a mesh refinement method for ax λbx abstact the aim of this paper is to introduce a simple but efficient mesh refinement strategy for use with inverse iteration for finding one or a few solutions of an ordinary or partial differential eigenproblem of the form ax xbx the focus is upon the case where a and b are symmetric and b is positive definite although the approaches have a very broad application a discussion of the combined use of mesh refinement and a correction scheme multigrid technique is also provided the methods are illustrated by numerical results from experiments with two point boundary value problems

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

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

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

finite element galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems abstract this paper establishes some refined estimates for the approximation of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems by finite element or more generally galerkin methods suppose lambda is an eigenvalue of multiplicity q of a selfajoint problem and let m lambda denote the space of eigenvectors corresponding to lambda keywords hilbert space

mimetic seismic wave modeling including topography on deformed staggered grids abstractfinite difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography as a consequence alternatives to finite difference methods have been proposed to circumvent these issues we present a new numerical scheme for 3d elastic wave propagation in the presence of strong topography this finite difference scheme is based upon a staggered grid of the lebedev type or fully staggered grid fsg it uses a grid deformation strategy to make a regular cartesian grid conform to a topographic surface in addition the scheme uses a mimetic approach to accurately solve the free surface condition and hence allows for a less restrictive grid spacing criterion in the computations the scheme can use high order operators for the spatial derivatives and obtain low dispersion results with as few as six points pe

a kohn sham equation solver based on hexahedral finite elements we design a kohn sham equation solver based on hexahedral finite element discretizations the solver integrates three schemes proposed in this paper the first scheme arranges one a priori locally refined hexahedral mesh with appropriate multiresolution the second one is a modified mass lumping procedure which accelerates the diagonalization in the self consistent field iteration the third one is a finite element recovery method which enhances the eigenpair approximations with small extra work we carry out numerical tests on each scheme to investigate the validity and efficiency and then apply them to calculate the ground state total energies of nanosystems c 6 0 c 1 2 0 and c 2 7 5h 1 7 2 it is shown that our solver appears to be computationally attractive for finite element applications in electronic structure study

locally adapted tetrahedral meshes using bisection we present an algorithm for the construction of locally adapted conformal tetrahedral meshes the algorithm is based on bisection of tetrahedra a new data structure is introduced which simplifies both the selection of the refinement edge of a tetrahedron and the recursive refinement to conformity of a mesh once some tetrahedra have been bisected we prove that repeated application of the algorithm leads to only finitely many tetrahedral shapes up to similarity and we bound the amount of additional refinement that is needed to achieve conformity numerical examples of the effectiveness of the algorithm are presented

local and parallel finite element discretizations for eigenvalue problems based on the work of xu and zhou math comp 69 2000 pp 881 909 this paper combines the local defect correction technique and the shifted inverse power method to establish new local and parallel finite element three scale schemes for a class of eigenvalue problems it is proved that with these schemes the solution of an eigenvalue problem on a fine grid pi h is reduced to the solution of an eigenvalue problem on a much coarser grid pi h the solution of a linear algebraic system on a globally mesoscopic grid pi w and the solutions of linear systems on several locally fine grids in parallel the principle to determine the diameters of three different scale grids is given especially this paper devises a new local and parallel finite element multiscale discretization scheme theoretical analysis and numerical experiments show that the computational approach proposed in this paper is simple and easy to carry out and can be used to solve singular eigenvalue problems efficiently

broyden s method for self consistent field convergence acceleration broyden s method 1965 for self consistent field convergence acceleration recently used by bendt and zunger 1982 in the band structure calculation of solids is reviewed and a simpler computational scheme is presented

on the essential spectrum of schrodinger operators abstract an extension of the theorem of hunziker van winter zhislin on the location of the essential spectrum of many body hamiltonians

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

adaptive multigrid techniques for large scale eigenvalue problems solutions of the schrodinger problem in two and three dimensions multigrid mg algorithms for large scale eigenvalue problems ep obtained from discretizations of partial differential ep have often been shown to be more efficient than single level eigenvalue algorithms this paper describes a robust and efficient adaptive mg eigenvalue algorithm the robustness of the present approach is a result of a combination of mg techniques introduced here i e the completion of clusters the adaptive treatment of clusters the simultaneous treatment of solutions in each cluster the miltigrid projection mgp coupled with backrotations and robustness tests due to the mgp the algorithm achieves a better computational complexity and better convergence rates than previous mg eigenvalue algorithms that use only fine level projections these techniques overcome major computational difficulties related to equal and closely clustered eigenvalues some of these difficulties were not treated in previous mg algorithms computational examples for the schr odinger eigenvalue problem in two and three dimensions are demonstrated for cases of special computational difficulties which are due to equal and closely clustered eigenvalues for these cases the algorithm requires o qn operations for the calculation of q eigenvectors of size n using a second order approximation the total computational cost is equivalent to only a few gause seidel relaxations per eigenvector

