a new analysis of discontinuous galerkin methods for a fourth order variational inequality abstract we study a family of discontinuous galerkin methods for the displacement obstacle problem of kirchhoff plates on two and three dimensional convex polyhedral domains which are characterized as fourth order elliptic variational inequalities of the first kind we prove that the error in an h 2 like energy norm is o h for the quadratic method where 1 2 1 is determined by the geometry of the domain under additional assumptions on the contact set such that the solution has improved regularity we derive the optimal error estimate with 1 3 2 for the cubic method numerical experiments demonstrate the performance of the methods and confirm the theoretical results

on the quadratic finite element approximation to the obstacle problem in this paper we obtain the error bound o left h frac 32 epsilon right for any epsilon 0 for the piecewise quadratic finite element approximation to the obstacle problem without the hypothesis that the free boundary has finite length see 3

a class of global fractional order projective dynamical systems involving set valued perturbations this paper studies a class of global fractional order projective dynamical systems involving set valued perturbations in real separable hilbert spaces we prove that the set of solutions for this type of systems is nonempty and closed under some suitable conditions furthermore we show that the set of solutions is continuous with respect to initial value in the sense of hausdorff metric finally an interesting numerical example is given to illustrate the validity of the main theorem presented in this paper

effect of local dispersion on solute transport in randomly heterogeneous media recently an exact eulerian lagrangian theory of advective transport in space time random velocity fields was developed by one of us we present a formal extension of this theory that accounts for anisotropic local dispersion the resultant conditional mean transport equation is generally nonlocal in space time to assess the impact of local dispersion on the prediction of transport under uncertainty we adopt a first order pseudo fickian approximation for this equation we then solve it numerically by galerkin finite elements for two dimensional transport from an instantaneous square source in a uniform unconditional mean flow field subject to isotropic local dispersion we use a higher order approximation to compute explicitly the standard deviation and coefficient of variation of the predicted concentrations our theory shows in an exact manner and our numerical results demonstrate under the above closure approximations that the effect of local dispersion on first and second concentration moments varies monotonically with the magnitude of the local dispersion coefficient when this coefficient is small relative to macrodispersion its effect on the prediction of nonreactive transport under uncertainty can for all practical purposes be disregarded this is contrary to some recent assertions in the literature that local dispersion must always be taken into account no matter how small

numerical analysis of stationary variational hemivariational inequalities variational hemivariational inequalities refer to the inequality problems where both convex and nonconvex functions are involved in this paper we consider the numerical solution of a family of stationary variational hemivariational inequalities by the finite element method for a variational hemivariational inequality of a general form we prove convergence of numerical solutions for some particular variational hemivariational inequalities we provide error estimates of numerical solutions which are of optimal order for the linear finite element method under appropriate solution regularity assumptions numerical results are reported on solving a variational hemivariational inequality modeling the contact between an elastic body and a foundation with the linear finite element illustrating the theoretically predicted optimal first order convergence and providing their mechanical interpretations

a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis a discontinuous galerkin formulation is developed and analyzed for the cases of classical and gradient plasticity the model of gradient plasticity is based on the von mises yield function in which dependence is on the isotropic hardening parameter and its laplacian the problem takes the form of a variational inequality of the second kind the discontinuous galerkin formulation is shown to be consistent and convergent error estimates are obtained for the cases of semi and fully discrete formulations these mimic the error estimates obtained for classical plasticity with the conventional galerkin formulation

optimal a priori error estimates for the hp version of the local discontinuous galerkin method for convection diffusion problems we study the convergence properties of the hp version of the local discontinuous galerkin finite element method for convection diffusion problems we consider a model problem in a one dimensional space domain we allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh width h in the polynomial degree p and in the regularity of the exact solution we identify a special numerical flux for which the estimates are optimal in both h and p the theoretical results are confirmed in a series of numerical examples

semilinear elliptic equations admitting similarity transformations in this paper we study the equation u 2 h u 0 in a smooth bounded domain where x dist x 0 and h is a nondecreasing function which satisfies keller osserman condition we introduce a condition on h which implies that the equation is subcritical i e the corresponding boundary value problem is well posed with respect to data given by finite measures under additional assumptions on h we show that this condition is necessary as well as sufficient we also discuss b v problems with data given by positive unbounded measures our results extend results of 13 treating equations of the form u uq 0 with q 1 2

a two level algorithm for an obstacle problem abstract due to the inequality feature of the obstacle problem the standard quadratic finite element method for solving the problem can only achieve an error bound of the form o n 3 4 n being the total number of degrees of freedom and 0 arbitrary to achieve a better error bound the key lies in how to capture the free boundary accurately in this paper we propose a two level algorithm for solving the obstacle problem the first part of the algorithm is through the use of the linear elements on a quasi uniform mesh then information on the approximate free boundary from the linear element solution is used in the construction of a quadratic finite element method under some assumptions it is shown that the numerical solution from the two level algorithm is expected to have a nearly optimal error bound of o n 1 0 arbitrary such an expected convergence order is observed numerically in numerical examples

numerical analysis of elliptic hemivariational inequalities this paper is devoted to a study of the numerical solution of elliptic hemivariational inequalities with or without convex constraints by the finite element method for a general family of elliptic

