a new paradigm for parallel adaptive meshing algorithms we present a new approach to the use of parallel computers with adaptive finite element methods this approach addresses the load balancing problem in a new way requiring far less communication than current approaches it also allows existing sequential adaptive pde codes such as pltmg and mc to run in a parallel environment without a large investment in recoding in this new approach the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor using a sequential adaptive solver without requiring any modifications to the sequential solver the small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh which are then used in a weighted recursive spectral bisection of the initial mesh the bulk of the calculation then takes place independently on each processor with no communication using possibly the same sequential adaptive solver each processor adapts its region of the mesh independently and a nearly load balanced mesh distribution is usually obtained as a result of the initial weighted spectral bisection only the initial fan out of the mesh decomposition to the processors requires communication two additional steps requiring boundary exchange communication may be employed after the individual processors reach an adapted solution namely the construction of a global conforming mesh from the independent subproblems followed by a final smoothing phase using the subdomain solutions as an initial guess we present a series of convincing numerical experiments which illustrate the effectiveness of this approach the justification of the initial refinement prediction step as well as the justification of skipping the two communication intensive steps is supported by some recent j xu and a zhou math comp to appear and not so recent j a nitsche and a h schatz math comp 28 1974 pp 937 958 a h schatz and l b wahlbin math comp 31 1977 pp 414 442 a h schatz and l b wahlbin math comp 64 1995 pp 907 928 results on local a priori and a posteriori error estimation

pointwise error estimates for differences in piecewise linear finite element approximations we consider piecewise linear finite element approximations uh to u the solution of an elliptic boundary value problem new estimates for the differences e x1 e x2 where e x u x u h x is the error and x1 and x2 are any two points in the domain are obtained in terms of weighted l infty norms as a consequence so called asymptotic expansion inequalities are derived that have been applied to obtain asymptotically exact a posteriori estimators for the gradient on each element

least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

a parallel oseen linearized algorithm for the stationary navier stokes equations abstract based on two grid discretizations and domain decomposition a parallel oseen linearized finite element algorithm for the stationary navier stokes equations with moderate or large viscosity parameter is proposed and analyzed the key idea of the algorithm is to first solve a nonlinear problem by picard iterative method on a coarse grid and then to solve an oseen problem in parallel on a fine grid to correct the coarse grid solution by using local a priori error estimate for the finite element solution and under the uniqueness condition error bounds of the corresponding finite element solution are analyzed numerical results are also given to demonstrate the high efficiency of the algorithm

an analysis of some high accuracy finite element methods for hyperbolic problems some high accuracy finite element methods for hyperbolic problems are studied in this paper it is proven that over a finite element mesh the convergence order of linear finite element solutions for both linear and nonlinear equations can be higher than one even if the exact solutions are discontinuous the theoretical tools for the convergence analysis are some superclose error estimates that are also developed in this paper for nonsmooth solutions

a posteriori error estimates for elliptic problems with dirac delta source terms the aim of this paper is to introduce residual type a posteriori error estimators for a poisson problem with a dirac delta source term in l p norm and w1 p seminorm the estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error they are used to guide adaptive procedures which are experimentally shown to lead to optimal orders of convergence

a new finite element gradient recovery method superconvergence property this is the first in a series of papers in which a new gradient recovery method is introduced and analyzed it is proved that the method is superconvergent for translation invariant finite element spaces of any order the method maintains the simplicity efficiency and superconvergence properties of the zienkiewicz zhu patch recovery method in addition for uniform triangular meshes the method is superconvergent for the linear element under the chevron pattern and ultraconvergent at element edge centers for the quadratic element under the regular pattern applications of this new gradient recovery technique will be discussed in forthcoming papers

model for coarsening froths and foams the coarsening process of froths and foams is described statistically in two dimensions by a random neighbor model this nonlinear dynamical model contains no free parameters all rates are determined dynamically it describes normal growth agrees with experimental results throws light on old data and hopefully inspires new simulations and measurements lewis s law anat rec 38 341 1928 am j bot 30 74 1943 and some other properties can be derived analytically from the model

