Analysis of an exactly mass conserving space-time hybridized discontinuous Galerkin method for the time-dependent Navier--Stokes equations

We introduce and analyze a space-time hybridized discontinuous Galerkin method for the evolutionary Navier--Stokes equations. Key features of the numerical scheme include point-wise mass conservation, energy stability, and pressure robustness. We prove that there exists a solution to the resulting nonlinear algebraic system in two and three spatial dimensions, and that this solution is unique in two spatial dimensions under a small data assumption. A priori error estimates are derived for the velocity in a mesh-dependent energy norm

A Divergence-Free and H(div)H(div)-Conforming Embedded-Hybridized DG Method for the Incompressible Resistive MHD equations

We proposed a divergence-free and $H(div)$-conforming embedded-hybridized discontinuous Galerkin (E-HDG) method for solving stationary incompressible viso-resistive magnetohydrodynamic (MHD) equations. In particular, the E-HDG method is computationally far more advantageous over the hybridized discontinuous Galerkin (HDG) counterpart in general. The benefit is even significant in the three-dimensional/high-order/fine mesh scenario. On a simplicial mesh, our method with a specific choice of the approximation spaces is proved to be well-posed for the linear case. Additionally, the velocity and magnetic fields are divergence-free and $H(div)$-conforming for both linear and nonlinear cases. Moreover, the results of well-posedness analysis, divergence-free property, and $H(div)$-conformity can be directly applied to the HDG version of the proposed approach. The HDG or E-HDG method for the linearized MHD equations can be incorporated into the fixed point Picard iteration to solve the nonlinear MHD equations in an iterative manner. We examine the accuracy and convergence of our E-HDG method for both linear and nonlinear cases through various numerical experiments including two- and three-dimensional problems with smooth and singular solutions. For smooth problems, the results indicate that convergence rates in the $L^2$ norm for the velocity and magnetic fields are optimal in the regime of low Reynolds number and magnetic Reynolds number. Furthermore, the divergence error is machine zero for both smooth and singular problems. Finally, we numerically demonstrated that our proposed method is pressure-robust

Hypothalamic Agrp Neurons Drive Stereotypic Behaviors beyond Feeding

SummaryThe nervous system evolved to coordinate flexible goal-directed behaviors by integrating interoceptive and sensory information. Hypothalamic Agrp neurons are known to be crucial for feeding behavior. Here, however, we show that these neurons also orchestrate other complex behaviors in adult mice. Activation of Agrp neurons in the absence of food triggers foraging and repetitive behaviors, which are reverted by food consumption. These stereotypic behaviors that are triggered by Agrp neurons are coupled with decreased anxiety. NPY5 receptor signaling is necessary to mediate the repetitive behaviors after Agrp neuron activation while having minor effects on feeding. Thus, we have unmasked a functional role for Agrp neurons in controlling repetitive behaviors mediated, at least in part, by neuropeptidergic signaling. The findings reveal a new set of behaviors coupled to the energy homeostasis circuit and suggest potential therapeutic avenues for diseases with stereotypic behaviors.PaperCli