improving the accuracy of risk prediction from particle based breakthrough curves reconstructed with kernel density estimators an edited version of this paper was published by agu copyright 2015 american geophysical union

impact of sampling volume on the probability density function of steady state concentration 1 in recent years statistical theory has been used to compute the ensemble mean and variance of solute concentration in aquifer formations with second order stationary velocity fields the merit of accurately estimating the mean and variance of concentration however remains unclear without knowing the shape of the probability density function pdf in a setup where a conservative solute is continuously injected into a domain the concentration is bounded between zero and the concentration value in the injected solution at small travel distances close to the fringe of the plume an observation point may fall into the plume or outside so that the statistical concentration distribution clusters at the two limiting values obviously this results in non gaussian pdf s of concentration with increasing travel distance the lateral plume boundaries are smoothed resulting in increased probability of intermediate concentrations likewise averaging the concentration in a larger sampling volume as typically done in field measurements leads to higher probabilities of intermediate concentrations we present semianalytical results of concentration pdf s for measurements with point like or larger support volumes based on stochastic theory applied to stationary media to this end we employ a reversed auxiliary transport problem in which we use analytical expressions for first and second central spatial lateral moments with an assumed gaussian pdf for the uncertainty of the first lateral moment and gauss like shapes in individual cross sections the resulting concentration pdf can be reasonably fitted by beta distributions the results are compared to monte carlo simulations of flow and steady state transport in 3 d heterogeneous domains in both methods the shape of the concentration pdf changes with distance to the contaminant source near the source the distribution is multimodal whereas it becomes a unimodal beta distribution far away from the contaminant source the semianalytical and empirical pdf s differ slightly which we contribute to the numerical artifacts in the monte carlo simulations but also to hard assumptions made in the semianalytical approach our results imply that geostatistical techniques for interpolation and other statistical inferences based on gaussian distributions such as kriging and cokriging may be feasible only far away from the contaminant source for calculations near the source the beta like distribution of concentration should be accounted for

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

the method of distributions for dispersive transport in porous media with uncertain hydraulic properties predictions of solute transport in subsurface environments are notoriously unreliable due to aquifer heterogeneity and uncertainty about the values of hydraulic parameters probabilistic framework which treats the relevant parameters and solute concentrations as random fields allows for quantification of this predictive uncertainty by providing deterministic equations for either probability density function or cumulative distribution function cdf of predicted concentrations the method of distributions enables one to estimate e g the probability of a contaminant s concentration exceeding a safe dose we derive a deterministic equation for the cdf of solute concentration which accounts for uncertainty in flow velocity and initial conditions the coefficients in this equation are expressed in terms of the mean and variance of concentration the accuracy and robustness of the cdf equations are analyzed by comparing their predictions with those obtained with monte carlo simulations and an assumed beta cdf

discontinuous galerkin methods for solving a quasistatic contact problem we consider the numerical solution of a nonlinear evolutionary variational inequality arising in the study of quasistatic contact problems we study spatially semi discrete and fully discrete schemes for the problem with several discontinuous galerkin discretizations in space and finite difference discretization in time under appropriate regularity assumptions on the solution a unified error analysis is established for the schemes reaching the optimal convergence order for linear elements numerical results are presented on a two dimensional test problem to illustrate numerical convergence orders

inhomogeneous dirichlet boundary condition in the a posteriori error control of the obstacle problem abstract we study a posteriori error control of finite element approximation of the elliptic obstacle problem with nonhomogeneous dirichlet boundary condition the results in the article are two fold firstly we address the influence of the inhomogeneous dirichlet boundary condition in residual based a posteriori error control of the elliptic obstacle problem secondly by rewriting the obstacle problem in an equivalent form we derive a posteriori error bounds which are in simpler form and efficient to accomplish this we construct and use a post processed solution u h of the discrete solution u h which satisfies the exact boundary conditions sharply although the discrete solution u h may not satisfy we propose two post processing methods and analyze them namely the harmonic extension and a linear extension the theoretical results are illustrated by the numerical results

the weber seifert dodecahedral space is non haken in this paper we settle thurston s old question of whether the weber seifert dodecahedral space is non haken a problem that has been a benchmark for progress in computational 3 manifold topology over recent decades we resolve this question by combining recent significant advances in normal surface enumeration new heuristic pruning techniques and a new theoretical test that extends the seminal work of jaco and oertel

approximating the common fixed points of two sequences of uniformly quasi lipschitzian mappings in convex metric spaces in this paper a kind of ishikawa type iterative scheme with errors for approximating a common fixed point of two sequences of uniformly quasi lipschitzian mappings is introduced and studied in convex metric spaces under some suitable conditions the convergence theorems concerned with the ishikawa type iterative scheme with errors to approximate a common fixed point of two sequences of uniformly quasi lipschitzian mappings were proved in convex metric spaces the results presented in the paper generalize and improve some recent results of wang and liu c wang l w liu convergence theorems for fixed points of uniformly quasi lipschitzian mappings in convex metric spaces nonlinear anal tma 70 2009 2067 2071