self similarity in a cut and paste model of coarsening we study certain redistribution processes for a resource distributed among a large number of resource holders clusters in the processes we consider the smallest clusters are cut up into pieces according to a given statistical law and the pieces are randomly redistributed among the remaining clusters we derive an evolution equation for the cluster size distribution show that self similar solutions exist and characterize their structure in a limiting case when the pieces are small we show that in general solutions approach a singular self similar form

high effective finite element algorithm for elliptic partial differential equation in this paper we will discuss elliptic partial differential equation whose coefficients are constants on basis of a new computing technique for green function we will propose a new effective finite element algorithm for that problem finally we will give some numerical examples to investigate that finite element algorithm

coarsening in the q state potts model and the ising model with globally conserved magnetization we study the nonequilibrium dynamics of the q state potts model following a quench from the high temperature disordered phase to zero temperature the time dependent two point correlation functions of the order parameter field satisfy dynamic scaling with a length scale l t t 1 2 in particular the autocorrelation function decays as l t q we illustrate these properties by solving exactly the kinetic potts model in d 1 we then analyze a langevin equation of an appropriate field theory to compute these correlation functions for general q and d we establish a correspondence between the two point correlations of the q state potts model and those of a kinetic ising model evolving with a fixed magnetization 2 q 1 the dynamics of this ising model is solved exactly in the large q limit and in the limit of a large number of components n for the order parameter for general q and in any dimension we introduce a gaussian closure approximation and calculate within this approximation the scaling functions and the exponent q these are in good agreement with the direct numerical simulations of the potts model as well as the kinetic ising model with fixed magnetization we also discuss the existing and possible experimental realizations of these models

sharply local pointwise a posteriori error estimates for parabolic problems we prove pointwise a posteriori error estimates for semi and fully discrete finite element methods for approximating the solution u to a parabolic model problem our estimates may be used to bound the finite element error u u h l d where d is an arbitrary subset of the space time domain of the definition of the given pde in contrast to standard global error estimates these estimators de emphasize spatial error contributions from space time regions removed from d our results are valid on arbitrary shape regular simplicial meshes which may change in time and also provide insight into the contribution of mesh change to local errors when implemented in an adaptive method these estimates require only enough spatial mesh refinement away from d in order to ensure that local solution quality is not polluted by global effects

a space time finite element method for the nonlinear schro dinger equation the discontinuous galerkin method the convergence of the discontinuous galerkin method for the nonlinear cubic schrodinger equation is analyzed in this paper we show the existence of the resulting approximations and prove optimal order error estimates in l l 2 these estimates are valid under weak restrictions on the space time mesh

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

statistical dynamics of interacting kinks ii abstract a statistical mechanical theory is developed for a gas of kinks or one dimensional domain walls interacting with exponential attractive forces which models long time behaviors of the bistable tdgl field equation employing a factorization approximation in the bbgky like hierarchical equations for the size distribution functions of consecutive domains we obtain a kinetic equation for the size distribution function of a single domain the scaling solution of thr latter yields the logarithmic growth law of domain sizes with the coefficient related to the width of the size distribution scattering structure function is also obtained the results are compared with those of molecular dynamics and experiments on layered magnets

a priori l 2 discretization error estimates for the state in elliptic optimization problems with pointwise inequality state constraints in this paper an elliptic optimization problem with pointwise inequality constraints on the state is considered the main contributions of this paper are a priori l 2 error estimates for the discretization error in the optimal states due to the non separability of the space for the lagrange multipliers for the inequality constraints the problem is tackled by separation of the discretization error into two components first the state constraints are discretized second with discretized inequality constraints a duality argument for the error due to the discretization of the pde is employed for the second stage an a priori error estimate is derived with constants depending on the regularity of the dual problem finally we discuss two cases in which these constants can be bounded in a favorable way leading to higher order estimates than those induced by the known l 2 error in the control variable more precisely we consider a given fixed number of pointwise inequality constraints and a case of infinitely many but only weakly active constraints

