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1. A simple virtual element-based flux recovery on quadtree

   https://doi.org/10.3934/era.2021054  

   Cao, Shuhao (January 2021, Electronic Research Archive)

   In this paper, we introduce a simple local flux recovery for \begin{document}$$ \mathcal{Q}_k $$\end{document} finite element of a scalar coefficient diffusion equation on quadtree meshes, with no restriction on the irregularities of hanging nodes. The construction requires no specific ad hoc tweaking for hanging nodes on \begin{document}$ l $$\end{document}-irregular (\begin{document}$$ l\geq 2 $$\end{document}$) meshes thanks to the adoption of virtual element families. The rectangular elements with hanging nodes are treated as polygons as in the flux recovery context. An efficient a posteriori error estimator is then constructed based on the recovered flux, and its reliability is proved under common assumptions, both of which are further verified in numerics. 

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2. Feedback particle filter for collective inference

   https://doi.org/10.3934/fods.2021018  

   Kim, Jin-Won; Taghvaei, Amirhossein; Chen, Yongxin; Mehta, Prashant G. (January 2021, Foundations of Data Science)

   The purpose of this paper is to describe the feedback particle filter algorithm for problems where there are a large number (\begin{document}$ M $$\end{document}) of non-interacting agents (targets) with a large number (\begin{document}$$ M $$\end{document}) of non-agent specific observations (measurements) that originate from these agents. In its basic form, the problem is characterized by data association uncertainty whereby the association between the observations and agents must be deduced in addition to the agent state. In this paper, the large-\begin{document}$$ M $$\end{document} limit is interpreted as a problem of collective inference. This viewpoint is used to derive the equation for the empirical distribution of the hidden agent states. A feedback particle filter (FPF) algorithm for this problem is presented and illustrated via numerical simulations. Results are presented for the Euclidean and the finite state-space cases, both in continuous-time settings. The classical FPF algorithm is shown to be the special case (with \begin{document}$$ M = 1 $$\end{document}) of these more general results. The simulations help show that the algorithm well approximates the empirical distribution of the hidden states for large \begin{document}$$ M $$\end{document}$. 

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3. Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation

   https://doi.org/10.1090/mcom/3842  

   Wang, Haijin; Li, Fengyan; Shu, Chi-Wang; Zhang, Qiang (November 2023, Mathematics of Computation)

   In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the $L^2$norm of the numerical solution does not increase in time, under the time step condition $\mathrm{\tau <\#comment/>}\le <\#comment/>\mathcal{F}(h/c,d/c^2)$, with the convection coefficient $c$, the diffusion coefficient $d$, and the mesh size $h$. The function $\mathcal{F}$depends on the specific IMEX temporal method, the polynomial degree $k$of the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomes $\mathrm{\tau <\#comment/>}\lesssim <\#comment/>h/c$in the convection-dominated regime and it becomes $\mathrm{\tau <\#comment/>}\lesssim <\#comment/>d/c^2$in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions. 

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4. Mimetic finite difference operators and higher order quadratures

   https://doi.org/10.1007/s13137-023-00230-z  

   Srinivasan, Anand; Dumett, Miguel; Paolini, Christopher; Miranda, Guillermo F.; Castillo, José E. (December 2023, GEM - International Journal on Geomathematics)

   Abstract Mimetic finite difference operators$$\textbf{D}$$ $D$,$$\textbf{G}$$ $G$are discrete analogs of the continuous divergence (div) and gradient (grad) operators. In the discrete sense, these discrete operators satisfy the same properties as those of their continuum counterparts. In particular, they satisfy a discrete extended Gauss’ divergence theorem. This paper investigates the higher-order quadratures associated with the fourth- and sixth- order mimetic finite difference operators, and show that they are indeed numerical quadratures and satisfy the divergence theorem. In addition, extensions to curvilinear coordinates are treated. Examples in one and two dimensions to illustrate numerical results are presented that confirm the validity of the theoretical findings. 

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5. A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility

   https://doi.org/10.1090/mcom/3843  

   Hou, Dianming; Ju, Lili; Qiao, Zhonghua (November 2023, Mathematics of Computation)

   In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with a general mobility. The proposed fully-discrete scheme is carefully constructed based on the combination of first and second-order backward differentiation formulas with nonuniform time steps for temporal approximation and the central finite difference for spatial discretization. The discrete maximum bound principle is proved of the proposed scheme by using the kernel recombination technique under certain mild constraints on the time steps and the ratios of adjacent time step sizes. Furthermore, we rigorously derive the discrete $H^1$error estimate and energy stability for the classic constant mobility case and the $L^{\mathrm{\infty <\#comment/>}}$error estimate for the general mobility case. Various numerical experiments are also presented to validate the theoretical results and demonstrate the performance of the proposed method with a time adaptive strategy. 

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