Adini‐Q1‐P3 FEM for general elastic multi‐structure problems

An Adini‐Q₁‐P₃ finite element method is introduced to solve general elastic multi‐structure problems, where displacements on bodies, longitudinal displacements on plates, longitudinal displacements and rotational angles on rods are discretized by conforming linear (bilinear or trilinear) elements, and transverse displacements on plates and rods are discretized by Adini elements and Hermite elements of third order, respectively. The unique solvability and optimal error estimates in the energy norm are established for the discrete method, whose numerical performance is illustrated by some numerical examples. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1092–1112, 2011     

The Menger property for infinite ordered sets

It is shown that, if an ordered set P contains at most k pairwise disjoint maximal chains, where k is finite, then every finite family of maximal chains in P has a cutset of size at most k. As a corollary of this, we obtain the following Menger-type result that, if in addition, P contains k pairwise disjoint complete maximal chains, then the whole family, M (P), of maximal chains in P has a cutset of size k. We also give a direct proof of this result. We give an example of an ordered set P in which every maximal chain is complete, P does not contain infinitely many pairwise disjoint maximal chains (but arbitrarily large finite families of pairwise disjoint maximal chains), and yet M (P) does not have a cutset of size <x, where x is any given (infinite) cardinal. This shows that the finiteness of k in the above corollary is essential and disproves a conjecture of Zaguia.     

Finding multiple solutions to elliptic systems with polynomial nonlinearity

Elliptic systems with polynomial nonlinearity usually possess multiple solutions. In order to find multiple solutions, such elliptic systems are discretized by eigenfunction expansion method (EEM). Error analysis of the discretization is presented, which is different from the error analysis of EEM for scalar elliptic equations in three aspects: first, the choice of framework for the nonlinear operator and the corresponding isomorphism of the linearized operator; second, the definition of an auxiliary problem in deriving the relation between the L² norm and H¹ norm of the Ritz projection error; third, the bilinearity/nonbilinearity of the linearized variational forms. The symmetric homotopy for the discretized equations preserves not only D₄ symmetry, but also structural symmetry. With the symmetric homotopy, a filter strategy and a finite element Newton refinement, multiple solutions to a system of semilinear elliptic equations arising from Bose–Einstein condensate are found.     

Hypergroup deformations and Markov chains

Persi Diaconis and Phil Hanlon in their interesting paper⁽⁴⁾ give the rates of convergence of some Metropolis Markov chains on the cubeZ ^d (2). Markov chains on finite groups that are actually random walks are easier to analyze because the machinery of harmonic analysis is available. Unfortunately, Metropolis Markov chains are, in general, not random walks on group structure. In attempting to understand Diaconis and Hanlon's work, the authors were led to the idea of a hypergroup deformation of a finite groupG, i.e., a continuous family of hypergroups whose underlying space isG and whose structure is naturally related to that ofG. Such a deformation is provided forZ ^d (2), and it is shown that the Metropolis Markov chains studied by Diaconis and Hanlon can be viewed as random walks on the deformation. A direct application of the Diaconis-Shahshahani Upper Bound Lemma, which applies to random walks on hypergroups, is used to obtain the rate of convergence of the Metropolis chains starting at any point. When the Markov chains start at 0, a result in Diaconis and Hanlon⁽⁴⁾ is obtained with exactly the same rate of convergence. These results are extended toZ ^d (3).Research supported in part by the Office of Research and Sponsored Programs, University of Oregon.     

Multigrid methods for H(div) in three dimensions with nonoverlapping domain decomposition smoothers

We design and analyze V‐cycle multigrid methods for an H(div) problem discretized by the lowest‐order Raviart–Thomas hexahedral element. The smoothers in the multigrid methods involve nonoverlapping domain decomposition preconditioners that are based on substructuring. We prove uniform convergence of the V‐cycle methods on bounded convex hexahedral domains (rectangular boxes). Numerical experiments that support the theory are also presented.     

Field induced aggregation in electrorheological suspension: kernel form and self similar solutions

Within the framework of the study of the fibrillation mechanism in an electrorheological (ER) suspension, this work presents a comparison between the self similar solutions when the kernel is K_i,j ~ (i⁻¹ + j⁻¹) and the behaviour of the chains growth. Till now, the field induced chains formation has only been studied by numerical or experimental methods. The work of Fournier and Lauren?ot (Communications in Mathematical Physics 256 2005) on the Smoluchowski’s equation allows us to present an analytical solution for the field induced pearl chains in a colloidal ER suspension.     