pl equivariant surgery and invariant decompositions of 3 manifolds using normal surface theory h j2 we introduce the notion of least weight normal surfaces the weight of a normal surface is a nonnegative integer invariant of the normal isotopy class of the surface if we focus on a particular class of normal surfaces and choose representatives which minimize the weight over the class then we have least weight normal surfaces it is remarkable how these least weight normal surfaces exhibit many of the same useful properties as least area minimal surfaces they provide a piecewise linear pl environment to obtain the recent topological results coming from the analysis and geometry of least area surfaces in an impressive series of papers meeks and yau m y i m y z m y 3 m y m y meeks simon and yau m s y scott s meeks and scott m s and freedman hass and scott f h s introduced the analysis and geometry of least area surfaces into the study of topological questions about 3 manifolds the consequences have led to the resolution of many outstanding problems in the topology of 3 manifolds since most of the problems being solved with these new techniques are topological in nature it has been felt that there should be a more topological approach to the proofs of these results besides the bulk of the work in the theory of

korn s inequalities for piecewise h 1 vector fields korn s inequalities for piecewise h 1 vector fields are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

error estimates of the weakly over penalized symmetric interior penalty method for two variational inequalities in this paper we apply the weakly over penalized symmetric interior penalty method to solve some variational inequalities which include the signorini problem and the obstacle problem optimal a priori error estimates in energy norm are derived some numerical tests are provided to confirm our theoretical analysis

discrete ordinate discontinuous galerkin methods for solving the radiative transfer equation the radiative transfer equation rte occurs in a wide variety of applications in this paper we study discrete ordinate discontinuous galerkin methods for solving the rte the numerical methods are formed in two steps in the first step the discrete ordinate technique is applied to discretize the integral operator for the angular variable resulting in a semidiscrete hyperbolic system in the second step the spatial discontinuous galerkin method is applied to discretize the semidiscrete system a stability and error analysis is performed on the numerical methods some numerical examples are included to demonstrate the convergence behavior of the methods

stabilized hp finite element methods for first order hyperbolic problems we analyze the hp version of the streamline diffusion finite element method sdfem and of the discontinuous galerkin finite element method dgfem for first order linear hyperbolic problems for both methods we derive new error estimates on general finite element meshes which are sharp in the mesh width h and in the spectral order p of the method assuming that the stabilization parameter is o h p for piecewise analytic solutions exponential convergence is established on quadrilateral meshes for the dgfem we admit very general irregular meshes and for the sdfem we allow meshes which contain hanging nodes numerical experiments confirm the theoretical results

a morley finite element method for the displacement obstacle problem of clamped kirchhoff plates abstract we study a morley finite element method for the displacement obstacle problem of clamped kirchhoff plates on polygonal domains error estimates are derived in the energy norm and the l norm the performance of the method is illustrated by numerical experiments

heegaard surfaces and measured laminations i the waldhausen conjecture we give a proof of the so called generalized waldhausen conjecture which says that an orientable irreducible atoroidal 3 manifold has only finitely many heegaard splittings in each genus up to isotopy jaco and rubinstein have announced a proof of this conjecture using different methods

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments

superconvergence analysis of low order nonconforming finite element methods for variational inequality problem with displacement obstacle abstract superconvergence analysis of nonconforming finite element methods fems are discussed for solving the second order variational inequality problem with displacement obstacle the elements employed have a common typical character i e the consistency error can reach order o h 3 2 nearly 1 2 order higher than their interpolation error when the exact solution of the considered problem belongs to h 5 2 for any e 0 by making full use of special properties of the element s interpolations and bramble hilbert lemma the superconvergence error estimates of order o h 3 2 in the broken h1 norm are derived finally some numerical results are provided to confirm the theoretical results

asymptotic dispersion in 2d heterogeneous porous media determined by parallel numerical simulations we determine the asymptotic dispersion coefficients in 2d exponentially correlated lognormally distributed permeability fields by using parallel computing fluid flow is computed by solving the flow equation discretized on a regular grid and transport triggered by advection and diffusion is simulated by a particle tracker to obtain a well defined asymptotic regime under ergodic conditions initial plume size much larger than the correlation length of the permeability field the characteristic dimension of the simulated computational domains was of the order of 103 correlation lengths with a resolution of ten cells by correlation length we determine numerically the asymptotic effective longitudinal and transverse dispersion coefficients over 100 simulations for a broad range of heterogeneities s 2 0 9 where s 2 is the lognormal permeability variance for purely advective transport the asymptotic longitudinal dispersion coefficient depends linearly on s 2 for s 2 1 and the asymptotic transverse dispersion coefficient is zero addition of homogeneous isotropic diffusion induces an increase of transverse dispersion and a decrease of longitudinal dispersion

conditional estimation of solute travel time in heterogeneous formations impact of transmissivity measurements solute transport through heterogeneous formations is modeled by the travel time approach i e the time it takes a solute particle to travel from the source to a control plane due to uncertainty is a random variable characterized by its cumulative probability density function g this function is used by regulatory agencies in pollution site assessment the main aim of the present study is to examine the impact of data namely measured transmissivities upon g and reduction of uncertainty this is achieved for two dimensional steady flow of average uniform head gradient by using a lagrangian approach developed in the past dagan 1982 1984 1989 rubin 1990 1991 the impact of data is seen in conditioning on measured values and in estimation of parameters characterizing flow and formation properties the approach is illustrated by a simulation based on synthetic data the results show how uncertainty is reduced by increasing the density of measurements for a control plane sufficiently far from the source