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

local pointwise a posteriori gradient error bounds for the stokes equations we consider the standard taylor hood finite element method for the stationary stokes system on polyhedral domains we prove local a posteriori error estimates for the maximum error in the gradient of the velocity field because the gradient of the velocity field blows up near reentrant corners and edges such local error control is necessary when pointwise control of the gradient error is desirable computational examples confirm the utility of our estimates in adaptive codes

double crises in two parameter dynamical systems a crisis is a sudden discontinuous change in a chaotic attractor as a system parameter is varied we investigate phenomena observed when two parameters of a dissipative system are varied simultaneously following a crisis along a curve in the parameter plane two such curves intersect at a point we call a double crisis vertex the phenomena we study include the double crisis vertex at which an interior and a boundary crisis coincide and related forms of double crisis we show how an experimenter can infer a crisis from observations of other related crises at a vertex

distribution of repetitions of ancestors in genealogical trees we calculate the probability distribution of repetitions of ancestors in a genealogical tree for simple neutral models of a closed population with sexual reproduction and non overlapping generations each ancestor at generation g in the past has a weight w which is up to a normalization the number of times this ancestor appears in the genealogical tree of an individual at present the distribution pg w of these weights reaches a stationary shape p1 w for large g i e for a large number of generations back in the past for small w p1 w is a power law p1 w w with a non trivial exponent which can be computed exactly using a standard procedure of the renormalization group approach some extensions of the model are discussed and the eect of these variants on the shape of p1 w are analysed c 2000 elsevier science b v all rights reserved

exact exponent for the number of persistent spins in the zero temperature dynamics of the one dimensional potts model for the zero temperature glauber dynamics of theq state potts model the fractionr q t of spins which never flip up to timet decays like a power lawr q t t q when the initial condition is random by mapping the problem onto an exactly soluble one species coagulation model a a a or alternatively by transforming the problem into a free fermion model we obtain the exact expression of q for all values ofq the exponent q is in general irrational 3 0 53795082 4 0 63151575 with the exception ofq 2 andq for which 2 3 8 and 1

model of cluster growth and phase separation exact results in one dimension we present exact results for a lattice model of cluster growth in one dimension the growth mechanism involves interface hopping and pairwise annihilation supplemented by spontaneous creation of the stable phase 1 regions by overturning the unstable phase 1 spins with probabilityp for cluster coarsening at phase coexistence p 0 the conventional structure factor scaling applies in this limit our model falls in the class of diffusion limited reactions a a inert the 1 cluster size grows diffusively t and the two point correlation function obeys scaling however forp 0 i e for the dynamics of formation of stable phase from unstable phase we find that structure factor scaling breaks down the length scale associated with the size of the growing 1 clusters reflects only the short distance properties of the two point correlations

the effect of numerical integration on the finite element approximation of linear functionals in this paper we have studied the effect of numerical integration on the finite element method based on piecewise polynomials of degree k in the context of approximating linear functionals which are also known as quantities of interest we have obtained the optimal order of convergence mathcal o h 2k of the error in the computed functional when the integrals in the stiffness matrix and the load vector are computed with a quadrature rule of algebraic precision 2k 1 however this result was obtained under an increased regularity assumption on the data which is more than required to obtain the optimal order of convergence of the energy norm of the error in the finite element solution with quadrature we have obtained a lower bound of the error in the computed functional for a particular problem which indicates the necessity of the increased regularity requirement of the data numerical experiments have been presented indicating that over integration may be necessary to accurately approximate the functional when the data lack the increased regularity

convergence of a continuous galerkin method with mesh modification for nonlinear wave equations we consider space time continuous galerkin methods with mesh modification in time for semilinear second order hyperbolic equations we show a priori estimates in the energy norm without mesh conditions under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates estimates of the jump in the riesz projection in two successive time steps are also derived

local and parallel finite element algorithms based on two grid discretization for the stream function form of navier stokes equations abstract a new local and parallel discretization finite element algorithms are proposed and analyzed in this paper for the stream function form of navier stokes equations according to 1 the theoretical tools for analyzing these methods are some local priori estimate that are also obtained in this paper for finite element solutions on general shape regular grids some numerical experiments are also presented to support the theory

