multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

bubble stabilization of spectral legendre methods for the advection diffusion equation abstract an advection diffusion model problem is discretized by a spectral legendre method augmented by trial test functions with local support bubbles the numerical analysis shows that the correction is equivalent to adding a spectrally accurate streamline upwind stabilization term the performances of two strategies for tuning the bubble stabilization are investigated

bubble stabilization of finite element methods for the linearized incompressible navier stokes equations abstract in this paper we discuss the stabilization via bubble functions of a finite element method for stationary linearized incompressible navier stokes equations it is shown that residual free bubbles can reproduce supg

covering of a finite lattice by a random walk abstract we study the covering process by a simple random walk of a d dimensional periodic hypercubic lattice of n sites in d 1 the probability l n x for site x to be the last site visited in this covering process does not depend on x as long as x is not the starting point of the walk we argue that in dimensions d 2 the probability l n x approaches a constant value according to a coulomb law l n x 1 2 1 const x d 2 valid for x small on the scale n 1 d whereas it behaves logarithmically in d 2 also there is a dimension dependent characteristic time scale on which the last site is visited the structure of the set of sites not yet visited on this characteristic time scale is fractal like in d 2 in d 3 on the other hand this set is essentially distributed randomly through the lattice

enforcement of constraints and maximum principles in the variational multiscale method abstract we present a new theoretical framework for the enforcement of constraints in variational multiscale vms analysis the theory is first presented in an abstract operator format and subsequently specialized for the steady advection diffusion equation the approach borrows heavily from results in constrained and convex optimization an exact expression for the fine scales is derived in terms of variational derivatives of the constraints lagrange multipliers and a fine scale green s function the methodology described enables the development of numerical methods which satisfy predefined attributes a practical and effective procedure for solving the steady advection diffusion equation is presented based on a vms inspired stabilized method weakly enforced dirichlet boundary conditions and enforcement of a maximum principle and conservation constraint

a field theoretic approach to the wiener sausage the wiener sausage the volume traced out by a sphere attached to a brownian particle is a classical problem in statistics and mathematical physics initially motivated by a range of field theoretic technical questions we present a single loop renormalised perturbation theory of a stochastic process closely related to the wiener sausage which however proves to be exact for the exponents and some amplitudes the field theoretic approach is particularly elegant and very enjoyable to see at work on such a classic problem while we recover a number of known classical results the field theoretic techniques deployed provide a particularly versatile framework which allows easy calculation with different boundary conditions even of higher momenta and more complicated correlation functions at the same time we provide a highly instructive non trivial example for some of the technical particularities of the field theoretic description of stochastic processes such as excluded volume lack of translational invariance and immobile particles the aim of the present work is not to improve upon the well established results for the wiener sausage but to provide a field theoretic approach to it in order to gain a better understanding of the field theoretic obstacles to overcome

on the survival probability of a random walk in a finite lattice with a single trap we consider the survival of a random walker in a finite lattice with periodic boundary conditions the initial position of the random walker is uniformly distributed on the lattice with respect to the trap we show that the survival of a random walker un can be exactly related to the expected number of distinct sites visted on a trap free lattice by un 1 sn nd wherend is the number of lattice points ind dimensions we then analyze the behavior of sn in any number of dimensions by using tauberian methods we find that at sufficiently long times sn decays exponentially withn in all numbers of dimensions ind 1 and 2 dimensions there is an intermediate behavior which can be calculated and is valid forn2 n 1 whend 1 andn lnn n 1 whend 2 no such crossover exists when z 3 the form of suggests that the single trap approximation is indeed a valid low concentration limit for survival on an infinite lattice with a finite concentration of traps

information loss in coarse graining of polymer configurations via contact matrices contact matrices provide a coarse grained description of the configuration of a linear chain polymer or random walk on n ij 1 when the distance between the positions of the ith and jth steps are less than or equal to some distance a and ij 0 otherwise we consider models in which polymers of length n have weights corresponding to simple and self avoiding random walks srw and saw with a the minimal permissible distance we prove that to leading order in n the number of matrices equals the number of walks for srw but not for saw the coarse grained shannon entropies for srw agree with the fine grained ones for n 2 but differs for n 3