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

a posteriori error estimates of discontinuous galerkin methods for the signorini problem a reliable and efficient a posteriori error estimator is derived for a class of discontinuous galerkin dg methods for the signorini problem a common property shared by many dg methods leads to a unified error analysis with the help of a constraint preserving enriching map the error estimator of dg methods is comparable with the error estimator of the conforming methods numerical experiments illustrate the performance of the error estimator

analysis of a damped nonlinear multilevel method in this paper we present a new algorithm that is obtained by introducing a damping parameter in the classical nonlinear multilevel method we analyse this damped nonlinear multilevel method in particular we prove global convergence and local efficiency for a suitable class of problems

evolutionary problems driven by variational inequalities abstract in this paper we introduce the differential system obtained by mixing an evolution equation and a variational inequality eevi for short first by using kkm theorem and monotonicity arguments we prove the superpositional measurability and upper semicontinuity for the solution set of a general variational inequality then we establish that the solution set of eevi is nonempty and compact our approach is based on the theory of semigroups filippov implicit function lemma and fixed point theory for set valued mappings

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

z 2 thurston norm and complexity of 3 manifolds a new lower bound on the complexity of a 3 manifold is given using the mathbb z 2 thurston norm this bound is shown to be sharp and the minimal triangulations realising it are characterised using normal surfaces consisting entirely of quadrilateral discs

solvability of parabolic variational hemivariational inequalities involving space fractional laplacian abstract the paper is devoted to investigate a new kind of variational hemivariational inequality of parabolic type driven by a generalized space fractional laplace operator and two multivalued terms which are expressed by the clarke generalized gradient and convex subgradient respectively an existence theorem of weak solutions to the problem is established by applying a surjectivity theorem for the sum of operators combined with results from nonsmooth analysis

expected residual minimization method for stochastic variational inequality problems with nonlinear perturbations in this paper we consider stochastic affine variational inequality problems with nonlinear perturbation for short svipp firstly we formulate svipp as an erm problem that minimizes the expected residual of the so called gap function furthermore we study some properties of the erm problem under some suitable conditions by means of quasi monte carlo method we obtain a discrete approximation of erm problem finally we consider the convergence of optimal solutions and stationary points of the approximation problem as sample size increases

monotone multigrid methods for elliptic variational inequalities i we derive fast solvers for discrete elliptic variational inequalities of the first kind obstacle problems as resulting from the approximation of related continuous problems by piecewise linear finite elements using basic ideas of successive subspace correction we modify well known relaxation methods by extending the set of search directions extended underrelaxations are called monotone multigrid methods if they are quasioptimal in a certain sense by construction all monotone multigrid methods are globally convergent we take a closer look at two natural variants the standard monotone multigrid method and a truncated version for the considered model problems the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarse grid transport while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case

finite element methods for the displacement obstacle problem of clamped plates we study finite element methods for the displacement obstacle problem of clamped kirchhoff plates a unified convergence analysis is provided for c1 finite element methods classical nonconforming finite element methods and c0 interior penalty methods under the condition that the obstacles are sufficiently smooth and that they are separated from each other and the zero displacement boundary constraint we prove that the convergence in the energy norm is o h for convex domains

multilevel projection algorithm for solving obstacle problems abstract obstacle problems are nonlinear free boundary problems and the computation of approximate solutions can be difficult and expensive little work has been done on effective numerical methods of such problems this paper addresses some aspects of this issue discretizing the problem in a continuous piecewise linear finite element space gives a quadratic programming problem with inequality constraints a new method called the multilevel projection mp method is established in this paper the mp algorithm extends the multigrid method for linear equations to nonlinear obstacle problems the convergence theorems of this method are also proved a numerical example presented shows our error estimate is sharp and the mp algorithm is robust

inflations of ideal triangulations abstract starting with an ideal triangulation of m the interior of a compact 3 manifold m with boundary no component of which is a 2 sphere we provide a construction called an inflation of the ideal triangulation to obtain a strongly related triangulation of m itself besides a step by step algorithm for such a construction we provide examples of an inflation of the two tetrahedra ideal triangulation of the complement of the figure eight knot in s 3 giving a minimal triangulation having ten tetrahedra of the figure eight knot exterior as another example we provide an inflation of the one tetrahedron gieseking manifold giving a minimal triangulation having seven tetrahedra of a non orientable compact 3 manifold with klein bottle boundary several applications of inflations are discussed

optimizing the double description method for normal surface enumeration many key algorithms in 3 manifold topology involve the enumeration of normal surfaces which is based upon the double description method for finding the vertices of a convex polytope typically we are only interested in a small subset of these vertices many key algorithms in 3 manifold topology involve the enumeration of normal surfaces which is based upon the double description method for finding the vertices of a convex polytope typically we are only interested in a small subset of these vertices thus opening the way for substantial optimization here we give an account of the vertex enumeration problem as it applies to normal surfaces and present new optimizations that yield strong improvements in both running time and memory consumption the resulting algorithms are tested using the freely available software package regina