Optimal Error Estimate of the Penalty Finite Element Method for the Micropolar Fluid Equations

We present an optimal error estimate of the numerical velocity, pressure, and angular velocity for the fully discrete penalty finite element method of the micropolar equations when the parameters ?, Δ t, and h are sufficiently small. In order to obtain this estimate, we present the time discretization of the penalty micropolar equation that is based on the backward Euler scheme; the spatial discretization of the time discretized penalty micropolar equation is based on a finite elements space pair (X _h , M _h ) that satisfies some approximations properties.     

Quantized CP approximation and sparse tensor interpolation of function‐generated data

In this article, we consider the iterative schemes to compute the canonical polyadic (CP) approximation of quantized data generated by a function discretized on a large uniform grid in an interval on the real line. This paper continues the research on the quantics‐tensor train (QTT) method (“O(d log N)‐quantics approximation of N‐d tensors in high‐dimensional numerical modeling” in Constructive Approximation, 2011) developed for the tensor train (TT) approximation of the quantized images of function related data. In the QTT approach, the target vector of length 2^L is reshaped to a Lth‐order tensor with two entries in each mode (quantized representation) and then approximated by the QTT tensor including 2r²L parameters, where r is the maximal TT rank. In what follows, we consider the alternating least squares (ALS) iterative scheme to compute the rank‐r CP approximation of the quantized vectors, which requires only 2rL?2^L parameters for storage. In the earlier papers (“Tensors‐structured numerical methods in scientific computing: survey on recent advances” in Chemom Intell Lab Syst, 2012), such a representation was called Q_Can format, whereas in this paper, we abbreviate it as the QCP (quantized canonical polyadic) representation. We test the ALS algorithm to calculate the QCP approximation on various functions, and in all cases, we observed the exponential error decay in the QCP rank. The main idea for recovering a discretized function in the rank‐r QCP format using the reduced number of the functional samples, calculated only at O(2rL) grid points, is presented. The special version of the ALS scheme for solving the arising minimization problem is described. This approach can be viewed as the sparse QCP‐interpolation method that allows to recover all 2rL representation parameters of the rank‐r QCP tensor. Numerical examples show the efficiency of the QCP‐ALS‐type iteration and indicate the exponential convergence rate in r.     

Field induced aggregation in electrorheological suspension: kernel form and self similar solutions

Within the framework of the study of the fibrillation mechanism in an electrorheological (ER) suspension, this work presents a comparison between the self similar solutions when the kernel is K_i,j ~ (i⁻¹ + j⁻¹) and the behaviour of the chains growth. Till now, the field induced chains formation has only been studied by numerical or experimental methods. The work of Fournier and Lauren?ot (Communications in Mathematical Physics 256 2005) on the Smoluchowski’s equation allows us to present an analytical solution for the field induced pearl chains in a colloidal ER suspension. René Limage: Chercheur indépendant, dipl?mé de l’Université de Liége.     

On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

In this paper, we couple regularization techniques of nondifferentiable optimization with the h‐version of the boundary element method (h‐BEM) to solve nonsmooth variational problems arising in contact mechanics. As a model example, we consider the delamination problem. The variational formulation of this problem leads to a hemivariational inequality with a nonsmooth functional defined on the contact boundary. This problem is first regularized and then discretized by an h‐BEM. We prove convergence of the h‐BEM Galerkin solution of the regularized problem in the energy norm, provide an a priori error estimate and give a numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Maximum norm a posteriori error estimate for a 3d singularly perturbed semilinear reaction-diffusion problem

A singularly perturbed semilinear reaction-diffusion problem in the unit cube, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio condition is imposed. This result is obtained by combining (i) sharp bounds on the Green’s function of the continuous differential operator in the Sobolev W ^1,1 and W ^2,1 norms and (ii) a special representation of the residual in terms of an arbitrary current mesh and the current computed solution. Numerical results on a priori chosen meshes are presented that support our theoretical estimate.     

Linear and Discontinuous Approximations for Optimal Control Problems

Abstract

An optimal control problem for 2D and 3D elliptic equations is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise linear but discontinuous functions. The state and the adjoint state are discretized by linear finite elements. The paper is focused on similarities and differences to piecewise constant and piecewise linear (continuous) approximation of the controls. Approximation of order h in the L ^∞-norm is proved in the main result.     

