a multi level method for transmission eigenvalues of anisotropic media in this paper we propose a multi level finite element method for the transmission eigenvalue problem of anisotropic media the problem is non standard and non self adjoint with important applications in inverse scattering theory we employ a suitable finite element method to discretize the problem the resulting generalized matrix eigenvalue problem is large sparse and non hermitian to compute the smallest real transmission eigenvalue which is usually an interior eigenvalue we devise a multi level method using arnoldi iteration at the coarsest mesh the eigenvalue is obtained using arnoldi iteration with an adaptive searching technique this value is used as the initial guess for arnoldi iteration at the next mesh level this procedure is then repeated until the finest mesh level numerical examples are presented to show the viability of the method

a new a posteriori error estimate for the morley element in this paper we present a posteriori error analysis for the nonconforming morley element of the fourth order elliptic equation we propose a new residual based a posteriori error estimator and prove its reliability and efficiency these results refine those of beirao da veiga et al numer math 106 165 179 2007 by dropping two edge jump terms in both the energy norm of the error and the estimator and those of wang and zhang local a priori and a posteriori error estimates of finite elements for biharmonic equation research report 13 2006 by showing the efficiency in the sense of verfurth a review of a posteriori error estimation and adaptive mesh refinement techniques wiley teubner new york 1996 moreover the normal component in the estimators of beirao da veiga et al numer math 106 165 179 2007 and wang and zhang local a priori and a posteriori error estimates of finite elements for biharmonic equation research report 13 2006 is dropped and therefore only the tangential component of the stress on each edge comes into the estimator in addition we generalize these results to three dimensional case

a projection method for generalized eigenvalue problems using numerical integration in this paper we propose a method for finding certain eigenvalues of a generalized eigenvalue problem that lie in a given domain of the complex plane the proposed method projects the matrix pencil onto a subspace associated with the eigenvalues that are located in the domain via numerical integration

finite element methods for maxwell s transmission eigenvalues the transmission eigenvalue problem plays a critical role in the theory of qualitative methods for inhomogeneous media in inverse scattering theory efficient computational tools for transmission eigenvalues are needed to motivate improvements to theory and more importantly are parts of inverse algorithms for estimating material properties in this paper we propose two finite element methods to compute a few lowest maxwell s transmission eigenvalues which are of interest in applications since the discrete matrix eigenvalue problem is large sparse and in particular non hermitian due to the fact that the problem is neither elliptic nor self adjoint we devise an adaptive method which combines the arnoldi iteration and estimation of transmission eigenvalues exact transmission eigenvalues for balls are derived and used as a benchmark numerical examples are provided to show the viability of the proposed methods and to test the accuracy of recently derived inequalities for transmission eigenvalues

a robust numerical algorithm for computing maxwell s transmission eigenvalue problems we study a robust and efficient eigensolver for computing a few smallest positive eigenvalues of the three dimensional maxwell s transmission eigenvalue problem the discretized governing equations by the nedelec edge element result in a large scale quadratic eigenvalue problem qep for which the spectrum contains many zero eigenvalues and the coefficient matrices consist of patterns in the matrix form xy 1 z both of which prevent existing eigenvalue solvers from being efficient to remedy these difficulties we rewrite the qep as a particular nonlinear eigenvalue problem and develop a secant type iteration together with an indefinite locally optimal block preconditioned conjugate gradient lobpcg method to sequentially compute the desired positive eigenvalues furthermore we propose a novel method to solve the linear systems in each iteration of lobpcg intensive numerical experiments show that our proposed method is robust although the desired real eigenvalues are surrounded by complex eigenv

a two grid discretization scheme of non conforming finite elements for transmission eigenvalues abstract in this paper for the helmholtz transmission eigenvalue problem we propose a two grid discretization scheme of non conforming finite elements with this scheme the solution of the transmission eigenvalue problem on a fine grid h is reduced to the solution of the primal and dual eigenvalue problem on a much coarser grid h and the solutions of two linear algebraic systems with the same positive definite hermitian and block diagonal coefficient matrix on the fine grid h we prove the resulting solution still maintains an asymptotically optimal accuracy and we report some numerical examples in two dimension and three dimension on the modified zienkiewicz element to validate the efficiency of our approach for solving transmission eigenvalues