convergence analysis of symmetric dual wind discontinuous galerkin approximation methods for the obstacle problem abstract this paper formulates and analyzes symmetric dual wind discontinuous galerkin dg methods for second order elliptic obstacle problem these new methods follow from the dg differential calculus framework that defines discrete differential operators to replace the continuous differential operators when discretizing a partial differential equation pde we establish optimal a priori error estimates for both linear and quadratic elements provided the exact solution is sufficiently regular these results are also shown to hold for some non positive penalty parameters with the emphasis on zero penalization across all interior and boundary edges numerical experiments are provided to validate the theoretical results and gauge the performance of the proposed methods

increasing the number of fibered faces of arithmetic hyperbolic 3 manifolds we exhibit a closed hyperbolic 3 manifold which satisfies a very strong form of thurston s virtual fibration conjecture in particular this manifold has finite covers which fiber over the circle in arbitrarily many ways more precisely it has a tower of finite covers where the number of fibered faces of the thurston norm ball goes to infinity in fact faster than any power of the logarithm of the degree of the cover and we give a more precise quantitative lower bound the example manifold m is arithmetic and the proof uses detailed number theoretic information at the level of the hecke eigenvalues to drive a geometric argument based on fried s dynamical characterization of the fibered faces the origin of the basic fibration m s 1 is the modular elliptic curve e x 0 49 which admits multiplication by the ring of integers of q 7 we first base change the holomorphic differential on e to a cusp form on gl 2 over k q 3 and then transfer over to a quaternion algebra d k ramified only at the primes above 7 the fundamental group of m is a quotient of the principal congruence subgroup of o d of level 7 to analyze the topological properties of m we use a new practical method for computing the thurston norm which is of independent interest we also give a noncompact finite volume hyperbolic 3 manifold with the same properties by using a direct topological argument

0 efficient triangulations of 3 manifolds efficient triangulations of 3 manifolds are defined and studied it is shown that any triangulation of a closed orientable irreducible 3 manifold m can be modified to a 0 efficient triangulation orm can be shown to be one of the manifolds s 3 rp 3 or l 3 1 similarly any triangulation of a com pact orientable irreducible irreducible 3 manifold can be modified to a 0 efficient triangulation the notion of a 0 efficient ideal tr is de fined it is shown if m is a compact orientable irreducible irreducible 3 manifold having no essential annuli and distinct from the 3 cell then m admits an ideal triangulation furthermore it is shown that any ideal trian gulation of such a 3 manifold can be modified to a 0 efficient ideal triangula tion a 0 efficient triangulation of a closed manifold has only one vertex or the manifold is s 3 and the triangulation has precisely two vertices 0 efficient tri angulations of 3 manifolds with boundary and distinct from the 3 cell have all their vertices in the boundary and then just one vertex in each bound ary component as tools we introduce the concepts of barrier surface and shrinking as well as the notion of crushing a triangulation along a normal sur face a number of applications are given including an algorithm to construct an irreducible decomposition of a closed orientable 3 manifold an algorithm to construct a maximal collection of pairwise disjoint normal 2 spheres in a closed 3 manifold an alternate algorithm for the 3 sphere recognition prob lem results on edges of low valence in minimal triangulations of 3 manifolds and a construction of irreducible knots in closed 3 manifolds

embeddability in the 3 sphere is decidable we show that the following algorithmic problem is decidable given a 2 dimensional simplicial complex can it be embedded topologically or equivalently piecewise linearly in r3 by a known reduction it suffices to decide the embeddability of a given triangulated 3 manifold x into the 3 sphere s3 the main step which allows us to simplify x and recurse is in proving that if x can be embedded in s3 then there is also an embedding in which x has a short meridian that is an essential curve in the boundary of x bounding a disk in s3 b x with length bounded by a computable function of the number of tetrahedra of x

volume optimization normal surfaces and thurston s equation on triangulated 3 manifolds we propose a finite dimensional variational principle on triangulated 3 manifolds so that its critical points are related to solutions to thurston s gluing equation and haken s normal surface equation the action functional is the volume

computing trisections of 4 manifolds we describe an algorithm to compute trisections of orientable four manifolds using arbitrary triangulations as input this results in explicit complexity bounds for the trisection genus of a 4 manifold in terms of the number of pentachora 4 simplices in a triangulation

a mixed discontinuous galerkin method with symmetric stress for brinkman problem based on the velocity pseudostress formulation abstract the brinkman equations can be regarded as a combination of the stokes and darcy equations which model transitions between the fast flow in channels governed by stokes equations and the slow flow in porous media governed by darcy s law the numerical challenge for this model is the designing of a numerical scheme which is stable for both the stokes dominated high permeability and the darcy dominated low permeability equations in this paper we solve the brinkman model in n dimensions n 2 3 by using the mixed discontinuous galerkin mdg method which meets this challenge this mdg method is based on the pseudostress velocity formulation and uses a discontinuous piecewise polynomial pair p k 1 s p k k 0 where the stress field is symmetric the main unknowns are the pseudostress and the velocity whereas the pressure is easily recovered through a simple postprocessing a key step in the analysis is to establish the parameter robust inf sup stability through specific parameter dependent norms at both continuous and discrete levels therefore the stability results presented here are uniform with respect to the permeability thanks to the parameter robust stability analysis we obtain optimal error estimates for the stress in broken h div norm and velocity in l 2 norm furthermore the l 2 error estimate for pseudostress is derived under certain conditions finally numerical experiments are provided to support the theoretical results and to show the robustness accuracy and flexibility of the mdg method

