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   https://doi.org/10.1016/j.jcp.2019.06.058  

   Nguyen, Lam H. ; Schillinger, Dominik ( November 2019 , Journal of Computational Physics)
2. A learning-based multiscale method and its application to inelastic impact problems

   https://doi.org/10.1016/j.jmps.2021.104668  

   Liu, Burigede ; Kovachki, Nikola ; Li, Zongyi ; Azizzadenesheli, Kamyar ; Anandkumar, Anima ; Stuart, Andrew M. ; Bhattacharya, Kaushik ( January 2022 , Journal of the Mechanics and Physics of Solids)
3. A Continuous-Discontinuous Galerkin Method for the Modeling and Simulation of Electromagnetic Multiscale Problems

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   Yan, Su ( August 2021 , ICEAA-IEEE APWC-USNC URSI RSM 2021)
4. A dynamic variational multiscale method on unstructured meshes for stationary transport problems

   https://doi.org/10.1002/fld.5184  

   Xu, Zelu ; Sahni, Onkar ( March 2023 , International Journal for Numerical Methods in Fluids)

   This paper presents a variational multiscale (VMS) based finite element method where the stabilization parameter is computed dynamically. The current dynamic procedure takes in a general structure/form of the stabilization parameter with unknown coefficients and computes them dynamically in a local fashion resulting in a dynamic VMS‐based finite element method. Thus, a static stabilization parameter with pre‐defined coefficients is not needed. A variational Germano identity (VGI) based local procedure suitable for unstructured meshes is developed to perform the dynamic computation in a local fashion. The local VGI based procedure is applied for each interior vertex in the mesh and unknown coefficients are first determined locally at each vertex, and subsequently, for each element a maximum value is taken over the vertices of the element. To make the current procedure practical, a coarser secondary solution is constructed from the primary coarse‐scale solution, which is done locally over a patch of elements around each interior vertex. Further, averaging steps are employed to make the local dynamic procedure robust. Currently, the new dynamic VMS formulation is applied to steady problems governed by the advection‐diffusion and incompressible Navier‐Stokes equations in both 1D and 2D to demonstrate its efficacy and effectiveness.

    

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5. Mixed Generalized Multiscale Finite Element Method for a Simplified Magnetohydrodynamics Problem in Perforated Domains

   https://doi.org/10.3390/computation8020058  

   Alekseev, Valentin ; Tang, Qili ; Vasilyeva, Maria ; Chung, Eric T. ; Efendiev, Yalchin ( June 2020 , Computation)

   In this paper, we consider a coupled system of equations that describes simplified magnetohydrodynamics (MHD) problem in perforated domains. We construct a fine grid that resolves the perforations on the grid level in order to use a traditional approximation. For the solution on the fine grid, we construct approximation using the mixed finite element method. To reduce the size of the fine grid system, we will develop a Mixed Generalized Multiscale Finite Element Method (Mixed GMsFEM). The method differs from existing approaches and requires some modifications to represent the flow and magnetic fields. Numerical results are presented for a two-dimensional model problem in perforated domains. This model problem is a special case for the general 3D problem. We study the influence of the number of multiscale basis functions on the accuracy of the method and show that the proposed method provides a good accuracy with few basis functions. 

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