Two-Level Newton Iteration Methods for Navier-Stokes Type Variational Inequality Problem

This paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality problem. We solve a small Navier-Stokes problem on the coarse mesh with mesh size $H$ and solve a large linearized Navier-Stokes problem on the fine mesh with mesh size $h$. The error estimates derived show that if we choose $h=\mathcal{O}(|\log h|^{1/2}H^3)$, then the two-level method we provide has the same $H^1$ and $L^2$ convergence orders of the velocity and the pressure as the one-level stabilized method. However, the $L^2$ convergence order of the velocity is not consistent with that of one-level stabilized method. Finally, we give the numerical results to support the theoretical analysis.     

Two-Scale Picard Stabilized Finite Volume Method for the Incompressible Flow

In this paper, we consider a two-scale stabilized finite volume method for the two-dimensional stationary incompressible flow approximated by the lowest equal-order element pair $P_1-P_1$ which does not satisfy the inf-sup condition. The two-scale method consists of solving a small non-linear system on the coarse mesh and then solving a linear Stokes equations on the fine mesh. Convergence of the optimal order in the $H^1$-norm for velocity and the $L^2$-norm for pressure is obtained. The error analysis shows there is the same convergence rate between the two-scale stabilized finite volume solution and the usual stabilized finite volume solution on a fine mesh with relation $h =\mathcal{O}(H^2)$. Numerical experiments completely confirm theoretic results. Therefore, this method presented in this paper is of practical importance in scientific computation.     

Finite Element $θ$-Schemes for the Acoustic Wave Equation

In this paper, we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave equation. The schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation method. We derive general stability conditions for the family of implicit schemes covering some well-known CFL conditions. Optimal error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the $L^2$-norm error over a finite time interval converges optimally as $\mathcal{O}(h^{p+1}+∆t^s)$, where $p$ denotes the polynomial degree, $s$=2 or 4, $h$ the mesh size, and $∆t$ the time step.     

Two-Grid Discretization Scheme for Nonlinear Reaction Diffusion Equation by Mixed Finite Element Methods

In this paper, we study an efficient scheme for nonlinear reaction-diffusion equations discretized by mixed finite element methods. We mainly concern the case when pressure coefficients and source terms are nonlinear. To linearize the nonlinear mixed equations, we use the two-grid algorithm. We first solve the nonlinear equations on the coarse grid, then, on the fine mesh, we solve a linearized problem using Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy $H =\mathcal{O}(h^{\frac{1}{2}})$. As a result, solving such a large class of nonlinear equations will not be much more difficult than getting solutions of one linearized system.     

${ \mathcal P }{ \mathcal T }$-symmetry berry phases,topology and ${ \mathcal P }{ \mathcal T }$-symmetry breaking

We study the complex Berry phases in non-Hermitian systems with parity- and time-reversal $\left({ \mathcal P }{ \mathcal T }\right)$ symmetry. We investigate a kind of two-level system with ${ \mathcal P }{ \mathcal T }$ symmetry. We find that the real part of the the complex Berry phases have two quantized values and they are equal to either 0 or π, which originates from the topology of the Hermitian eigenstates. We also find that if we change the relative parameters of the Hamiltonian from the unbroken-${ \mathcal P }{ \mathcal T }$-symmetry phase to the broken-${ \mathcal P }{ \mathcal T }$-symmetry phase, the imaginary part of the complex Berry phases are divergent at the exceptional points. We exhibit two concrete examples in this work, one is a two-level toys model, which has nontrivial Berry phases; the other is the generalized Su–Schrieffer–Heeger (SSH) model that has physical loss and gain in every sublattice. Our results explicitly demonstrate the relation between complex Berry phases, topology and ${ \mathcal P }{ \mathcal T }$-symmetry breaking and enrich the field of the non-Hermitian physics.     

Two-Level Defect-Correction Method for Steady Navier-Stokes Problem with Friction Boundary Conditions

In this paper, we present two-level defect-correction finite element method for steady Navier-Stokes equations at high Reynolds number with the friction boundary conditions, which results in a variational inequality problem of the second kind. Based on Taylor-Hood element, we solve a variational inequality problem of Navier-Stokes type on the coarse mesh and solve a variational inequality problem of Navier-Stokes type corresponding to Newton linearization on the fine mesh. The error estimates for the velocity in the $H^1$ norm and the pressure in the $L^2$ norm are derived. Finally, the numerical results are provided to confirm our theoretical analysis.     