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

hp version discontinuous galerkin methods for hyperbolic conservation laws abstract the development of hp version discontinuous galerkin methods for hyperbolic conservation laws is presented in this work a priori error estimates are derived for a model class of linear hyperbolic conservation laws these estimates are obtained using a new mesh dependent norm that reflects the dependence of the approximate solution on the local element size and the local order of approximation the results generalize and extend previous results on mesh dependent norms to hp version discontinuous galerkin methods a posteriori error estimates which provide bounds on the actual error are also developed in this work numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a posteriori estimates in providing reliable estimates of the actual error in the numerical solution

concentration variance and spatial covariance in second order stationary heterogeneous conductivity fields the concentration variance and spatial covariance resulting from convective dispersive transport driven by a uniform mean flow in second order stationary conductivity fields was derived in a lagrangian framework based on a first order approximation of solute particle trajectories the approximation of the concentration co variances for large injection volumes is considerably simplified by defining the concentration at a certain location x and time t in terms of the backward solute trajectory probability distribution this is the probability that the trajectory of a fictitious microscopic and indivisible solute particle which is in a volume x centered around x at time t was at the time of solute injection t0 in the injection volume v0 the approximated concentration co variances were validated against concentration covariances derived from transport simulations in generated second order stationary conductivity fields the approximate solutions reproduced fairly well the effects of local scale dispersion and of the spatial variability of the hydraulic conductivity on the concentration co variance the effect of the spatial structure of the hydraulic conductivity field on the spatial covariance of the concentrations was investigated in order to identify parameters that can be unequivocally determined from the structure of the concentration field for a given spreading of the solute plume in the mean flow direction x11 t a given spatial correlation length of the loge transformed hydraulic conductivity in the transverse to flow direction i 2 and a given local scale dispersion dd the concentration covariance was nearly invariant and hardly influenced by the anisotropy of the covariance function e and the variance f2 of the loge transformed conductivity as a result e and f2 cannot be unequivocally determined from the spatial structure of the concentration field the concentration variance and spatial covariance in the transverse to mean flow direction are predominantly determined by the lateral component of the local scale dispersion dd22 and by the spatial correlation length of the loge transformed conductivity in the transverse to flow direction i 2 these two parameters might be unequivocally derived from the concentration co variance

discontinuous galerkin methods for solving elliptic variational inequalities we study discontinuous galerkin methods for solving elliptic variational inequalities of both the first and second kinds analysis of numerous discontinuous galerkin schemes for elliptic boundary value problems is extended to the variational inequalities we establish a priori error estimates for the discontinuous galerkin methods which reach optimal order for linear elements results from some numerical examples are reported

large time behavior of concentration variance and dilution in heterogeneous formations consider the advection and dispersion of a conservative nonsorbing solute in a spatially variable but statistically homogeneous velocity field a lagrangian approach leads to expressions for the large time mean and variance of concentration the expressions require only the macroscopic velocity vector um the macrodispersion tensor dm and an additional tensor the macroscopic velocities and macrodispersivities are well known from numerous previous studies but is introduced here for the first time the tensor is needed to describe the kinetics of dilution of a plume two cases with the same um anddm have different dilution characteristics depending on the characteristic times of dilution are given by tensor d 1m it is demonstrated for the first time through a lagrangian approach that the coefficient of variation of concentration at the center of the plume becomes proportionate to 1 t as was previously shown in the eulerian theory ofkapoor and gelhar 1994a b expressions for the geometric mean of the dilution index and the reactor ratio are derived numerical simulations support the validity of the approach

a posteriori error control of discontinuous galerkin methods for elliptic obstacle problems in this article we derive an a posteriori error estimator for various discontinuous galerkin dg methods that are proposed in wang han and cheng siam j numer anal 48 708 733 2010 for an elliptic obstacle problem using a key property of dg methods we perform the analysis in a general framework the error estimator we have obtained for dg methods is comparable with the estimator for the conforming galerkin cg finite element method in the analysis we construct a non linear smoothing function mapping dg finite element space to cg finite element space and use it as a key tool the error estimator consists of a discrete lagrange multiplier associated with the obstacle constraint it is shown for non over penalized dg methods that the discrete lagrange multiplier is uniformly stable on non uniform meshes finally numerical results demonstrating the performance of the error estimator are presented

gradient schemes for the signorini and the obstacle problems and application to hybrid mimetic mixed methods gradient schemes is a framework which enables the unified convergence analysis of many different methods such as finite elements conforming non conforming and mixed and finite volumes methods for 2nd order diffusion equations we show in this work that the gradient schemes framework can be extended to variational inequalities involving mixed dirichlet neumann and signorini boundary conditions this extension allows us to provide error estimates for numerical approximations of such models recovering known convergence rates for some methods and establishing new convergence rates for schemes not previously studied for variational inequalities the general framework we develop also enables us to design a new numerical method for the obstacle and signorini problems based on hybrid mimetic mixed schemes we provide numerical results that demonstrate the accuracy of these schemes and confirm our theoretical rates of convergence