eine aussage zurl stabilitat und zur genauen konvergenzordnung derh 0 0 projektionen in this paper we settle a question concerning the necessity of the logh factor appearing in error estimates of linear finite element solutions let 2 be a bounded convex polygon and t h a regular family of subdivisions of into triangles be furthers 0 h h 0 1 the corresponding linear finite element spaces andp h h 0 1 s 0 h theh 0 1 projections with respect to the uniform norm asymptotically estimates of the form parallel u p h u parallel infty leqq c logh dist u s 0 h u in c bar omega cap h 0 1 omega hold as pointed out by schatz 1980 for functionsu with bounded second derivatives this implies parallel u p h u parallel infty leqq ch 2 log h u 2 infty a result which was proved by nitsche 1975 and scott 1976 with different methods it was an open question if the factor logh is necessary or only due to the methods of proof in this paper it is shown by an example that the cited estimates are sharp in the very strong sense that in general they are no longer valid if logh is replaced by a termo logh

the bandcount increment scenario i basic structures bifurcation structures in two dimensional parameter spaces formed only by chaotic attractors are still far away from being understood completely in a series of three papers we investigate the chaotic domain without periodic inclusions for a map which is considered by many authors as some kind of one dimensional canonical form for discontinuous maps in the first part we report a novel bifurcation scenario formed by crises bifurcations which includes multi band chaotic attractors with arbitrary high bandcounts and determines the basic structure of the chaotic domain

stable spins in the zero temperature spinodal decomposition of 2d potts models we present the results of zero temperature monte carlo simulations of the q state potts model on a square lattice with either four or eight neighbors and for the triangular lattice with six neighbors in agreement with previous works we observe that the domain growth process gets blocked for the nearest neighbor square lattice when q is large enough whereas for the eight neighbor square lattice and for the triangular lattice no blocking is observed our simulations indicate that the number of spins which never flipped from the beginning of the simulation up to time t follows a power law as a function of the energy even in the case of blocking the exponent of this power law varies from less than 1 2 for the ising case q 2 to 2 for q o0 and seems to be universal the effect of blocking on this exponent is invisible at least up to q 7

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

exponents appearing in the zero temperature dynamics of the 1d potts model for the glauber dynamics of the one dimensional q state potts model at zero temperature the fraction of spins which have never flipped up to time t decreases as t theta q where the exponent theta q varies continuously with q the calculation of the q dependence of this exponent can be reduced to the study of a simple reaction diffusion model using finite size scaling this allows one to obtain accurate estimates of b q for all values of q one can also show that for 1 1 epsilon the expansion of theta y is given by theta 1 2 pi epsilon o epsilon 2

codimension two bifurcations of nonlinear systems driven by fuzzy noise abstract by means of a fuzzy generalized cell mapping method a duffing van der pol oscillator in the presence of fuzzy noise is studied in a regime where two symmetrically related fuzzy period one attractors grow and merge as the intensity of the fuzzy noise is increased by introducing a small symmetry breaking parameter to break the symmetry the merging explosion bifurcation unfolds to a pattern of two catastrophic and explosive bifurcations considering both the intensity of the fuzzy noise and the symmetry breaking parameter together as controls a codimension two bifurcation of fuzzy attractors is defined and examples of additive and multiplicative fuzzy noise are given such a codimension two bifurcation is fuzzy noise induced effects which cannot be seen in the deterministic systems

non trivial exponents in coarsening phenomena abstract one of the simplest examples of stochastic automata is the glauber dynamics of ferromagnetic spin models such as ising or potts models at zero temperature if the initial condition is random one observes a pattern of growing domains with a characteristic size which increases with time like t 1 2 in this self similar regime the fraction of spins which never flip up to time t decreases like t where the exponent is non trivial and depends both on the number q of states of the potts model and on the dimension of space this exponent can be calculated exactly in one dimension similar non trivial exponents are also present in even simpler models of coarsening where the dynamical rule is deterministic

