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Jun Hu

Publications and source records attributed to Jun Hu.

At least 271 records · Page 15Linked to original sources
arXiv2014

Superconvergence of both the Crouzeix-Raviart and Morley elements

In this paper, a new method is proposed to prove the superconvergence of both the Crouzeix-Raviart and Morley elements. The main idea is to fully employ equivalences with the first order Raviart-Thomas element and the first order Hellan-Herrmann-Johnson element, respectively. In this way, some special conformity of discrete stresses is explored and superconvergence of mixed elements can be used to analyze superconvergence of nonconforming elements. Finally, a half order superconvergence by postprocessing is proved for both nonconforming elements.

math.NA↗
arXiv2014

Superconvergence of simple conforming mixed finite elements for linear elasticity on rectangular grids in any space dimension

This paper is to prove superconvergence of a family of simple conforming mixed finite elements of first orderfor the linear elasticity problem with the Hellinger--Reissner variational formulation. The analysis is based on three main ingredients: a new interpolation operator, a new expansion method, and a new iterative argument for superconvergence analysis.

math.NA↗
arXiv2014

Guaranteed Lower and upper bounds for eigenvalues of second order elliptic operators in any dimension

In this paper, a new method is proposed to produce guaranteed lower bounds for eigenvalues of general second order elliptic operators in any dimension. Unlike most methods in the literature, the proposed method only needs to solve one discrete eigenvalue problem but not involves any base or intermediate eigenvalue problems, and does not need any a priori information concerning exact eigenvalues either. Moreover, it just assumes basic regularity of exact eigenfunctions. This method is defined by a novel generalized Crouzeix-Raviart element which is proved to yield asymptotic lower bounds for eigenvalues of general second order elliptic operators, and a simple post-processing method. As a byproduct, a simple and cheap method is also proposed to obtain guaranteed upper bounds for eigenvalues, which is based on generalized Crouzeix-Raviart element approximate eigenfunctions, an averaging interpolation from the the generalized Crouzeix-Raviart element space to the conforming linear element space, and an usual Rayleigh-Ritz procedure. The ingredients for the analysis consist of a crucial projection property of the canonical interpolation operator of the generalized Crouzeix-Raviart element, explicitly computable constants for two interpolation operators. Numerics are provided to demonstrate the theoretical results.

math.NA↗
arXiv2014

Runge-Kutta Central Discontinuous Galerkin BGK Method for the Navier-Stokes Equations

In this paper, we propose a Runge-Kutta (RK) central discontinuous Galerkin (CDG) gas-kinetic BGK method for the Navier-Stokes equations. The proposed method is based on the CDG method defined on two sets of overlapping meshes to avoid discontinuous solutions at cell interfaces, as well as the gas-kinetic BGK model to evaluate fluxes for both convection and diffusion terms. Redundant representation of the numerical solution in the CDG method offers great convenience in the design of gas-kinetic BGK fluxes. Specifically, the evaluation of fluxes at cell interfaces of one set of computational mesh is right inside the cells of the staggered mesh, hence the corresponding particle distribution function for flux evaluation is much simpler than that in existing gas-kinetic BGK methods. As a central scheme, the proposed CDG-BGK has doubled the memory requirement as the corresponding DG scheme; on the other hand, {for the convection part,} the CFL time step constraint of the CDG method for numerical stability is relatively large compared with that for the DG method. Numerical boundary conditions have to be treated with special care. Numerical examples for 1D and 2D viscous flow simulations are presented to validate the accuracy and robustness of the proposed RK CDG-BGK method.

math.NA↗
arXiv2014

The enriched Crouzeix-Raviart elements are equivalent to the Raviart-Thomas elements

For both the Poisson model problem and the Stokes problem in any dimension, this paper proves that the enriched Crouzeix-Raviart elements are actually identical to the first order Raviart-Thomas elements in the sense that they produce the same discrete stresses. This result improves the previous result in literature which, for two dimensions, states that the piecewise constant projection of the stress by the first order Raviart-Thomas element is equal to that by the Crouzeix-Raviart element. For the eigenvalue problem of Laplace operator, this paper proves that the error of the enriched Crouzeix-Raviart element is equivalent to that of the Raviart-Thomas element up to higher order terms.