大雷诺数Navier-Stokes方程的两水平亚格子模型稳定化方法

基于两重网格离散方法,提出三种求解大雷诺数定常Navier-Stokes方程的两水平亚格子模型稳定化有限元算法.其基本思想是首先在一粗网格上求解带有亚格子模型稳定项的Navier-Stokes方程,然后在细网格上分别用三种不同的校正格式求解一个亚格子模型稳定化的线性问题,以校正粗网格解.通过适当的稳定化参数和粗细网格尺寸的选取,这些算法能取得最优渐近收敛阶的有限元解.最后,用数值模拟验证三种算法的有效性.     

Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods

In this paper, we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method. We use two Newton iterations on the fine grid in our methods. Firstly, we solve an original nonlinear problem on the coarse nonlinear grid, then we use Newton iterations on the fine grid twice. The two-grid idea is from Xu$'$s work [SIAM J. Numer. Anal., 33 (1996), pp. 1759-1777] on standard finite method. We also obtain the error estimates for the algorithms of the two-grid method. It is shown that the algorithm achieves asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy $h=\mathcal{O}(H^{(4k+1)/(k+1)})$.     

Oscillatory stability of quantum droplets in $\text{}{ \mathcal P }{ \mathcal T }$-symmetric optical lattice

We study stability and collisions of quantum droplets (QDs) forming in a binary bosonic condensate trapped in parity-time (${ \mathcal P }{ \mathcal T }$)-symmetric optical lattices. It is found that the stability of QDs in the ${ \mathcal P }{ \mathcal T }$-symmetric system depends strongly on the values of the imaginary part W₀ of the ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices, self-repulsion strength g, and the condensate norm N. As expected, the ${ \mathcal P }{ \mathcal T }$-symmetric QDs are entirely unstable in the broken ${ \mathcal P }{ \mathcal T }$-symmetric phase. However, the ${ \mathcal P }{ \mathcal T }$-symmetric QDs exhibit oscillatory stability with the increase of N and g in the unbroken ${ \mathcal P }{ \mathcal T }$-symmetric phase. Finally, collisions between ${ \mathcal P }{ \mathcal T }$-symmetric QDs are considered. The collisions of droplets with unequal norms are completely different from that in free space. Besides, a stable ${ \mathcal P }{ \mathcal T }$-symmetric QDs collides with an unstable ones tend to merge into breathers after the collision.     

Solving forward and inverse problems of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via PINN deep learning

In this paper, based on physics-informed neural networks (PINNs), a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations (PDEs) and other types of nonlinear physical models, we study the nonlinear Schrödinger equation (NLSE) with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential, which is an important physical model in many fields of nonlinear physics. Firstly, we choose three different initial values and the same Dirichlet boundary conditions to solve the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via the PINN deep learning method, and the obtained results are compared with those derived by the traditional numerical methods. Then, we investigate the effects of two factors (optimization steps and activation functions) on the performance of the PINN deep learning method in the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential. Ultimately, the data-driven coefficient discovery of the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential or the dispersion and nonlinear items of the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential can be approximately ascertained by using the PINN deep learning method. Our results may be meaningful for further investigation of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential in the deep learning.     

不可压缩流的并行两水平稳定有限元算法

提出一种数值求解定常不可压缩Stokes方程的并行两水平Grad-div稳定有限元算法。首先在粗网格中求解Grad-div稳定化的全局解, 再在相互重叠的细网格子区域上并行纠正。通过对稳定化参数、粗细网格尺寸恰当的选取, 该方法可得到最优收敛率, 数值结果验证了算法的高效性。     

