More Like this

1. A Dilation-Based Seamless Multiscale Method for Elliptic Problems

   https://doi.org/10.1137/24M1668755  

   Chen, Ziheng; Engquist, Björn (June 2025, Multiscale Modeling & Simulation)
2. A Space-Time Multiscale Mortar Mixed Finite Element Method for Parabolic Equations

   https://doi.org/10.1137/21M1447945  

   Jayadharan, Manu; Kern, Michel; Vohralík, Martin; Yotov, Ivan (April 2023, SIAM Journal on Numerical Analysis)
3. A macroelement stabilization for mixed finite element/finite volume discretizations of multiphase poromechanics

   https://doi.org/10.1007/s10596-020-09964-3  

   Camargo, Julia T.; White, Joshua A.; Borja, Ronaldo I. (January 2020, Computational Geosciences)

   Strong coupling between geomechanical deformation and multiphase fluid flow appears in a variety of geoscience applications. A common discretization strategy for these problems is a continuous Galerkin finite element scheme for the momentum balance equation and a finite volume scheme for the mass balance equations. When applied within a fully implicit solution strategy, however, this discretization is not intrinsically stable. In the limit of small time steps or low permeabilities, spurious oscillations in the piecewise-constant pressure field, i.e., checkerboarding, may be observed. Further, eigenvalues associated with the spurious modes will control the conditioning of the matrices and can dramatically degrade the convergence rate of iterative linear solvers. Here, we propose a stabilization technique in which the mass balance equations are supplemented with stabilizing flux terms on a macroelement basis. The additional stabilization terms are dependent on a stabilization parameter. We identify an optimal value for this parameter using an analysis of the eigenvalue distribution of the macroelement Schur complement matrix. The resulting method is simple to implement and preserves the underlying sparsity pattern of the original discretization. Another appealing feature of the method is that mass is exactly conserved on macroelements, despite the addition of artificial fluxes. The efficacy of the proposed technique is demonstrated with several numerical examples. 

   more » « less
4. 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. 

   more » « less
5. A Continuous-Discontinuous Galerkin Method for the Modeling and Simulation of Electromagnetic Multiscale Problems

   https://doi.org/10.1109/ICEAA52647.2021.9539862  

   Yan, Su (August 2021, ICEAA-IEEE APWC-USNC URSI RSM 2021)