math.NA↗
arXiv2014

Giant magnetic anisotropy of transition-metal dimers on defected graphene

Continuous miniaturization of magnetic units in spintronics and quantum computing devices inspires efforts to search for magnetic nanostructures with large magnetic anisotropy energy (MAE). Typical nanostructures including molecular magnets, magnetic nanoclusters and magnetic nanowires have MAEs of a few meV so their blocking temperature is mostly lower than 50 K. In this work, we demonstrated the feasibility of achieving giant MAE in systems with transition metal dimers on defected and decorated graphene, based on density functional theory calculations. In particular, either a Pt-Ir dimer on a single vacancy or an Os-Ru dimer on a nitrogen-decorated divacancy possesses an MAE larger than 60 meV and high structural stability. Interestingly, their magnetic anisotropy can be conveniently manipulated by using external electric field. These features make them good candidates for the use in room temperature spintronics and quantum computing devices.

cond-mat.mtrl-sci↗
arXiv2014

Magnetism and Magnetic Anisotropy of Transition Metal-Phthalocyanine Molecules

Search for single-molecule magnets (SMMs) with high blocking temperature (TB) is urgent for practical applications in magnetic recording, molecular spintronics and quantum computing. Based on the First-principles calculations, magnetic anisotropy energies (MAE) of the transition metal-Phthalocyanine (TM-Pc) molecules are investigated and the mechanism that determines the MAE of TM-Pc molecules is established. In particular, colossal MAE > 100 meV can be obtained by adding an Os atom on RuPc and OsPc, so these molecules may offer ultrahigh thermal stability in devices.

cond-mat.mtrl-sci↗
arXiv2014

Control of the magnetism and magnetic anisotropy of a single-molecule magnet with an electric field

Through systematics density functional calculations, the mechanism of the substrate induced spin reorientation transition in FePc/O-Cu(110) was explained in terms of charge transfer and rearrangement of Fe-d orbitals. Moreover, we found giant magnetoelectric effects in this system, manifested by the sensitive dependences of its magnetic moment and magnetic anisotropy energy on external electric field. In particular, the direction of magnetization of FePc/O-Cu(110) is switchable between in-plane and perpendicular axes, simply by applying an external electric field of 0.5 eV/Å along the surface normal.

cond-mat.mtrl-sci↗
arXiv2013

Characterization of the asymptotic Teichmuller space of the open unit disk through shears

We give a parametrization to the asymptotic Teichmuller space of the open unit disk through equivalent classes of shear functions induced by quasisymmetric homeomorphisms on the Farey tesselation of the unit disk. Then using the parametrization, we define a new metric on the asymptotic Teichmuller space. Two other related metrics are also introduced on the asymptotic Teichmuller space by using cross-ratio distortions or quadrilateral dilatations under the boundary maps on degenerating sequences of quadruples or quadrilaterals. We show that the topologies induced by the three metrics are equivalent to the one induced by the Teichmuller metric on the asymptotic Teichmuller space. Before proving our main results, we revisit and rectify a mistake in the proof in the reference [21] on the characterization of quasisymmetric homeomorphisms in terms of shear functions.

math.GT↗
arXiv2013

Convergence and optimality of the adaptive Morley element method

This paper is devoted to the convergence and optimality analysis of the adaptive Morley element method for the fourth order elliptic problem. A new technique is developed to establish a quasi-orthogonality which is crucial for the convergence analysis of the adaptive nonconforming method. By introducing a new parameter-dependent error estimator and further establishing a discrete reliability property, sharp convergence and optimality estimates are then fully proved for the fourth order elliptic problem.

math.NA↗
arXiv2013

Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem

In this paper, we analyze the convergence and optimality of a standard adaptive nonconforming linear element method for the Stokes problem. After establishing a special quasi--orthogonality property for both the velocity and the pressure in this saddle point problem, we introduce a new prolongation operator to carry through the discrete reliability analysis for the error estimator. We then use a specially defined interpolation operator to prove that, up to oscillation, the error can be bounded by the approximation error within a properly defined nonlinear approximate class. Finally, by introducing a new parameter-dependent error estimator, we prove the convergence and optimality estimates.

math.NA↗
arXiv2013

The lower bound of the error estimate in the L2 norm for the Adini element of the biharmonic equation

This paper is devoted to the $L^2$ norm error estimate of the Adini element for the biharmonic equation. Surprisingly, a lower bound is established which proves that the $ L^2$ norm convergence rate can not be higher than that in the energy norm. This proves the conjecture of [Lascaux and Lesaint, Some nonconforming finite elements for the plate bending problem, RAIRO Anal. Numer. 9 (1975), pp. 9--53.] that the convergence rates in both $L^2$ and $H^1$ norms can not be higher than that in the energy norm for this element.