residual based stabilized higher order fem for advection dominated problems abstract we reconsider the numerical solution of linear ized advection diffusion reaction problems using higher order finite elements together with stabilized galerkin methods of streamline diffusion type supg and with shock capturing stabilization the analysis improves the a priori analysis in our previous paper t knopp g lube g rapin stabilized finite element methods with shock capturing for advection diffusion problems comput methods appl mech engrg 191 2002 2997 3013 the theoretical results are supported by some numerical experiments

numerical evidence against a conjecture on the cover time of planar graphs we investigate a conjecture on the cover times of planar graphs by means of large monte carlo simulations the conjecture states that the cover time g n of a planar graph g n of n vertices and maximal degree d is lower bounded by g n c d n lnn 2 with c d d 4 tan d with equality holding for some geometries we tested this conjecture on the regular honeycomb d 3 regular square d 4 regular elongated triangular d 5 and regular triangular d 6 lattices as well as on the nonregular union jack lattice dmin 4 dmax 8 indeed the monte carlo data suggest that the rigorous lower bound may hold as an equality for most of these lattices with an interesting issue in the case of the union jack lattice the data for the honeycomb lattice however violate the bound with the conjectured constant the empirical probability distribution function of the cover time for the square lattice is also briefly presented since very little is known about cover time probability distribution functions in general

fluctuation results for the wiener sausage on demontre des resultats de fluctuation pour le volume de la saucisse de wiener associe au mouvement brownien a d dimensions et a un ensemble compact de capacite positive

the aspherity of random walks the asphericity a of a random walk is defined and calculated for non self avoiding walks the definition of a generalises to self avoiding walks percolating clusters and other fractal objects and thus provides a generalised quantitative measure of the departure from spherical symmetry of the gross shape of these objects

stabilized finite element methods for steady advection diffusion with production abstract finite element methods for solving equations of steady advection diffusion with production are constructed employing a linear source to model production the galerkin method requires relatively fine meshes to retain an acceptable degree of accuracy for wide ranges of values of the physical coefficients numerous criteria for improved accuracy via galerkin least squares are proposed and examined in the entire parameter space of these several provide the lowest error in nodal amplification factors each in a certain range of ratios of physical coefficients guidelines for required mesh resolutions are presented for galerkin and galerkin least squares showing that in large parts of the parameter space substantial savings may be obtained by employing galerkin least squares galerkin least squares gradient least squares a new variation of the galerkin method is designed to provide exact nodal amplification factors offering accurate solutions at any resolution numerical tests validate these conclusions

analysis of a new stabilized higher order finite element method for advection diffusion equations abstract we consider a singularly perturbed advection diffusion two point boundary value problem whose solution has a single boundary layer based on piecewise polynomial approximations of degree k 1 a new stabilized finite element method is derived in the framework of a variation multiscale approach the method coincides with the supg method for k 1 but differs from it for k 2 estimates for the error to an appropriate interpolant are given in several norms on different types of meshes for k 1 enhanced accuracy is achieved by superconvergence postprocessing guarantees the same estimates for the error to the solution itself

probability distribution of the number of distinct sites visited by a random walk on the finite size fully connected lattice the probability distribution of the number s of distinct sites visited up to time t by a random walk on the fully connected lattice with n sites is first obtained by solving the eigenvalue problem associated with the discrete master equation then using generating function techniques we compute the joint probability distribution of s and r where r is the number of sites visited only once up to time t mean values variances and covariance are deduced from the generating functions and their finite size scaling behaviour is studied introducing properly centered and scaled variables u and v for r and s and working in the scaling limit with w t n fixed the joint probability density of u and v is shown to be a bivariate gaussian density it follows that the fluctuations of r and s around their mean values in a finite size system are gaussian in the scaling limit the same type of finite size scaling is expected to hold on periodic lattices above the critical dimension

