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1. Virtual finite element method for two-dimensional Maxwell interface problems with a background unfitted mesh

   https://doi.org/10.1142/S0218202521500652  

   Cao, S.; Chen, L.; Guo, R (March 2021, Mathematical models and methods in applied sciences)

   A virtual element method (VEM) with the first order optimal convergence order is developed for solving two dimensional Maxwell interface problems on a special class of polygonal meshes that are cut by the interface from a background unfitted mesh. A novel virtual space is introduced on a virtual triangulation of the polygonal mesh satisfying a maximum angle condition, which shares exactly the same degrees of freedom as the usual H(curl)-conforming virtual space. This new virtual space serves as the key to prove that the optimal error bounds of the VEM are independent of high aspect ratio of the possible anisotropic polygonal mesh near the interface. 

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2. The VEM for time-harmonic Maxwell equations in inhomogeneous media with Lipschitz interface

   https://doi.org/10.1142/S0218202525500289  

   Chen, Chunyu; Guo, Ruchi; Wei, Huayi (May 2025, Mathematical Models and Methods in Applied Sciences)

   It has been extensively studied in the literature that solving Maxwell equations is very sensitive to mesh structures, space conformity and solution regularity. Roughly speaking, for almost all the methods in the literature, optimal convergence for low-regularity solutions heavily relies on conforming spaces and highly regular simplicial meshes. This can be a significant limitation for many popular methods based on broken spaces and non-conforming or polytopal meshes often used for inhomogeneous media, as the discontinuity of electromagnetic parameters can lead to quite low regularity of solutions near media interfaces. This very issue can be potentially worsened by geometric singularities, making those methods particularly challenging to apply. In this paper, we present a lowest-order virtual element method for solving an indefinite time-harmonic Maxwell equation in 2D inhomogeneous media with quite arbitrary polytopal meshes, and the media interface is allowed to have geometric singularity to cause low regularity. We employ the “virtual mesh” technique originally invented in [S. Cao, L. Chen and R. Guo, A virtual finite element method for two-dimensional Maxwell interface problems with a background unfitted mesh, Math. Models Methods Appl. Sci. 31 (2021) 2907–2936] for error analysis. This work admits three key novelties: (i) the proposed method is theoretically guaranteed to achieve robust optimal convergence for solutions with merely [Formula: see text] regularity, [Formula: see text]; (ii) the polytopal element shape can be highly anisotropic and shrinking, and an explicit formula is established to describe the relationship between the shape regularity and solution regularity; (iii) we show that the stabilization term is needed to produce optimal convergent solutions for indefinite problems. Extensive numerical experiments will be given to demonstrate the effectiveness of the proposed method. 

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3. Immersed virtual element methods for electromagnetic interface problems in three dimensions

   https://doi.org/10.1142/S0218202523500112  

   Cao, Shuhao; Chen, Long; Guo, Ruchi (March 2023, Mathematical Models and Methods in Applied Sciences)

   Finite element methods for electromagnetic problems modeled by Maxwell-type equations are highly sensitive to the conformity of approximation spaces, and non-conforming methods may cause loss of convergence. This fact leads to an essential obstacle for almost all the interface-unfitted mesh methods in the literature regarding the application to electromagnetic interface problems, as they are based on non-conforming spaces. In this work, a novel immersed virtual element method for solving a three-dimensional (3D) H(curl) interface problem is developed, and the motivation is to combine the conformity of virtual element spaces and robust approximation capabilities of immersed finite element spaces. The proposed method is able to achieve optimal convergence. To develop a systematic framework, the [Formula: see text], H(curl) and H(div) interface problems and their corresponding problem-orientated immersed virtual element spaces are considered all together. In addition, the de Rham complex will be established based on which the Hiptmair–Xu (HX) preconditioner can be used to develop a fast solver for the H(curl) interface problem. 

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4. A mixed quasi-trace surface finite element method for the Laplace–Beltrami problem

   https://doi.org/10.1515/jnma-2025-0124  

   Demlow, Alan (March 2026, Journal of Numerical Mathematics)

   Abstract Trace finite element methods have become a popular option for solving surface partial differential equations, especially in problems where surface and bulk effects are coupled. In such methods a surface mesh is formed by approximately intersecting the continuous surface on which the PDE is posed with a three-dimensional (bulk) tetrahedral mesh. InH¹-conforming trace methods, the surface finite element space is obtained by restricting a bulk finite element space to the surface mesh. It is not clear how to carry out a similar procedure in order to obtain other important types of finite element spaces such asH(div)-conforming spaces. Following previous work of Olshanskii, Reusken, and Xu onH¹-conforming methods, we develop a “quasi-trace” mixed method for the Laplace–Beltrami problem. The finite element mesh is taken to be the intersection of the surface with a regular tetrahedral bulk mesh as previously described, resulting in a surface triangulation that is highly unstructured and anisotropic but satisfies a classical maximum angle condition. The mixed method is then employed on this mesh. Optimal error estimates with respect to the bulk mesh size are proved along with superconvergent estimates for the projection of the scalar error and a postprocessed scalar approximation. Higher-order surface approximations are also considered. 

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5. Optimal Geometric Multigrid Preconditioners for HDG-P0 Schemes for the reaction-diffusion equation and the Generalized Stokes equations

   https://doi.org/10.1051/m2an/2023025  

   Fu, Guosheng; Kuang, Wenzheng (May 2023, ESAIM: Mathematical Modelling and Numerical Analysis)

   We present the lowest-order hybridizable discontinuous Galerkin schemes with numerical integration (quadrature), denoted as HDG-P0 for the reaction-diffusion equation and the generalized Stokes equations on conforming simplicial meshes in two- and three-dimensions. Here by lowest order, we mean that the (hybrid) finite element space for the global HDG facet degrees of freedom (DOFs) is the space of piecewise constants on the mesh skeleton. A discontinuous piecewise linear space is used for the approximation of the local primal unknowns. We give the optimal a priori error analysis of the proposed HDG-P0 schemes, which hasn’t appeared in the literature yet for HDG discretizations as far as numerical integration is concerned. Moreover, we propose optimal geometric multigrid preconditioners for the statically condensed HDG-P0 linear systems on conforming simplicial meshes. In both cases, we first establish the equivalence of the statically condensed HDG system with a (slightly modified) nonconforming Crouzeix–Raviart (CR) discretization, where the global (piecewise-constant) HDG finite element space on the mesh skeleton has a natural one-to-one correspondence to the nonconforming CR (piecewise-linear) finite element space that live on the whole mesh. This equivalence then allows us to use the well-established nonconforming geometry multigrid theory to precondition the condensed HDG system. Numerical results in two- and three-dimensions are presented to verify our theoretical findings. 

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