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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. 

Abstract Branched DNA motifs serve as the basic construction elements for all synthetic DNA nanostructures. However, precise control of branching orientation remains a key challenge to further heighten the overall structural order. In this study, we use two strategies to control the branching orientation. The first one is based on immobile Holliday junctions which employ specific nucleotide sequences at the branch points which dictate their orientation. The second strategy is to use angle‐enforcing struts to fix the branching orientation with flexible spacers at the branch points. We have also demonstrated that the branching orientation control can be achieved dynamically, either by canonical Watson–Crick base pairing or non‐canonical nucleobase interactions (e.g., i‐motif and G‐quadruplex). With precise angle control and feedback from the chemical environment, these results will enable novel DNA nanomechanical sensing devices, and precisely‐ordered three‐dimensional architectures.