convergence of adaptive finite element methods for general second order linear elliptic pdes we prove convergence of adaptive finite element methods afems for general nonsymmetric second order linear elliptic pdes thereby extending the result of morin nochetto and siebert it siam j numer anal 38 2000 pp 466 488 it siam rev 44 2002 pp 631 658 the proof relies on quasi orthogonality which accounts for the bilinear form not being a scalar product together with novel error and oscillation reduction estimates which now do not decouple we show that afems are a contraction for the sum of energy error plus oscillation numerical experiments including oscillatory coefficients and both coercive and noncoercive convection diffusion pde illustrate the theory and yield optimal meshes

a new class of zienkiewicz type non conforming element in any dimensions in this paper a new class of zienkiewicz type non conforming finite element in n spatial dimensions with n 2 is proposed the new finite element is proved to be convergent for the biharmonic equation

a new multigrid finite element method for the transmission eigenvalue problems numerical methods for the transmission eigenvalue problems are hot topics in recent years based on the work of lin and xie 2015 we build a multigrid method to solve the problems with our method we only need to solve a series of primal and dual eigenvalue problems on a coarse mesh and the associated boundary value problems on the finer and finer meshes theoretical analysis and numerical results show that our method is simple and easy to implement and is efficient for computing real and complex transmission eigenvalues

a type of adaptive c0 non conforming finite element method for the helmholtz transmission eigenvalue problem abstract the transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering in this paper for the helmholtz transmission eigenvalue problem in r d d 2 3 we discuss the a posteriori error estimates and adaptive algorithm of a type of c 0 non conforming finite elements including the morley zienkiewicz element and the modified zienkiewicz element we give the a posteriori error estimators for approximate eigenpair and prove their reliability and efficiency for eigenfunctions we also analyze their reliability for eigenvalues numerical results not only in r 2 but also in r 3 indicate that the a posteriori error estimators are sharp and the adaptive algorithm is efficient for computing real and complex transmission eigenvalues as expected

non conforming finite element methods for transmission eigenvalue problem abstract the transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory in this paper for the helmholtz transmission eigenvalue problem we give a weak formulation which is a nonselfadjoint linear eigenvalue problem based on the weak formulation we first discuss the non conforming finite element approximation and prove the error estimates of the discrete eigenvalues obtained by the adini element the morley zienkiewicz element the modified zienkiewicz element et al and we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem

the adaptive ciarlet raviart mixed method for biharmonic problems with simply supported boundary condition abstract in this paper we study the adaptive fashion of the ciarlet raviart mixed method for biharmonic equation eigenvalue problem with simply supported boundary condition in r d we propose an a posteriori error indicator of the ciarlet raviart approximate solution for the biharmonic equation and an a posteriori error indicator of the ciarlet raviart approximate eigenfuction and prove the reliability and efficiency of the indicators we also give an a posteriori error indicator for the approximate eigenvalue and prove its reliability we design an adaptive ciarlet raviart mixed method with piecewise polynomials of degree less than or equal to m and numerical experiments show that numerical eigenvalues obtained by the method can achieve the optimal convergence order o d o f 2 m d d 2 m 2 3 d 3 m 3

mixed methods for the helmholtz transmission eigenvalues the ciarlet raviart mixed finite element method is popular for the biharmonic equations in this paper we apply this method to the helmholtz transmission eigenvalue problem which is quadratic and

a c0ipg method and its error estimates for the helmholtz transmission eigenvalue problem abstract the interior penalty methods using c 0 lagrange elements c 0 ipg developed in the last decade for the fourth order problems are an interesting topic in academia at present in this paper we apply the methods to the helmholtz transmission eigenvalue problem which is quadratic and non self adjoint and has important applications in the inverse scattering theory we propose a discrete scheme present a complete error analysis and first prove the c 0 ipg method can capture smooth eigenpairs efficiently using the argument in this paper we can prove that the c 0 ipg methods for the fourth order eigenvalue problems in the existing literature can also capture smooth eigenpairs efficiently numerical examples confirm our theoretical analysis

a mixed fem for the quad curl eigenvalue problem the quad curl problem arises in the study of the electromagnetic interior transmission problem and magnetohydrodynamics in this paper we study the quad curl eigenvalue problem and propose a mixed method using edge elements assuming stringent regularity of the solution of the quad curl source problem we prove the convergence and show that the divergence free condition can be bypassed