Dark Energy Phenomenon from Backreaction Effect

In this paper, we interpret the dark energy phenomenon as an averaged effect caused by small scale inhomogeneities of the universe with the use of the spatial averaged approach of Buchert. Two models are considered here, one of which assumes that the backreaction term ${\cal Q}_\mathcal{D}$ and the averaged spatial Ricci scalar $\langle\mathcal{R}\rangle_\mathcal{D}$ obey the scaling laws of the volume scale factor $a_\mathcal{D}$ at adequately late times, and the other one adopts the ansatz that the backreaction term ${\cal Q}_\mathcal{D}$ is a constant in the recent universe. Thanks to the effective geometry introduced by Larena et al. in their previous work, we confront these two backreaction models with latest type Ia supernova and Hubble parameter observations, coming out with the results that the constant backreaction model is slightly favoured over the other model and the best fitting backreaction term in the scaling backreaction model behaves almost like a constant. Also, the numerical results show that the constant backreaction model predicts a smaller expansion rate and decelerated expansion rate than the other model does at redshifts higher than about 1, and both backreaction terms begin to accelerate the universe at a redshift around 0.5.     

A spin-less particle on a rotating curved surface in Minkowski space

In Minkowski space ${ \mathcal M }$, we derive the effective Schrödinger equation describing a spin-less particle confined to a rotating curved surface ${ \mathcal S }$. Using the thin-layer quantization formalism to constrain the particle on ${ \mathcal S }$, we obtain the relativity-corrected geometric potential ${V}_{g}^{{\prime} }$, and a novel effective potential ${\tilde{V}}_{g}$ related to both the Gaussian curvature and the geodesic curvature of the rotating surface. The Coriolis effect and the centrifugal potential also appear in the equation. Subsequently, we apply the surface Schrödinger equation to a rotating cylinder, sphere and torus surfaces, in which we find that the interplays between the rotation and surface geometry can contribute to the energy spectrum based on the potentials they offer.     

A High-Accuracy Finite Difference Scheme for Solving Reaction-Convection-Diffusion Problems with a Small Diffusivity

This paper is devoted to a new high-accuracy finite difference scheme for solving reaction-convection-diffusion problems with a small diffusivity $\varepsilon$. With a novel treatment for the reaction term, we first derive a difference scheme of accuracy $\mathcal{O}(\varepsilon^2 h + \varepsilon h^2 + h^3)$ for the 1-D case. Using the alternating direction technique, we then extend the scheme to the 2-D case on a nine-point stencil. We apply the high-accuracy finite difference scheme to solve the 2-D steady incompressible Navier-Stokes equations in the stream function-vorticity formulation. Numerical examples are given to illustrate the effectiveness of the proposed difference scheme. Comparisons made with some high-order compact difference schemes show that the newly proposed scheme can achieve good accuracy with better stability.     

One-dimensional $\mathcal{PT}$-symmetric acoustic heterostructure

The explorations of parity-time ($\mathcal{PT}$)-symmetric acoustics have resided at the frontier in physics, and the pre-existing accessing of exceptional points typically depends on Fabry-Perot resonances of the coupling interlayer sandwiched between balanced gain and loss components. Nevertheless, the concise $\mathcal{PT}$-symmetric acoustic heterostructure, eliminating extra interactions caused by the interlayer, has not been researched in depth. Here we derive the generalized unitary relation for one-dimensional (1D) $\mathcal{PT}$-symmetric heterostructure of arbitrary complexity, and demonstrate four disparate patterns of anisotropic transmission resonances (ATRs) accompanied by corresponding spontaneous phase transitions. As a special case of ATR, the occasional bidirectional transmission resonance reconsolidates the ATR frequencies that split when waves incident from opposite directions, whose spatial profiles distinguish from a unitary structure. The derived theoretical relation can serve as a predominant signature for the presence of $\mathcal{PT}$ symmetry and $\mathcal{PT}$-symmetry-breaking transition, which may provide substantial support for the development of prototype devices with asymmetric acoustic responses.     

Fragmentation function and hadronic production of the heavy supersymmetric hadrons