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

error analysis on bi parameter finite elements abstract a bi parameter finite element is introduced in this paper in which a new local grid refinement technique is presented for a class of second order elliptic equations the new approach is based on a bi parameter finite element space defined respectively on the whole domain with size h and on a subdomain with size h h h the stability and convergence analysis of the bi parameter finite element approximation is studied the behavior of approximation with respect to the parameters h and h is presented in particular the error estimates for the finite element green s function expressed by h and h are derived

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

some new error estimates for ritz galerkin methods with minimal regularity assumptions new uniform error estimates are established for finite element approximations u h of solutions u of second order elliptic equations lu f using only the regularity assumption u 1 c f 1 using an aubin nitsche type duality argument we show for example that for arbitrary fixed e sufficiently small there exists an h 0 such that for 0 h h 0 u u h 0 e u u h 1 here s denotes the norm on the sobolev space h s other related results are established

theory of phase ordering kinetics the theory of phase ordering dynamics that is the growth of order through domain coarsening when a system is quenched from the homogeneous phase into a broken symmetry phase is reviewed with the emphasis on recent developments interest will focus on the scaling regime that develops at long times after the quench how can one determine the growth laws that describe the time dependence of characteristic length scales and what can be said about the form of the associated scaling functions particular attention will be paid to systems described by more complicated order parameters than the simple scalars usually considered for example vector and tensor elds the latter are needed for example to describe phase ordering in nematic liquid crystals on which there have been a number of recent experiments the study of topological defects domain walls vortices strings and monopoles provides a unifying framework for discussing coarsening in these dia erent systems

the multilevel mixed finite element discretizations based on local defect correction for the stokes eigenvalue problem abstract based on the work of xu and zhou 2000 this paper combines mixed finite element method and the local defect correction technique to establish new local and parallel multilevel discretization schemes for the stokes eigenvalue problem theoretical analysis and numerical experiments show that the computational approach proposed in this paper is simple and easy to carry out in parallel and can be used to solve singular stokes eigenvalue problem efficiently this paper also discusses local error estimates of mixed finite element approximations it is a new feature here that the estimates apply to the local domains containing corner points

an improved vacuum formulation for 2d finite difference modeling of rayleigh waves including surface topography and internal discontinuities rayleigh waves are generated along the free surface and their propagation can be strongly influenced by surface topography modeling of rayleigh waves in the near surface in the presence of topography is fundamental to the study of surface waves in environmental and engineering geophysics for simulation of rayleigh waves the traction free boundary condition needs to be satisfied on the free surface avacuum formulation naturally incorporates surface topographyinfinite difference fd modeling by treating the surface grid nodes as the internal grid nodes however theconventionalvacuumformulationdoesnotcompletely fulfill the free surface boundary condition and becomes unstableformodelingusinghigh orderfdoperators wedeveloped a stable vacuum formulation that fully satisfies the free surface boundary condition by choosing an appropriate combination of the staggered grid form and a parameter averaging scheme the elastic parameters on the topographic free surface are updated with exactly the same treatment as internal grid nodes the improved vacuum formulation can accurately and stably simulate rayleigh waves along the topographic surface for homogeneous and heterogeneous elastic models with high poisson s ratios 0 4 this method requires fewer grid points per wavelength than the stress image based methods internal discontinuities in a model can be handled without modification of the algorithm onlyminorchangesarerequiredtoimplementtheimproved vacuum formulation in existing 2d fd modeling codes

a defect correction scheme for finite element eigenvalues with applications to quantum chemistry to produce highly accurate approximations for self adjoint elliptic eigenvalues we propose a finite element defect correction scheme its efficiency is proven by theoretical and numerical evidence very satisfying applications in computational quantum chemistry are also presented