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates this part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in mathbb r n global estimates were considered in part i in the sense to be discussed below these sharpen known interior quasi optimal l infty and w 1 infty estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point the higher the order of the finite element the more local the behavior of the finite element approximation as a consequence of these estimates new types of local error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in mathbb r n and have applications to superconvergence extrapolation and a posteriori estimates for both smooth and nonsmooth problems

maximum norm stability and maximal l p lp regularity of fems for parabolic equations with lipschitz continuous coefficients in this paper we study the semi discrete galerkin finite element method for parabolic equations with lipschitz continuous coefficients we prove the maximum norm stability of the semigroup generated by the corresponding elliptic finite element operator and prove the space time stability of the parabolic projection onto the finite element space in l infty q t l qt and l p 0 t l q omega lp 0 t lq 1

boundary crisis in quasiperiodically forced systems abstract we study boundary crisis in quasiperiodically forced dissipative systems using the henon map as a characteristic example the crisis is due to a homoclinic tangency of the stable and unstable manifolds of an accessible invariant circle of saddle type on the basin boundary of the attractor numerical evidence shows that the type of boundary crisis can change as the saddle circle loses its accessibility due to the quasiperiodic forcing this implies the existence of codimension two double crisis vertices where a curve of boundary crisis and a curve of interior crisis meet we argue that bifurcation points of higher codimension must exist in the full parameter space

extrapolation techniques for reducing the pollution effect or reentrant corners in the finite element method standard finite element schemes applied to boundary value problems on domains with reentrant corners suffer from a global loss of accuracy caused by the influence of the corner singularities for a model situation it is shown that this pollution effect can be completely described in terms of an asymptotic error expansion with respect to fractional powers of the mesh size parameter this provides the basis of an extrapolation procedure which recovers the full order of accuracy for the solution as well as for the corresponding stress intensity factors

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

distribution of domain sizes in the zero temperature glauber dynamics of the one dimensional potts model for the zero temperature glauber dynamics of the q state potts model we calculate the exact distribution of domain sizes by mapping the problem on an exactly soluble one species coagulation model a1a a in the long time limit this distribution is universal and from its complicated exact expression we extract its behavior in various regimes our results are tested in a simulation and compared to the predictions of a simple approximation proposed recently considering the dynamics of domain walls as a reaction diffusion model a1a a with probability q22 q21 and a1a f with probability 1 q21 we calculate the pair correlation function in the long time regime s1063 651x 96 12809 9

pointwise error estimates for finite element solutions of the stokes problem in this paper pointwise error estimates for general finite element approximations of the stokes problem are established on quasi uniform grids in r n the results obtained in this paper improve and extend the existing error estimates in the maximum norm for the stokes problem the new pointwise error estimates exhibit a more local dependence of the errors on the true solution and as a by product provide logarithm free bounds for all errors except the error of the velocity approximation of the lowest order

implicit explicit multistep methods for quasilinear parabolic equations efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in 1 in this note we present a refined analysis to allow more general nonlinearities the abstract theory is applied to a quasilinear parabolic equation

complex balancing motions of an inverted pendulum subject to delayed feedback control we show that an inverted pendulum that is balanced on a cart by linear delayed feedback control may exhibit small chaotic motion about the upside down position in periodic windows associated with this chaotic regime we find periodic orbits of arbitrarily high period that correspond to complex balancing motion of the pendulum with bounded velocity of the cart this result shows that complex balancing is possible in a controlled mechanical system with a geometric nonlinearity even when the control law is only linear this is in contrast to other proposed models that require a nonlinearity of the controller such as round off due to digitization we find the complex motion by studying homoclinic bifurcations of a reduced three dimensional vector field near a triple zero eigenvalue singularity more generally the dynamics we describe must be expected in any system with a triple zero singularity and reflection symmetry

superconvergence in finite element methods and meshes that are locally symmetric with respect to a point consider a second order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution consider also its numerical approximation by standard conforming finite element methods with for example fixed degree piecewise polynomials on a quasi uniform mesh family the h method it will be shown that if the finite element function spaces are locally symmetric about a point x 0 with respect to the antipodal map x to x 0 x x 0 then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of x 0 for piecewise polynomials of even degree superconvergence occurs in function values for piecewise polynomials of odd degree it occurs in derivatives