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1. A mortar method for the coupled Stokes-Darcy problem using the MAC scheme for Stokes and mixed finite elements for Darcy

   https://doi.org/10.1007/s10596-023-10267-6  

   Boon, Wietse M; Gläser, Dennis; Helmig, Rainer; Weishaupt, Kilian; Yotov, Ivan (June 2024, Computational Geosciences)
2. High-performance finite elements with MFEM

   https://doi.org/10.1177/10943420241261981  

   Andrej, Julian; Atallah, Nabil; Bäcker, Jan-Phillip; Camier, Jean-Sylvain; Copeland, Dylan; Dobrev, Veselin; Dudouit, Yohann; Duswald, Tobias; Keith, Brendan; Kim, Dohyun; et al (June 2024, The International Journal of High Performance Computing Applications)

   The MFEM (Modular Finite Element Methods) library is a high-performance C++ library for finite element discretizations. MFEM supports numerous types of finite element methods and is the discretization engine powering many computational physics and engineering applications across a number of domains. This paper describes some of the recent research and development in MFEM, focusing on performance portability across leadership-class supercomputing facilities, including exascale supercomputers, as well as new capabilities and functionality, enabling a wider range of applications. Much of this work was undertaken as part of the Department of Energy’s Exascale Computing Project (ECP) in collaboration with the Center for Efficient Exascale Discretizations (CEED). 

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3. Droplet formation simulation using mixed finite elements

   https://doi.org/10.1063/5.0089752  

   Nathawani, Darsh K.; Knepley, Matthew G. (June 2022, Physics of Fluids)

   Droplet formation happens in finite time due to the surface tension force. The linear stability analysis is useful to estimate the size of a droplet but fails to approximate the shape of the droplet. This is due to a highly nonlinear flow description near the point where the first pinch-off happens. A one-dimensional axisymmetric mathematical model was first developed by Eggers and Dupont [“Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] using asymptotic analysis. This asymptotic approach to the Navier–Stokes equations leads to a universal scaling explaining the self-similar nature of the solution. Numerical models for the one-dimensional model were developed using the finite difference [Eggers and Dupont, “Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] and finite element method [Ambravaneswaran et al., “Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops,” Phys. Fluids 14, 2606–2621 (2002)]. The focus of this study is to provide a robust computational model for one-dimensional axisymmetric droplet formation using the Portable, Extensible Toolkit for Scientific Computation. The code is verified using the Method of Manufactured Solutions and validated using previous experimental studies done by Zhang and Basaran [“An experimental study of dynamics of drop formation,” Phys. Fluids 7, 1184–1203 (1995)]. The present model is used for simulating pendant drops of water, glycerol, and paraffin wax, with an aspiration of extending the application to simulate more complex pinch-off phenomena. 

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4. Direct serendipity finite elements on cuboidal hexahedra

   https://doi.org/10.1016/j.cma.2024.117500  

   Arbogast, Todd; Wang, Chuning (January 2025, Computer Methods in Applied Mechanics and Engineering)

   We construct direct serendipity finite elements on general cuboidal hexahedra, which are 𝐻1-conforming and optimally approximate to any order. The new finite elements are direct in that the shape functions are directly defined on the physical element. Moreover, they are serendipity by possessing a minimal number of degrees of freedom satisfying the conformity requirement. Their shape function spaces consist of polynomials plus (generally nonpolynomial) supplemental functions, where the polynomials are included for the approximation property and supplements are added to achieve 𝐻1 -conformity. The finite elements are fully constructive. The shape function spaces of higher order 𝑟 ≥ 3 are developed first, and then the lower order spaces are constructed as subspaces of the third order space. Under a shape regularity assumption, and a mild restriction on the choice of supplemental functions, we develop the convergence properties of the new direct serendipity finite elements. Numerical results with different choices of supplements are compared on two mesh sequences, one regularly distorted and the other one randomly distorted. They all possess a convergence rate that aligns with the theory, while a slight difference lies in their performance. 

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5. Preconditioning mixed finite elements for tide models

   https://doi.org/10.1016/j.camwa.2020.11.002  

   Kirby, Robert C.; Kernell, Tate (January 2021, Computers & Mathematics with Applications)

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