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1. A coupled HDG/DG method for porous media with conducting/sealing faults

   https://doi.org/10.1016/j.camwa.2025.10.015  

   Cesmelioglu, Aycil; Kuchta, Miroslav; Lee, Jeonghun J; Rhebergen, Sander (December 2025, Computers & Mathematics with Applications)

   We introduce and analyze a coupled hybridizable discontinuous Galerkin/discontinuous Galerkin (HDG/DG) method for porous media in which we allow fully and partly immersed faults, and faults that separate the domain into two disjoint subdomains. We prove well-posedness and present an a priori error analysis of the discretization. Numerical examples verify our analysis. 

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2. Convergence and optimality of an adaptive modified weak Galerkin finite element method

   https://doi.org/10.1002/num.23027  

   Yingying, Xie; Cao, Shuhao; Chen, Long; Zhong, Liuqiang (April 2023, Numerical Methods for Partial Differential Equations)

   Abstract An adaptive modified weak Galerkin method (AmWG) for an elliptic problem is studied in this article, in addition to its convergence and optimality. The modified weak Galerkin bilinear form is simplified without the need of the skeletal variable, and the approximation space is chosen as the discontinuous polynomial space as in the discontinuous Galerkin method. Upon a reliable residual‐baseda posteriorierror estimator, an adaptive algorithm is proposed together with its convergence and quasi‐optimality proved for the lowest order case. The primary tool is to bridge the connection between the modified weak Galerkin method and the Crouzeix–Raviart nonconforming finite element. Unlike the traditional convergence analysis for methods with a discontinuous polynomial approximation space, the convergence of AmWG is penalty parameter free. Numerical results are presented to support the theoretical results. 

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3. Flow in porous media with low dimensional fractures by employing enriched Galerkin method

   https://doi.org/https://doi.org/10.1016/j.advwatres.2020.103620  

   Kadeethum, T; Nick, H.M; Lee, Sanghyun; Ballarin, F. (January 2020, Advances in water resources)

   This paper presents the enriched Galerkin discretization for modeling fluid flow in fractured porous media using the mixed-dimensional approach. The proposed method has been tested against published benchmarks. Since fracture and porous media discontinuities can significantly influence single- and multi-phase fluid flow, the heterogeneous and anisotropic matrix permeability setting is utilized to assess the enriched Galerkin performance in handling the discontinuity within the matrix domain and between the matrix and fracture domains. Our results illustrate that the enriched Galerkin method has the same advantages as the discontinuous Galerkin method; for example, it conserves local and global fluid mass, captures the pressure discontinuity, and provides the optimal error convergence rate. However, the enriched Galerkin method requires much fewer degrees of freedom than the discontinuous Galerkin method in its classical form. The pressure solutions produced by both methods are similar regardless of the conductive or non-conductive fractures or heterogeneity in matrix permeability. This analysis shows that the enriched Galerkin scheme reduces the computational costs while offering the same accuracy as the discontinuous Galerkin so that it can be applied for large-scale flow problems. Furthermore, the results of a time-dependent problem for a three-dimensional geometry reveal the value of correctly capturing the discontinuities as barriers or highly-conductive fractures. 

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4. Binary neutron star mergers using a discontinuous Galerkin-finite difference hybrid method

   https://doi.org/10.1088/1361-6382/ad88cf  

   Deppe, Nils; Foucart, Francois; Bonilla, Marceline_S; Boyle, Michael; Corso, Nicholas_J; Duez, Matthew_D; Giesler, Matthew; Hébert, François; Kidder, Lawrence_E; Kim, Yoonsoo; et al (November 2024, Classical and Quantum Gravity)

   Abstract We present a discontinuous Galerkin-finite difference hybrid scheme that allows high-order shock capturing with the discontinuous Galerkin method for general relativistic magnetohydrodynamics in dynamical spacetimes. We present several optimizations and stability improvements to our algorithm that allow the hybrid method to successfully simulate single, rotating, and binary neutron stars. The hybrid method achieves the efficiency of discontinuous Galerkin methods throughout almost the entire spacetime during the inspiral phase, while being able to robustly capture shocks and resolve the stellar surfaces. We also use Cauchy-characteristic evolution to compute the first gravitational waveforms at future null infinity from binary neutron star mergers. The simulations presented here are the first successful binary neutron star inspiral and merger simulations using discontinuous Galerkin methods. 

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5. Positivity and Maximum Principle Preserving Discontinuous Galerkin Finite Element Schemes for a Coupled Flow and Transport

   https://doi.org/10.1007/s10915-025-02830-3  

   Gong, Shihua; Lee, Young-Ju; Li, Yukun; Yu, Yue (March 2025, Journal of Scientific Computing)

   Abstract We introduce a new concept of the local flux conservation and investigate its role in the coupled flow and transports. We demonstrate how the proposed concept of the locally conservative flux can play a crucial role in obtaining the$$L^2$$ $L^2$norm stability of the discontinuous Galerkin finite element scheme for the transport in the coupled system with flow. In particular, the lowest order discontinuous Galerkin finite element for the transport is shown to inherit the positivity and maximum principle when the locally conservative flux is used, which has been elusive for many years in literature. The theoretical results established in this paper are based on the equivalence between Lesaint-Raviart discontinuous Galerkin scheme and Brezzi-Marini-Süli discontinuous Galerkin scheme for the linear hyperbolic system as well as the relationship between the Lesaint-Raviart discontinuous Galerkin scheme and the characteristic method along the streamline. Sample numerical experiments have then been performed to justify our theoretical findings. 

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