an integral method for solving nonlinear eigenvalue problems we propose a numerical method for computing all eigenvalues and the corresponding eigenvectors of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane the method uses complex integrals of the resolvent operator applied to at least k column vectors where k is the number of eigenvalues inside the contour the theorem of keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k no initial approximations of eigenvalues and eigenvectors are needed the method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension we also give an extension of the method to the case where k is larger than the matrix dimension the quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour

a unifying theory of a posteriori error control for nonconforming finite element methods residual based a posteriori error estimates were derived within one unifying framework for lowest order conforming nonconforming and mixed finite element schemes in carstensen numer math 100 617 637 2005 therein the key assumption is that the conforming first order finite element space v h c annulates the linear and bounded residual l written v h c subseteq rm ker ell that excludes particular nonconforming finite element methods ncfems on parallelograms in that v h c not subset rm ker ell the present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates also for mortar and discontinuous galerkin methods the key assumption is the existence of some bounded linear operator pi v h c rightarrow v h nc with some elementary properties it is conjectured that the more general hypothesis h1 h3 can be established for all known ncfems applications on various nonstandard finite element schemes for the laplace stokes and navier lame equations illustrate the presented unifying theory of a posteriori error control for ncfem

homogeneous almost quaternion hermitian manifolds an almost quaternion hermitian structure on a riemannian manifold m 4n g is a reduction of the structure group of m to mathrm sp n mathrm sp 1 subset text so 4n in this paper we show that a compact simply connected homogeneous almost quaternion hermitian manifold of non vanishing euler characteristic is either a wolf space or mathbb s 2 times mathbb s 2 or the complex quadric text so 7 mathrm u 3

mixed methods for the elastic transmission eigenvalue problem abstract the elastic transmission eigenvalue problem is quadratic in the eigenvalue parameter nonselfadjoint and of fourth order in this paper we apply the ciarlet raviart mixed method to this problem give a mixed variational form and establish two mixed methods using the classical lagrange finite element and the spectral element respectively we deduce the error estimates of the discrete eigenpairs theoretical analysis and numerical experiments show that these two methods are simple and easy to implement and can efficiently compute real and complex elastic transmission eigenvalues

stochastic monotonicity and stationary distributions for dynamic economies the existence and stability of invariant distributions for stochastically monotone processes is studied the knaster tarski fixed point theorem is applied to establish existence of fixed points of mappings on compact sets of measures that are increasing with respect to a stochastic ordering global convergence of a monotone markov process to its unique invariant distribution is established under an easily verified assumption topkis theory of supermodular functions is applied to stochastic dynamic optimization providing conditions under which optimal stationary decisions are monotone functions of the state and induce a monotone markov process applications of these results to investment theory stochastic growth and industry equilibrium dynamics are given

convergence and optimal complexity of adaptive finite element eigenvalue computations in this paper an adaptive finite element method for elliptic eigenvalue problems is studied both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved the analysis is based on a certain relationship between the finite element eigenvalue approximation and the associated finite element boundary value approximation which is also established in the paper

clifford structures on riemannian manifolds we introduce the notion of even clifford structures on riemannian manifolds which for rank r 2 and r 3 reduce to almost hermitian and quaternion hermitian structures respectively we give the complete classification of manifolds carrying parallel rank r even clifford structures kahler quaternion kahler and riemannian products of quaternion kahler manifolds for r 2 3 and 4 respectively several classes of 8 dimensional manifolds for 5 r 8 families of real complex and quaternionic grassmannians for r 8 6 and 5 respectively and rosenfeld s elliptic projective planes op 2 c o p 2 h o p 2 and o o p 2 which are symmetric spaces associated to the exceptional simple lie groups f4 e6 e7 and e8 for r 9 10 12 and 16 respectively as an application we classify all riemannian manifolds whose metric is bundle like along the curvature constancy distribution generalizing well known results in sasakian and 3 sasakian geometry 2000 mathematics subject classification primary 53c26 53c35 53c10 53c15