math.NA↗
arXiv2013

Asymmetric transportation induced by thermal noise at the nanoscale

Based on a simple model, we theoretically show that asymmetric transportation is possible in nanoscale systems experiencing thermal noise without the presence of external fluctuations. The key to this theoretical advance is that the correlation lengths of the thermal fluctuations become significantly long for nanoscale systems. This differs from macroscopic systems in which the thermal noises are usually treated as white noise. Our observation does not violate the second law of thermodynamics, since at nanoscale, extra energy is required to keep the asymmetric structure against thermal fluctuations.

cond-mat.stat-mech↗
arXiv2013

The Lower Bounds for Eigenvalues of Elliptic Operators --By Nonconforming Finite Element Methods

The aim of the paper is to introduce a new systematic method that can produce lower bounds for eigenvalues. The main idea is to use nonconforming finite element methods. The general conclusion herein is that if local approximation properties of nonconforming finite element spaces $V_h$ are better than global continuity properties of $V_h$, corresponding methods will produce lower bounds for eigenvalues. More precisely, under three conditions on continuity and approximation properties of nonconforming finite element spaces we first show abstract error estimates of approximate eigenvalues and eigenfunctions. Subsequently, we propose one more condition and prove that it is sufficient to guarantee nonconforming finite element methods to produce lower bounds for eigenvalues of symmetric elliptic operators. As one application, we show that this condition hold for most nonconforming elements in literature. As another important application, this condition provides a guidance to modify known nonconforming elements in literature and to propose new nonconforming elements. In fact, we enrich locally the Crouzeix-Raviart element such that the new element satisfies the condition; we propose a new nonconforming element for second order elliptic operators and prove that it will yield lower bounds for eigenvalues. Finally, we prove the saturation condition for most nonconforming elements.

math.NA↗
arXiv2013

The simplest mixed finite element method for linear elasticity in the symmetric formulation on $n$-rectangular grids

A family of mixed finite elements is proposed for solving the first order system of linear elasticity equations in any space dimension, where the stress field is approximated by symmetric finite element tensors. This family of elements has a perfect matching between the stress components and the displacement. The discrete spaces for the normal stress $σ_{ii}$, the shear stress $σ_{ij}$ and the displacement $u_i$ are $\operatorname{span}\{1,x_i\}$, $\operatorname{span}\{1,x_i,x_j\}$ and $\operatorname{span}\{1\}$, respectively, on rectangular grids. In particular, the definition remains the same for all space dimensions. As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. In 1D, this element is nothing else but the 1D Raviart-Thomas element, which is the only conforming element in this family. In 2D and higher dimensions, they are new elements but of the minimal degrees of freedom. The total degrees of freedom per element is 2 plus 1 in 1D, 7 plus 2 in 2D, and 15 plus 3 in 3D. The previous record of the least degrees of freedom is, 13 plus 4 in 2D, and 54 plus 12 in 3D, on the rectangular grid. These elements are the simplest element for any space dimension. The well-posedness condition and the optimal a priori error estimate of the family of finite elements are proved for both pure displacement and traction problems. Numerical tests in 2D and 3D are presented to show a superiority of the new element over others, as a superconvergence is surprisingly exhibited.

math.NA↗
arXiv2013

Nonconforming finite element methods on quadrilateral meshes

It is well-known that it is comparatively difficult to design nonconforming finite elements on quadrilateral meshes by using Gauss-Legendre points on each edge of triangulations. One reason lies in that these degrees of freedom associated to these Gauss-Legendre points are not all linearly independent for usual expected polynomial spaces, which explains why only several lower order nonconforming quadrilateral finite elements can be found in literature. The present paper proposes two families of nonconforming finite elements of any odd order and one family of nonconforming finite elements of any even order on quadrilateral meshes. Degrees of freedom are given for these elements, which are proved to be well-defined for their corresponding shape function spaces in a unifying way. These elements generalize three lower order nonconforming finite elements on quadrilaterals to any order. In addition, these nonconforming finite element spaces are shown to be full spaces which is somehow not discussed for nonconforming finite elements in literature before.

math.NA↗
arXiv2013

Learning Belief Networks in Domains with Recursively Embedded Pseudo Independent Submodels

A pseudo independent (PI) model is a probabilistic domain model (PDM) where proper subsets of a set of collectively dependent variables display marginal independence. PI models cannot be learned correctly by many algorithms that rely on a single link search. Earlier work on learning PI models has suggested a straightforward multi-link search algorithm. However, when a domain contains recursively embedded PI submodels, it may escape the detection of such an algorithm. In this paper, we propose an improved algorithm that ensures the learning of all embedded PI submodels whose sizes are upper bounded by a predetermined parameter. We show that this improved learning capability only increases the complexity slightly beyond that of the previous algorithm. The performance of the new algorithm is demonstrated through experiment.

cs.AI↗