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

adaptive discontinuous galerkin methods for solving an incompressible stokes flow problem with slip boundary condition of frictional type abstract we consider the numerical solution of a variational inequality arising in the study of an incompressible stokes flow with a nonlinear slip boundary condition of frictional type we study several discontinuous galerkin dg schemes for solving the problem and derive reliable and efficient residual type a posteriori error estimators an adaptive dg method is constructed based on the error estimators to solve the problem some numerical examples are presented to illustrate the efficiency of the adaptive dg method

partial differential variational inequalities involving nonlocal boundary conditions in banach spaces abstract in this paper we firstly introduce a complicated system obtained by mixing a nonlinear evolutionary partial differential equation and a mixed variational inequality in infinite dimensional banach spaces in the case where the set of constraints is not necessarily bounded and the problem is driven by nonlocal boundary conditions which is called partial differential variational inequality pdvi for short then we show that the solution set of the mixed variational inequality involved in problem pdvi is nonempty bounded closed and convex moreover the upper semicontinuity and measurability properties for set valued mapping u 0 t e 2 c b v e 1 see 3 7 below are also established finally several existence results for pdvi are obtained by using a fixed point theorem for condensing set valued operators and theory of measure of noncompactness

on the optimal control of variational hemivariational inequalities abstract the present paper represents a continuation of 26 there a continuous dependence result for the solution of an elliptic variational hemivariational inequality was obtained and then used to prove the existence of optimal pairs for two associated optimal control problems in the current paper we complete this study with more general results indeed we prove the continuous dependence of the solution with respect to a parameter which appears in all the data of the problem including the set of constraints the nonlinear operator and the two functionals which govern the variational hemivariational inequality this allows us to consider a general associated optimal control problem for which we prove the existence of optimal pairs together with a new convergence result the mathematical tools developed in this paper are useful in the analysis and control of a large class of boundary value problems which in a weak formulation lead to elliptic variational hemivariational inequalities to provide an example we illustrate our results in the study of an inequality which describes the equilibrium of an elastic body in frictional contact with a foundation made of a rigid body covered by a layer of soft material

impact of aquifer heterogeneity structure and local scale dispersion on solute concentration uncertainty in this paper we study the influence of high log conductivity variance sigma 2 y and local scale dispersion on the first two concentration moments as well as on higher order moments skewness

semi discrete stabilized finite element methods for navier stokes equations with nonlinear slip boundary conditions based on regularization procedure based on the pressure projection stabilized methods the semi discrete finite element approximation to the time dependent navier stokes equations with nonlinear slip boundary conditions is considered in this paper because this class of boundary condition includes the subdifferential property then the variational formulation is the navier stokes type variational inequality problem using the regularization procedure we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter varepsilon which means that the solution of the regularized problem converges to the solution of the navier stokes type variational inequality problem as the parameter varepsilon longrightarrow 0 moreover some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data the pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived

rothe method for parabolic variational hemivariational inequalities abstract the paper deals with the convergence analysis of the semidiscrete rothe scheme for the parabolic variational hemivariational inequality with the nonlinear pseudomonotone elliptic operator the problem involves both a discontinuous and nonmonotone multivalued term as well as a monotone term with potentials which assume infinite values and hence are not locally lipschitz we prove the existence of a solution and establish a convergence result of a numerical semidiscrete scheme the proof can be viewed both as the proof of solution existence as well as the proof of the convergence of a numerical semidiscrete scheme the numerical simulations to present the rate of convergence with respect to space and time for piecewise linear finite elements are presented as well

angle structures and normal surfaces let m be the interior of a compact 3 manifold with boundary and let t be an ideal triangulation of m this paper describes necessary and sufficient conditions for the existence of angle structures semi angle structures and generalised angle structures on m t respectively in terms of a generalised euler characteristic function on the solution space of the normal surface theory of m t this extends previous work of kang and rubinstein and is itself generalised to a more general setting for 3 dimensional pseudo manifolds

stochastic modeling of groundwater flow by unconditional and conditional probabilities 2 the solute transport the problem of solute transport in a heterogeneous formation whose transmissivity or hydraulic conductivity are subject to uncertainty is studied for two and three dimensional flows approximate closed form solutions are derived for the case of a solute pulse in an average uniform flow through a formation of unconditional stationary random transmissivity the solute concentration regarded as a random variable is determined in terms of its expectation and variance and is found to be subject to a high degree of uncertainty the uncertainty is greatly reduced if the solute input zone is large compared to the transmissivity integral scale in any case the concentration expectation does not obey a diffusion type equation in the case of two dimensional flows unless the solute body has traveled a distance larger than a few tens transmissivity integral scales this distance may be exceedingly large in many conceivable applications

