The interface conditions are incorporated through the introduction of the traces of the structure velocity and the Darcy pressure as Lagrange multipliers. Existence and uniqueness of a solution are established for the continuous weak formulation. Stability and error estimates are derived for the semi-discrete continuous-in-time mixed finite element approximation. Numerical experiments are presented to verify the theoretical results and illustrate the robustness of the method with respect to the physical parameters. 

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5. Trimmed serendipity finite element differential forms

   https://doi.org/10.1090/mcom/3354  

   Gillette, Andrew; Kloefkorn, Tyler (May 2018, Mathematics of Computation)

   We introduce the family of trimmed serendipity finite element differential form spaces, defined on cubical meshes in any number of dimensions, for any polynomial degree, and for any form order. The relation between the trimmed serendipity family and the (non-trimmed) serendipity family developed by Arnold and Awanou [Math. Comp. 83(288) 2014] is analogous to the relation between the trimmed and (non-trimmed) polynomial finite element differential form families on simplicial meshes from finite element exterior calculus. We provide degrees of freedom in the general setting and prove that they are unisolvent for the trimmed serendipity spaces. The sequence of trimmed serendipity spaces with a fixed polynomial order r provides an explicit example of a system described by Christiansen and Gillette [ESAIM:M2AN 50(3) 2016], namely, a minimal compatible finite element system on squares or cubes containing order r-1 polynomial differential forms. 

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