The light top-squark may be the lightest squark, and its lifetime may be ‘long enough’ in a kind of SUSY models that have not been ruled out yet experimentally, so colorless ‘supersymmetric hadrons (superhadrons)’ (q is a quark excluding the t-quark) may be formed as long as the light top-squark can be produced. The fragmentation function of into heavy ‘supersymmetric hadrons (superhadrons)’ (Q̄=c̄ or b̄) and hadronic production of the superhadrons are investigated quantitatively. The fragmentation function is calculated precisely. Due to the difference in spin of the SUSY component, the asymptotic behavior of the fragmentation function is different from those of the existing ones. The fragmentation function is also applied to compute the production of heavy superhadrons at the hadronic colliders Tevatron and LHC in the so-called fragmentation approach. The resultant cross-section for the heavy superhadrons is too small to observe at Tevatron, but large enough at LHC, when all the relevant parameters in the SUSY models are taken within the favored region for the heavy superhadrons. The production of ‘light superhadrons’ (q=u,d,s) is also roughly estimated with the same SUSY parameters. It is pointed out that the production cross-sections of the light superhadrons may be much greater than those of the heavy superhadrons, so that even at Tevatron the light superhadrons may be produced in great quantities. PACS 12.38.Bx; 13.87.Fh; 12.60.Jv; 14.80.Ly     

Joint System Quantum Descriptions Arising from Local Quantumness

We study the entropy flux in the stationary state of a finite one-dimensional sample ${\mathcal{S}}$ connected at its left and right ends to two infinitely extended reservoirs ${\mathcal{R}_{l/r}}$ at distinct (inverse) temperatures ${\beta_{l/r}}$ and chemical potentials ${\mu_{l/r}}$ . The sample is a free lattice Fermi gas confined to a box [0, L] with energy operator ${h_{\mathcal{S}, L}= - \Delta + v}$ . The Landauer-Büttiker formula expresses the steady state entropy flux in the coupled system ${\mathcal{R}_l + \mathcal{S} + \mathcal{R}_r}$ in terms of scattering data. We study the behaviour of this steady state entropy flux in the limit ${L \to \infty}$ and relate persistence of transport to norm bounds on the transfer matrices of the limiting half-line Schrödinger operator ${h_\mathcal{S}}$ .     

Courant–Dorfman Algebras and their Cohomology

We introduce a new type of algebra, the Courant–Dorfman algebra. These are to Courant algebroids what Lie–Rinehart algebras are to Lie algebroids, or Poisson algebras to Poisson manifolds. We work with arbitrary rings and modules, without any regularity, finiteness or non-degeneracy assumptions. To each Courant–Dorfman algebra ${(\mathcal{R}, \mathcal{E})}$ we associate a differential graded algebra ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ in a functorial way by means of explicit formulas. We describe two canonical filtrations on ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ , and derive an analogue of the Cartan relations for derivations of ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ ; we classify central extensions of ${\mathcal{E}}$ in terms of ${H^2(\mathcal{E}, \mathcal{R})}$ and study the canonical cocycle ${\Theta \in \mathcal{C}^3(\mathcal{E}, \mathcal{R})}$ whose class ${[\Theta]}$ obstructs re-scalings of the Courant–Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ ; for Courant–Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ is isomorphic to the one constructed in Roytenberg (On the structure of graded symplectic supermanifolds and Courant algebroids. American Mathematical Society, Providence, 2002) using graded manifolds.     

Error Analysis for a Non-Monotone FEM for a Singularly Perturbed Problem with Two Small Parameters

In this paper, we consider a singularly perturbed convection-diffusion problem. The problem involves two small parameters that gives rise to two boundary layers at two endpoints of the domain. For this problem, a non-monotone finite element methods is used. A priori error bound in the maximum norm is obtained. Based on the a priori error bound, we show that there exists Bakhvalov-type mesh that gives optimal error bound of$\mathcal{O}(N^{−2})$ which is robust with respect to the two perturbation parameters. Numerical results are given that confirm the theoretical result.     

Reichenbachian Common Cause Systems

A partition C_{i i∈ I} of a Boolean algebra $\mathcal{S}$ in a probability measure space $(\mathcal{S},p)$ is called a Reichenbachian common cause system for the correlated pair A,B of events in $\mathcal{S}$ if any two elements in the partition behave like a Reichenbachian common cause and its complement, the cardinality of the index set I is called the size of the common cause system. It is shown that given any correlation in $(\mathcal{S},p)$ , and given any finite size n>2, the probability space $(\mathcal{S},p)$ can be embedded into a larger probability space in such a manner that the larger space contains a Reichenbachian common cause system of size n for the correlation. It also is shown that every totally ordered subset in the partially ordered set of all partitions of $\mathcal{S}$ contains only one Reichenbachian common cause system. Some open problems concerning Reichenbachian common cause systems are formulated.