eulerian lagrangian approach for modeling of flow and transport in heterogeneous geological formations this paper presents a new eulerian lagrangian method for modeling flow and transport of passive solutes in heterogeneous porous formations the physically plausible random velocity fields are generated by a geostatistical based model the ability of the method to correctly reproduce the velocity field statistics and to satisfy mass balance is tested and demonstrated for the case of two dimensional flow the method is found particularly useful in dealing with large fields where numerical fixed grid methods such as finite difference or finite elements become very demanding from a computational point of view the transport problem is solved in a lagrangian framework by the particle tracking approach which uses a suitable grid refinement to correctly handle the velocity fluctuation at any a priori defined resolution the new method is employed to compute the probability distribution functions of the concentration and travel times and to investigate the limitations of existing methods for predicting concentrations

random walk simulation of transport in heterogeneous porous media local mass conservation problem and implementation methods the random walk method for simulating solute transport in porous media is typically based on the assumption that the velocity and velocity dependent dispersion tensor vary smoothly in space however in cases where sharp interfaces separate materials with contrasting hydraulic properties these quantities may be discontinuous normally velocities are interpolated to arbitrary particle locations when finite difference or finite element methods are used to solve the flow equation the use of interpolation schemes that preserve discontinuities in velocity at material contacts can result in a random walk model that does not locally conserve mass unless a correction is applied at these contacts test simulations of random walk particle tracking with and without special treatment of material contacts demonstrate the problem techniques for resolving the problem including interpolation schemes and a reflection principle are reviewed and tested results from simulations of transport in porous media with discontinuities in the dispersion tensor show which methods satisfy continuity simulations of transport in two dimensional heterogeneous porous media demonstrate the potentially significant effect of using a nonconservative model to compute spatial moments and breakthrough of a solute plume

first order based cumulative distribution function for solute concentration in heterogeneous aquifers theoretical analysis and implications for human health risk assessment quantifying the uncertainty of solute concentration in heterogeneous aquifers is an important step in both human health and ecological risk analysis the need for a probabilistic representation of transport is justified by the incomplete characterization of the subsurface we derive the one point concentration cumulative distribution function cdf while taking into account the spatial statistical structure of the hydraulic conductivity space dimensionality the injection source size the peclet number and the sampling volume at the monitoring location the cdf is application oriented and derived at first order in the log conductivity variance we illustrate how several key parameters control the shape of the concentration cdf the cdf shape is important since it reflects both uncertainty and the dilution state of the plume the transition from a bimodal to a unimodal cdf is examined and results are further supported by analyzing the concentration coefficient of variation results indicate the significance of the statistical anisotropy ratio i e the ratio between the hydraulic conductivity correlation scales in determining the cdf shape the importance of the sampling volume in the tails of the concentration cdf and a comparison between the proposed model with the cdf approach i e beta distribution are also shown finally we illustrate how the framework could be used in applications by evaluating the human health risk cdf our results are formally valid for low to moderate heterogeneous aquifers and source sizes small as compared to the hydraulic conductivity correlation length the proposed approach can serve as a benchmark tool for other methods

a discontinuous galerkin formulation for classical and gradient plasticity part 2 algorithms and numerical analysis this work is the second of a two part investigation into the use of discontinuous galerkin methods for obtaining approximate solutions to problems of classical and gradient plasticity part i j k djoko f ebobisse a t mcbride b d reddy a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis comput methods appl mech engrg 196 2007 3881 3897 presented the formulation and analysis of such problems this part focusses on algorithmic and computational aspects of the problem in particular it is shown that the predictor corrector algorithms of classical plasticity are readily extended to the case of gradient plasticity and to discontinuous galerkin formulations conditions for convergence of the algorithms are presented for the elastic secant and consistent tangent predictors the form of the consistent tangent modulus is established for the case of gradient plasticity a selection of numerical examples is presented and discussed with a view to illustrating aspects of the approximation scheme and algorithms as well as features of the model of gradient plasticity adopted here

weak galerkin finite element methods for a second order elliptic variational inequality abstract a weak galerkin finite element method is used to determine approximate solutions of an elliptic variational inequality three sets of basis functions are employed the first has constant values inside each element and on each edge the second has constant values inside each element but is a linear polynomial on each edge and the third has linear polynomials inside each element and on each edge error estimates including convergence rates are derived for all three cases numerical results were provided to illustrate the theoretical results to show that the choice of how the obstacle function is approximated affects those rates and to show the super convergence for piecewise constant basis functions

word hyperbolic dehn surgery in the late 1970 s thurston dramatically changed the nature of 3 manifold theory with the introduction of his geometrisation conjecture and by proving it in the case of haken 3 manifolds 23 the conjecture for general closed orientable 3 manifolds remains perhaps the most important unsolved problem in the subject a weaker form of the conjecture 19 deals with the fundamental group of a closed orientable 3 manifold it proposes that either it contains z z as a subgroup or it is word hyperbolic in the sense of gromov 11 word hyperbolic groups are precisely those groups which satisfy a linear isoperimetric inequality they are of fundamental importance in geometric group theory and have very many useful properties

mixed hp dgfem for incompressible flows we consider several mixed discontinuous galerkin approximations of the stokes problem and propose an abstract framework for their analysis using this framework we derive a priori error estimates for hp approximations on tensor product meshes we also prove a new stability estimate for the discrete divergence bilinear form