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

the ciarlet raviart method for biharmonic problems on general polygonal domains mapping properties and preconditioning for biharmonic boundary value problems the ciarlet raviart mixed method is considered on polygonal domains without additional convexity assumptions mapping properties of the involved operators on the continuous as well as on the discrete level is studied based on this efficient preconditioners are constructed and numerical experiments are shown

attracting graphs of skew products with non contracting fiber maps we study attracting graphs of step skew products from the topological and ergodic points of view where the usual contracting like assumptions of the fiber dynamics are replaced by weaker merely topological conditions in this context we prove the existence of an attracting invariant graph and study its topological properties we prove the existence of globally attracting measures and we show that in some specific cases the rate of convergence to these measures is exponential

a posteriori error estimators for convection diffusion equations we derive a posteriori error estimators for convection diffusion equations with dominant convection the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh peclet number the estimators are either based on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

regularity results for elliptic equations in lipschitz domains abstract we develop a simple variational argument based on the usual nirenberg difference quotient technique to deal with the regularity of the solutions of dirichlet and neumann problems for some linear and quasilinear elliptic equation in lipschitz domains we obtain optimal regularity results in the natural family of sobolev spaces associated with the variational structure of the equations in the linear case we obtain in a completely different way some of the results of d jerison and c e kenig about the laplace equation

can we have superconvergent gradient recovery under adaptive meshes we study adaptive finite element methods for elliptic problems with domain corner singularities our model problem is the two dimensional poisson equation results of this paper are twofold first we prove that there exists an adaptive mesh gauged by a discrete mesh density function under which the recovered gradient by the polynomial preserving recovery ppr is superconvergent second we demonstrate by numerical examples that an adaptive procedure with an a posteriori error estimator based on ppr does produce adaptive meshes that asatisfy our mesh density assumption and the recovered gradient by ppr is indeed superconvergent in the adaptive process

a new error analysis for discontinuous finite element methods for linear elliptic problems the standard a priori error analysis of discontinuous galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm in this paper a new error analysis of discontinuous galerkin methods is developed using only the h k weak formulation of a boundary value problem of order 2k this is accomplished by replacing the galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in h k

a coupled bem and fem for the interior transmission problem in acoustics abstract the interior transmission problem itp is a boundary value problem arising in inverse scattering theory and it has important applications in qualitative methods in this paper we propose a coupled boundary element method bem and a finite element method fem for the itp in two dimensions the coupling procedure is realized by applying the direct boundary integral equation method to define the so called dirichlet to neumann dtn mappings we show the existence of the solution to the itp for the anisotropic medium numerical results are provided to illustrate the accuracy of the coupling method

efficient spectral galerkin methods iii polar and cylindrical geometries we present in this paper several extremely efficient and accurate spectral galerkin methods for second and fourth order equations in polar and cylindrical geometries these methods are based on appropriate variational formulations which incorporate naturally the pole condition s in particular the computational complexities of the chebyshev galerkin method in a disk and the chebyshev legendre galerkin method in a disk or a cylinder are quasi optimal optimal up to a logarithmic term as an indication of efficiency the cpu time for the poisson solver on a disk by our chebyshev galerkin method is only about 70 of the corresponding finite difference code in fishpack

discontinuous galerkin methods of the non selfadjoint steklov eigenvalue problem in inverse scattering abstract in this paper we apply discontinuous galerkin methods to the non selfadjoint steklov eigenvalue problem arising in inverse scattering the variational formulation of the problem is non selfadjoint and does not satisfy h1 elliptic condition by using the spectral approximation theory of compact operators we prove the spectral approximation and optimal convergence order for the eigenvalues finally some numerical experiments are reported to show that the proposed numerical schemes are efficient for real and complex steklov eigenvalues

a two grid discretization scheme for eigenvalue problems a two grid discretization scheme is proposed for solving eigenvalue problems including both partial differential equations and integral equations with this new scheme the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy

numerical algorithms based on analytic function values at roots of unity let f z be an analytic or meromorphic function in the closed unit disk sampled at the n th roots of unity based on these data how can we approximately evaluate f z or f m z at a point z in the disk how can we calculate the zeros or poles of f in the disk these questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations partial differential equations and large scale linear algebra we analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of cauchy integrals we then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension finally we highlight the power of rational in comparison with polynomial approximations for some of these problems

optimal solvers for fourth order pdes discretized on unstructured grids this paper provides a class of optimal algorithms for the linear algebraic systems arising from direct finite element discretization of the fourth order equation with different boundary conditions on any polygonal domains that are partitioned by unstructured grids several preconditioners are presented and the preconditioned conjugate gradient methods applied with these preconditioners are proved to converge uniformly with respect to the size of the linear systems as a result for the first time in the literature it is rigorously proved in this paper that a discretized finite element system for fourth order pdes on an unstructured grid of size n can be solved within mathcal o n log n operations one main ingredient in constructing these optimal preconditioners is a mixed form discretization of the fourth order problem such a mixed form discretization often leads to an approximation of the original solution that is either suboptimal or divergent but it nevertheless provides optimal preconditio