the p and hp versions of the finite element method for problems with boundary layers we study the uniform approximation of boundary layer functions exp x d for x 0 1 d 0 1 by the p and hp versions of the finite element method for the p version with fixed mesh we prove super exponential convergence in the range p 1 2 e 2d we also establish for this version an overall convergence rate of o p 1 ln p in the energy norm error which is uniform in d and show that this rate is sharp up to the ln p term when robust estimates uniform in d 0 1 are considered for the p version with variable mesh i e the hp version we show that exponential convergence uniform in d 0 1 is achieved by taking the first element at the boundary layer to be of size o pd numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates even when few degrees of freedom are used and when d is as small as e g 10 8 they also illustrate the superiority of the hp approach over other methods including a low order h version with optimal exponential mesh refinement the estimates established in this paper are also applicable in the context of corresponding spectral element methods

statistics of the one dimensional riemann walk the riemann walk is the lattice version of the levy flight for the one dimensional riemann walk of levy exponent 0 2 we study the statistics of the support i e set of visited sites after t steps we consider a wide class of support related observables m t including the number s t of visited sites and the number i t of sequences of adjacent visited sites for t we obtain the asymptotic power laws for the averages variances and correlations of these observables logarithmic correction factors appear for 2 3 and 1 bulk and surface observables have different power laws for 1 2 fluctuations are shown to be universal for 2 3 2 this means that in the limit t the deviations from average m t m t m 0304 t 0304 are fully described either by a single m independent stochastic process when 2 3 1 or by two such processes one for the bulk and one for the surface observables when 1 2

stabilization of spectral methods by finite element bubble functions abstract we show that a standard spectral collocation scheme can be stabilized by adding extra trial test functions with local support bubbles applications are given to a scalar advection diffusion problem and to the stokes problem

variational multiscale analysis the fine scale green s function projection optimization localization and stabilized methods we derive an explicit formula for the fine scale green s function arising in variational multiscale analysis the formula is expressed in terms of the classical green s function and a projector which defines the decomposition of the solution into coarse and fine scales the theory is presented in an abstract operator format and subsequently specialized for the advection diffusion equation it is shown that different projectors lead to fine scale green s functions with very different properties for example in the advection dominated case the projector induced by the h 1 0 seminorm produces a fine scale green s function which is highly attenuated and localized these are very desirable properties in a multiscale method and ones that are not shared by the l 2 projector by design the coarse scale solution attains optimality in the norm associated with the projector this property combined with a localized fine scale green s function indicates the possibility of effective methods with local character for dominantly hyperbolic problems the constructs lead to a new class of stabilized methods and the relationship between h 1 0 optimality and the streamline upwind petrov galerkin supg method is described

on the natural stabilization of convection dominated problems using high order bubnovgalerkin finite elements in the case of dominating convection standard bubnovgalerkin finite elements are known to deliver oscillating discrete solutions for the convectiondiffusion equation this paper demonstrates that increasing the polynomial degree p extension limits these artificial numerical oscillations this is contrary to a widespread notion that an increase of the polynomial degree destabilizes the discrete solution this treatise also provides explicit expressions as to which polynomial degree is sufficiently high to obtain stable solutions for a given peclet number at the nodes of a mesh

study of the shape of random walks ii inertia moment ratios and the two dimensional asphericity a study of some features of the quantities used to describe the shape of random walks focusing on the probability distribution of the asphericity of discrete random walks in two dimensions is presented a connection is established between the asphericity distribution and the probability distribution corresponding to the ratio of the principal inertia moments the probability distributions of arbitrary inertia moment ratios are analysed for varying spatial dimension and an analytic expression for them is found which presents excellent agreement with monte carlo data in all the cases considered this function is then used to obtain the corresponding two dimensional asphericity distribution which also proved to be a very good approximation to the true distribution obtained from an independent monte carlo simulation

wiener sausage volume moments the statistical characteristics of a spatial region visited by a spherical brownian particle during timet wiener sausage are investigated the expectation value and dispersion of this quantity are obtained for a space of arbitrary dimension in the one dimensional case the distribution of probability density and the moments of any order are determined for this quantity

variance of the range of a random walk tn the expectation of the square of the number of distinct sites occupied by a random walk in steps 1 throughn is obtained from its relation to the dual first occupancy probabilityfij x x and the latter quantity is obtained from a recursion with the first occupancy probabilityf k x the variancevn of the number of distinct sites occupied is calculated directly from tn the procedure is illustrated by the calculation ofvn 4096 n and the derivation of asymptotic expansions forvn for a particular random walk in dimensions 1 through 3