A priori error estimates for optimal control problems with pointwise constraints on the gradient of the state

We analyze a finite element approximation of an elliptic optimal control problem with pointwise bounds on the gradient of the state variable. We derive convergence rates if the control space is discretized implicitly by the state equation. In contrast to prior work we obtain these results directly from classical results for the W ^1,∞-error of the finite element projection, without using adjoint information. If the control space is discretized directly, we first prove a regularity result for the optimal control to control the approximation error, based on which we then obtain analogous convergence rates.     

Two-Grid Solution of Symm's Integral Equation

We propose two-grid iteration methods for Symm's integral equation discretized by quadrature-collocation or quadrature methods. Asymptotically the optimal order of error estimate is achieved already on the first iteration, for some modifications on the second iteration. This enables us to introduce some solvers which are of the optimal convergence order and cheap in a practical implementation; the cost varies between O(N²) and O(N log N) arithmetic operations. Numerical experiments confirm the approximation properties of the schemes.     

An analysis of the numerical solution of Fredholm integral equations of the first kind

Summary This paper analyzes the numerical solution of Fredholm integral equations of the first kindTx=y by means of finite rank and other approximation methods replacingTx=y byT _N x=y _N ,N=1,2, .... The operatorsT andT _N can be viewed as operators from eitherL ₂[a, b] toL ₂[c,d] or as operators fromL [a, b] toL [c, d]. A complete analysis of the fully discretized problem as compared with the continuous problemTx=y is also given. The filtered least squares minimum norm solutions (LSMN) to the discrete problem and toT _N x=y are compared with the LSMN solution ofTx=y. Rates of convergence are included in all cases and are in terms of the mesh spacing of the quadrature for the fully discretized problem.     

A Central Limit Theorem for Convex Chains in the Square

Points P ₁ ,... ,P _n in the unit square define a convex n -chain if they are below y=x and, together with P ₀ =(0,0) and P _n+1 =(1,1) , they are in convex position. Under uniform probability, we prove an almost sure limit theorem for these chains that uses only probabilistic arguments, and which strengthens similar limit shape statements established by other authors. An interesting feature is that the limit shape is a direct consequence of the method. The main result is an accompanying central limit theorem for these chains. A weak convergence result implies several other statements concerning the deviations between random convex chains and their limit. Received April 17, 1998, and in revised form December 4, 1998.     

Auxiliary space preconditioning in H 0(curl; Ω)

We adapt the principle of auxiliary space preconditioning as presented in [J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56 (1996), pp. 215–235.] to H (curl; ω)-elliptic variational problems discretized by means of edge elements. The focus is on theoretical analysis within the abstract framework of subspace correction. Employing a Helmholtz-type splitting of edge element vector fields we can establish asymptotic h-uniform optimality of the preconditioner defined by our auxiliary space method. This author was fully supported by Hong Kong RGC grant (Project No. 403403) This author acknowledges the support from a Direct Grant of CUHK during his visit at The Chinese University of Hong Kong.     

Perturbed Markov Chains with Damping Component

The paper is devoted to studies of regularly and singularly perturbed Markov chains with damping component. In such models, a matrix of transition probabilities is regularised by adding a special damping matrix multiplied by a small damping (perturbation) parameter ε. We perform a detailed perturbation analysis for such Markov chains, particularly, give effective upper bounds for the rate of approximation for stationary distributions of unperturbed Markov chains by stationary distributions of perturbed Markov chains with regularised matrices of transition probabilities, asymptotic expansions for approximating stationary distributions with respect to damping parameter, explicit coupling type upper bounds for the rate of convergence in ergodic theorems for n-step transition probabilities, as well as ergodic theorems in triangular array mode.

     

Infinitely dimensional control Markov branching chains in random environments

First of all we introduce the concepts of infinitely dimensional control Markov branching chains in random environments (β-MBCRE) and prove the existence of such chains, then we introduce the concepts of conditional generating functionals and random Markov transition functions of such chains and investigate their branching property. Base on these concepts we calculate the moments of the β-MBCRE and obtain the main results of this paper such as extinction probabilities, polarization and proliferation rate. Finally we discuss the classification of β-MBCRE according to the different standards.     

Lexicographic behaviour of chains

This paper introduces a different approach to the study of the existence of numerical representations of totally ordered sets (chains). We pay attention to the properties of non-representable chains showing that, under certain conditions, those chains must have a sort of lexicographic behaviour similar to that of the lexicographic plane. We prove that a countably bounded connected chain (Z, \prec )(Z, \prec ) admits a lexicographic decomposition as a subset of the lexicographic product \Bbb R ×Z\Bbb R \times Z. Then we apply our approach to state both a sufficient and a necessary condition for the lack of utility functions. The concept of planar chain is also introduced.