almost complex structures on quaternion kahler manifolds and inner symmetric spaces we prove that compact quaternionic kahler manifolds of positive scalar curvature admit no almost complex structure even in the weak sense except for the complex grassmannians gr2 n 2 we also prove that irreducible inner symmetric spaces m 4n of compact type are not weakly complex except for spheres and hermitian symmetric spaces

a c0 discontinuous galerkin formulation for kirchhoff plates abstract a particular discontinuous galerkin finite element formulation for the simulation of kirchhoff plates is presented it is rotation free and utilises standard c 0 lagrange finite element basis functions with the required continuity imposed in a weak sense across element boundaries the implications of the scheme in terms of coercivity and convergence of the galerkin problem in various norms are studied with the formulation shown to be stable for any positive value of a penalty parameter a priori error estimates are supported by a range of numerical examples properties of the approach for the important eigenvalue problems of plate buckling and vibration are also examined through numerical examples based on the results of the analysis and numerical examples it is concluded that the formulation is robust accurate and relatively simple

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

two grid methods for maxwell eigenvalue problems two new two grid algorithms are proposed for solving the maxwell eigenvalue problem the new methods are based on the two grid methodology recently proposed by xu and zhou math comp 70 2001 pp 17 25 and further developed by hu and cheng math comp 80 2011 pp 1287 1301 for elliptic eigenvalue problems the new two grid schemes reduce the solution of the maxwell eigenvalue problem on a fine grid to one linear indefinite maxwell equation on the same fine grid and an original eigenvalue problem on a much coarser grid the new schemes therefore save total computational cost the error estimates reveals that the two grid methods maintain asymptotically optimal accuracy and the numerical experiments presented confirm the theoretical results

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

recursive integral method for transmission eigenvalues the transmission eigenvalue problem arose in the inverse scattering theory for inhomogeneous media and has important applications in a variety of inverse problems for target identification and nondestructive testing the problem is numerically challenging because of the nonself adjointness and complicated spectrum in this talk we introduce a novel recursive contour integral method using the spectrum projection the method can compute both real and complex eigenvalues and does not require any a priori information numerical examples show that the method is effective and robust

iterative methods for transmission eigenvalues transmission eigenvalues have important applications in inverse scattering theory they can be used to obtain useful information of the physical properties such as the index of refraction of the scattering target despite considerable effort devoted to the existence and estimation for the transmission eigenvalues the numerical treatment is limited since the problem is nonstandard classical finite element methods result in non hermitian matrix eigenvalue problems in this paper we focus on the computation of a few lowest transmission eigenvalues which are of practical importance instead of a non hermitian problem we work on a series of generalized hermitian problems we first use a fourth order reformulation of the transmission eigenproblem to construct functions involving an associated generalized eigenvalue problem the roots of these functions are the transmission eigenvalues then we apply iterative methods to compute the transmission eigenvalues we show the convergence of the numerical schemes the effectiveness of the methods is demonstrated using various numerical examples

the shifted inverse iteration based on the multigrid discretizations for eigenvalue problems the shifted inverse iteration based on the multigrid discretizations developed in recent years is an efficient computation method for eigenvalue problems in this paper for general self adjoint ei

three scale finite element discretizations for quantum eigenvalue problems based on globally and locally coupled discretizations some three scale finite element schemes are proposed in this paper for a class of quantum eigenvalue problems it is shown that the solution of a quantum eigenvalue problem on a fine grid may be reduced to the solution of an eigenvalue problem on a relatively coarse grid and the solutions of linear algebraic systems on a globally mesoscopic grid and the locally fine grid and the resulting solution is still very satisfactory

spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem we first develop an efficient spectral galerkin method and a rigorous error analysis for the generalized eigenvalue problems associated to a transmission eigenvalue problem then we present an iterative scheme based on computation of the first transmission eigenvalue to estimate the index of refraction of an inhomogeneous medium we present ample numerical results to demonstrate the effectiveness and accuracy of our approach

einstein metrics via intrinsic or parallel torsion the classification of riemannian manifolds by the holonomy group of their levi civita connection picks out many interesting classes of structures several of which are solutions to the einstein equations the classification has two parts the first consists of isolated examples the riemannian symmetric spaces the second consists of geometries that can occur in continuous families these include the calabi yau structures and joyce manifolds of string theory one may ask how one can weaken the definitions and still obtain similar classifications we present two closely related suggestions the classifications for these give isolated examples that are isotropy irreducible spaces and known families that are the nearly kahler manifolds in dimension 6 and gray s weak holonomy g2 structures in dimension 7