universal fluctuations in the support of the random walk a random walk starts from the origin of ad dimensional lattice the occupation numbern x t equals unity if aftert steps site x has been visited by the walk and zero otherwise we study translationally invariant sumsm t of observables defined locally on the field of occupation numbers examples are the numbers t of visited sites the areae t of the appropriately defined surface of the set of visited sites and in dimension d 3 the euler index of this surface ind 3 theaverages m t all increase linearly witht ast we show that in d 3 to leading order in an asymptotic expansion int thedeviations from average m t m t m t are up to a normalization allidentical to a single universal random variable this result resembles an earlier one in dimensiond 2 we show that this universality breaks down ford 3

a priori error analysis of residual free bubbles for advection diffusion problems we develop an a priori error analysis of a finite element approximation to the elliptic advection diffusion equation eps delta u conv cdot nabla u f subject to a homogeneous dirichlet boundary condition based on the use of residual free bubble functions an optimal order error bound is derived in the so called stability norm biggl eps nabla v 2 l 2 omega sum t h t conv cdot nabla v 2 l 2 t biggr 1 2 where ht denotes the diameter of element t in the subdivision of the computational domain

statistical properties of the set of sites visited by the two dimensional random walk we study the support i e the set of visited sites of a t step random walk on a two dimensional square lattice in the large t limit a broad class of global properties m t of the support is considered including for example the number s t of its sites the length of its boundary the number of islands of unvisited sites that it encloses the number of such islands of given shape size and orientation and the number of occurrences in space of specific local patterns of visited and unvisited sites on a finite lattice we determine the scaling functions that describe the averages on appropriate lattice size dependent time scales on an infinite lattice we first observe that the all increase with t as where k is an m dependent positive integer we then consider the class of random processes constituted by the fluctuations around average we show that to leading order as t gets large these fluctuations are all proportional to a single universal random process normalized to for the probability law of tends to that of varadhan s renormalized local time of self intersections an implication is that in the long time limit all are proportional to

global and local error analysis for the residual free bubbles method applied to advection dominated problems we prove the stability and a priori global and local error analysis for the residual free bubbles rfb finite element method applied to advection dominated advection diffusion problems

a class of discontinuous petrov galerkin methods part i the transport equation considering a simple model transport problem we present a new finite element method while the new method fits in the class of discontinuous galerkin dg methods it differs from standard dg and streamline diffusion methods in that it uses a space of discontinuous trial functions tailored for stability the new method unlike the older approaches yields optimal estimates for the primal variable in both the element size h and polynomial degree p and outperforms the standard upwind dg method

how a random walk covers a finite lattice abstract a random walker is confined to a finite periodic d dimensional lattice of n initially white sites when visited by the walk a site is colored black after t steps of the walk for t scaled appropriately with n we determined the structure of the set of white sites the variance of their number has a line of critical points in the td plane which separates a mean field region from a region with enhanced fluctuations at d 2 the critical point becomes a critical interval moreover for d 2 the set of white sites is fractal with a fractal dimensionality whose t dependence we determine

random walks with spontaneous emission on lattices with periodically distributed imperfect traps abstract we study random walks on d dimensional lattices with periodically distributed traps in which the walker has a finite probability per step of disappearing from the lattice and a finite probability of escaping from a trap general expressions are derived for the total probability that the walk ends in a trap and for the moments of the number of steps made before this happens if it does happen the analysis is extended to lattices with more types of traps and to a model where the trapping occurs during special steps finally the green s function at the origin g 0 z for a finite lattice with periodic boundary conditions which enters into the main expressions is studied more closely a generalization of an expression for g 0 1 for the square lattice given by montroll to values of z different from but close to 1 is derived

repeated bond traversal probabilities for the simple random walk we consider the average number bm t of bonds traversed exactly m times by a t step simple random walk we determine bm t explicitly in the scaling limit t with m t fixed in dimension d 1 and m log t fixed in dimension d 2 the scaling function is an erfc in d 1 and an exponential in d 2

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100