a multilevel variational method for au bu on composite grids abstract there are two existing general approaches to the numerical solution of differential eigen problems by multigrid one which uses a linear multigrid solver for the inner loop of a linearization technique e g inverse iteration and the other which integrates multigrid directly into the problem solution e g fas the present paper developes a third multigridlike approach rqmg that appeals directly to the variational problem of minimizing the rayleigh quotient applying easily to both global and local grid implementations numerical results are presented for a simplified single group diffusion problem

two grid discretizations and a local finite element scheme for a non selfadjoint stekloff eigenvalue problem abstract in this paper for a new stekloff eigenvalue problem which is non selfadjoint and not h 1 elliptic we establish and analyze two kinds of two grid discretization scheme and a local finite element scheme we present the error estimates of approximations of two grid discretizations we also prove a local error estimate which is suitable for the case that the local refined region contains singular points lying on the boundary of domain numerical experiments are reported finally to show the efficiency of our schemes

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

superconvergence two grid scheme based on shifted inverse power method for eigenvalue problems by function value recovery abstract in the paper an improved two grid scheme based on shifted inverse power method is proposed to solve the elliptic eigenvalue problems with this new scheme the solution of the elliptic eigenvalue problem on a fine grid t h is reduced to the solution of the elliptic eigenvalue problem and the recovered eigenfunction on a much coarser grid t h and the solution of an elliptic boundary value problem and the recovered solution on the fine grid t h theoretical analysis shows that the scheme allows a much coarser mesh to achieve the superconvergence rate finally some numerical experiments are carried out to confirm the theoretical analysis

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

the adaptive finite element method based on multi scale discretizations for eigenvalue problems in this paper adaptive finite element methods for differential operator eigenvalue problems are discussed for multi scale discretization schemes based on rayleigh quotient iteration see scheme 3 in y yang h bi a two grid discretization scheme based on shifted inverse power method siam j numer anal 49 2011 1602 1624 a reliable and efficient a posteriori error indicator is given in addition a new adaptive algorithm based on the multi scale discretizations is proposed and we apply the algorithm to the schrodinger equation for hydrogen atoms the algorithm is performed under the package of chen and satisfactory numerical results are obtained

sign preserving of principal eigenfunctions in p1 finite element approximation of eigenvalue problems of second order elliptic operators this paper is concerned with the p1 finite element approximation of the eigenvalue problem of second order elliptic operators subject to the dirichlet boundary condition the focus is on the preservation of the basic properties of the principal eigenvalue and eigenfunctions of the continuous problem it is shown that when the stiffness matrix is an irreducible m matrix the discrete eigenvalue problem maintains almost all of the basic properties such as the smallest eigenvalue being real and simple and the corresponding eigenfunctions being either positive or negative inside the physical domain mesh conditions leading to such a stiffness matrix are also studied a sufficient condition is that the mesh is simplicial interiorly connected and acute when measured in the metric specified by the inverse of the diffusion matrix the acute requirement can be replaced by the delaunay condition in two dimensions numerical results show that when the stiffness matrix is not an m matrix a finite element approximation can be structurally different from the continuous eigenvalue problem the eigenfunctions corresponding to the smallest eigenvalue can change sign over the physical domain and the smallest eigenvalue in modulus can even be complex for the case with nonsymmetric operators

multilevel finite element discretizations based on local defect correction for nonsymmetric eigenvalue problems based on the work of xu and zhou 2000 we establish new three level and multilevel finite element discretizations by local defect correction techniques theoretical analysis and numerical experiments show that the discretizations are simple and easy to implement and can be used to solve nonsymmetric eigenvalue problems with non smooth eigenfunctions efficiently we also discuss the local error estimates of finite element approximations it is a new feature here that the estimates apply to the local domains containing corner points

local and parallel finite element algorithms based on two grid discretizations a number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems these algorithms are motivated by the observation that for a solution to some elliptic problems low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure the theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape regular grids some numerical experiments are also presented to support the theory in this paper we will propose some new parallel techniques for nite element computation these techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems simply speaking the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo nents the main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue which contains mostly high frequencies by some local parallel procedures let us now give a somewhat more detailed but informal and hopefully infor mative description of the main ideas and results in this paper we consider the following very simple model problem posed on a convex polygonal domain r 2 u bru f in

generation of finite difference formulas on arbitrarily spaced grids on etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d ordre arbitraire sur des grilles a une dimension d espacement arbitraire