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1. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

   https://doi.org/10.3934/dcdss.2022016  

   Wu, Yixuan ; Zhang, Yanzhi ( January 2022 , Discrete & Continuous Dynamical Systems - S)

   In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian \begin{document}$ (- \Delta)^\frac{{ \alpha}}{{2}} $\end{document} for \begin{document}$ \alpha \in (0, 2) $\end{document}. One main advantage is that our method can easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of \begin{document}$ {\mathcal O}(h^2) $\end{document}, while \begin{document}$ {\mathcal O}(h^4) $\end{document} for quadratic basis functions with \begin{document}$ h $\end{document} a small mesh size. This accuracy can be achieved for any \begin{document}$ \alpha \in (0, 2) $\end{document} and can be further increased if higher-degree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies \begin{document}$ u \in C^{m, l}(\bar{ \Omega}) $\end{document} for \begin{document}$ m \in {\mathbb N} $\end{document} and \begin{document}$ 0 < l < 1 $\end{document}, our method has an accuracy of \begin{document}$more »{\mathcal O}(h^{\min\{m+l, \, 2\}}) $\end{document} for constant and linear basis functions, while \begin{document}$ {\mathcal O}(h^{\min\{m+l, \, 4\}}) $\end{document} for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.

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2. Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian

   https://doi.org/10.3934/dcds.2022085  

   Mayer, Martin ; Ndiaye, Cheikh Birahim ( January 2022 , Discrete and Continuous Dynamical Systems)

   We study the asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian for conformal infinities of asymptotically hyperbolic manifolds. We derive sharp expansions of the Poisson kernel and Green's functions of the conformal Laplacian near their singularities. Our expansions of the Green's functions answer the first part of the conjecture of Kim-Musso-Wei[21] in the case of locally flat conformal infinities of Poincare-Einstein manifolds and together with the Poisson kernel asymptotic is used also in our paper [25] to show solvability of the fractional Yamabe problem in that case. Our asymptotics of the Green's functions on the general case of conformal infinities of asymptotically hyperbolic space is used also in [29] to show solvability of the fractional Yamabe problem for conformal infinities of dimension \begin{document}$ 3 $\end{document} and fractional parameter in \begin{document}$ (\frac{1}{2}, 1) $\end{document} corresponding to a global case left by previous works.
3. A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems

   https://doi.org/10.3934/jcd.2022019  

   Jones, Morgan ; Peet, Matthew M. ( January 2022 , Journal of Computational Dynamics)

   Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose \begin{document}$ 1 $\end{document}-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.
4. Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition

   https://doi.org/10.3934/krm.2022003  

   Dong, Hongjie ; Guo, Yan ; Yastrzhembskiy, Timur ( January 2022 , Kinetic and Related Models)

   We establish existence of finite energy weak solutions to the kinetic Fokker-Planck equation and the linear Landau equation near Maxwellian, in the presence of specular reflection boundary condition for general domains. Moreover, by using a method of reflection and the \begin{document}$ S_p $\end{document} estimate of [7], we prove regularity in the kinetic Sobolev spaces \begin{document}$ S_p $\end{document} and anisotropic Hölder spaces for such weak solutions. Such \begin{document}$ S_p $\end{document} regularity leads to the uniqueness of weak solutions.
5. 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.