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

on the determination of higher order terms of singular elastic stress fields near corners in two dimensional elasticity stresses at reentrant corners exhibit singular behavior the stress field is of the form sigma sigma rm k iota r lambda 2 1 f i theta lambda i where r are polar coordinates centered at the tip of the corner andf i i are smooth functions for practical use of this series the eigenvalues i which are generally complex numbers are required in order of ascending real part the problem then is to find the roots of a transcendental equation eigenequation in the complex plane and arranged in order of ascending real part a theorem is proved on the number location and nature of the roots of this equation with the real part in fixed intervals of length excellent initial estimates of the real part of the complex roots become available and so are bounds within which single real roots exist this enables the determination of any number of roots in ascending order of real part the critical angles at which the eigenvalues change nature are also determined it is shown that for certain cases and for the symmetric mode of deformation the eigenvalue 1 does not represent a rigid body rotation therefore it has to be included in the series representation of the stresses the coefficientsk i can be determined by recently developed extraction techniques thus allowing complete determination of the elastic field and enabling its correlation with experimental data on brittle fracture crack initiation plastic zone estimation etc

a discrete helmholtz decomposition with morley finite element functions and the optimality of adaptive finite element schemes the discrete reliability of a finite element method is a key ingredient to prove optimal convergence of an adaptive mesh refinement strategy and requires the interchange of a coarse triangulation and some arbitrary refinement of it one approach for this is the careful design of an intermediate triangulation with one level refinements and with the remaining difficulty to design some interpolation operator which maps a possibly nonconforming approximation into the finite element space based on the finer triangulation this paper enfolds the second possibility of some novel discrete helmholtz decomposition for the nonconforming morley finite element method this guarantees the optimality of a standard adaptive mesh refining algorithm for the biharmonic equation numerical examples illustrate the crucial dependence of the bulk parameter and the surprisingly short pre asymptotic range of the adaptive morley finite element method

estimation by simulation of monotone dynamical systems this paper offers a general proof of consistency for the simulated moments estimator in a parameterized family of stochastic models with monotone dynamics models with this monotonicity property are frequently encountered in economic applications the proof of consistency of the estimator draws upon a uniform law of large numbers over a continuum of invariant distributions indexed by the model s parameters

a c0ip method of transmission eigenvalues for elastic waves abstract this paper develops a discontinuous galerkin method to compute a few smallest elasticity transmission eigenvalues which are of practical importance in inverse elastic scattering theory for high order problems comparing with classical conforming finite element methods discontinuous galerkin methods use simpler basis functions which make the numerical implementation much easier in this paper we propose an interior penalty discontinuous galerkin method using c 0 lagrange elements c 0 ip for the elastic transmission eigenvalue problem and prove the optimal convergence numerical examples are presented to validate its effectiveness both real and complex eigenvalues can be obtained

an efficient spectral galerkin approximation based on dimension reduction scheme for transmission eigenvalues in polar geometries abstract in this paper we put forward an efficient spectral galerkin approximation in view of dimension reduction scheme for transmission eigenvalue problem in polar geometries firstly we turn the original problem into an equivalent fourth order nonlinear eigenvalue problem then the fourth order nonlinear eigenvalue problem is transformed into a coupled fourth order linear eigenvalue system by introducing an auxiliary poisson equation secondly based on polar coordinate transformation we further reduce the coupled fourth order linear eigenvalue system to a series of equivalent one dimensional eigenvalue systems thirdly we derive the essential polar condition and introduce the appropriate weighted sobolev space according to the polar condition and establish the weak form and the corresponding discrete form in addition by utilizing spectral theory of compact operators we prove the error estimates of approximation eigenvalues and eigenvectors for each one dimensional eigenvalue system finally we provide ample numerical experiments and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results

a posteriori error estimates for the morley plate bending element a local a posteriori error indicator for the well known morley element for the kirchhoff plate bending problem is presented the error indicator is proven to be both reliable and efficient the technique applied is general and it is shown to have also other applications

asymptotics of a class of markov processes which are not in general irreducible on obtient des conditions d existence de probabilites invariantes uniques pour des processus de markov qui ne sont pas en general irreductibles