viscosity versus vorticity stretching global well posedness for a family of navier stokes alpha like models abstract we study global well posedness and regularity of solutions for a family of incompressible three dimensional navier stokes alpha like models that employ fractional laplacian operators this family of equations depends on two parameters 1 and 2 which affect the strength of non linearity vorticity stretching and the degree of viscous smoothing varying 1 and 2 interpolates between the incompressible navier stokes equations and the incompressible lagrangian averaged navier stokes model our main result which contains previously established results of j l lions and others provides a relationship between 1 and 2 that is sufficient to guarantee global existence uniqueness and regularity of solutions

the h text p version of the finite element method for domains with curved boundaries this paper analyzes the convergence of the h text p version of the finite element method for elliptic problems on curved domains it proves that when the boundary of the domain is piecewise analytic boundary conditions are piecewise analytic and the sides of the elements are analytic the h text p version has an exponential rate of convergence numerical computations are included the paper generalizes the results previously obtained by the authors the h text p version of finite element method part 1 the basic approximation results comp mech 1 1986 pp 21 41 the h text p version of finite element method part 2 general results and applications comp mech 1 1986 pp 203 220

adaptive remeshing and h p domain decomposition abstract the h p version of the finite element method combines local mesh refinement with an increase of the shape functions polynomial degree this paper presents a method which uses an h p approximation for a domain decomposition with fully overlapping domains allowing c 0 continuous approximation for geometrically incompatible mesh refinement to be obtained the method is applied to a reaction diffusion problem resulting from semiconductor process simulation in numerical examples it is demonstrated that this approach is well suited for problems with sharp interior layers or shock like behaviour

possible problems of scale dependency in applications of the three dimensional fractional advection dispersion equation to natural porous media 1 a three dimensional 3 d analysis of transport and macrodispersion at the macrodispersion experiment made site boggs et al 1993 using the fractional advection dispersion equation fade developed by meerschaert et al 1999 2001 shows that the levy dispersion process is scale dependent levy dispersion may be superior to gaussian dispersion on a sufficiently small scale on larger scales both theories are likely to suffer from the fact that because of depositional structures most flow fields display an evolving nonstationary structure motion in such fields is advection dominated displays a lot of memory and therefore is not modeled well by markov random processes which underlie the derivation of both the gaussian and levy advection dispersion equations berkowitz et al 2002 to improve plume simulation of an advection dominated transport process one would have to bring in more advective irregularity while simultaneously decreasing the levy dispersion coefficient therefore on a 3 d basis first order levy dispersion has limitations similar to gaussian dispersion however this and related theories such as the continuous time random walk ctrw formalism are in the early stages of development and thus may be fruitful areas for further research

p version plate and curved shell element for geometrically non linear analysis this paper presents a p version geometrically non linear formulation based on the total lagrangian approach for a nine node three dimensional curved shell element the element geometry is defined by the coordinates of the nodes located on its middle surface and nodal vectors describing the bottom and top surfaces of the element the element displacement approximation can be of arbitrary and different polynomial orders in the plane of the element and in the transverse direction the element approximation functions and the corresponding nodal variables are derived from the lagrange family of interpolation functions the resulting approximation functions and the nodal variables are hierarchical and the element displacement approximation ensures c continuity the element properties are established using the principle of virtual work and the hierarchical element approximation in formulating the properties of the element complete three dimensional stresses and strains are considered hence the element is equally effective for very thin as well as extremely thick shells and plates incremental equations of equilibrium are derived and solved using the standard newton raphson method the total load is divided into increments and for each increment of load equilibrium iterations are performed until each component of the residuals is within a preset tolerance numerical examples are presented to show the accuracy efficiency and advantages of the present formulation the results obtained from the present formulation are compared with those available in the literature

joint probability distributions for a class of non markovian processes we consider joint probability distributions for the class of coupled langevin equations introduced by fogedby h c fogedby phys rev e 50 1657 1994 we generalize well known results for the single time probability distributions to the case of n time joint probability distributions it is shown that these probability distribution functions can be obtained by an integral transform from distributions of a markovian process the integral kernel obeys a partial differential equation with fractional time derivatives reflecting the non markovian character of the process

error estimates for the combinedh andp versions of the finite element method in theh version of the finite element method convergence is achieved by refining the mesh while keeping the degree of the elements fixed on the other hand thep version keeps the mesh fixed and increases the degree of the elements in this paper we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh in addition it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom even in the presence of corner singularities numerical results comparing theh version p version and combinedh p version for a one dimensional problem are presented

element free galerkin method convergence of the continuous and discontinuous shape functions abstract we consider numerical solutions of second order elliptic partial differential equations such as laplace s equation or linear elasticity in two dimensional non convex domains by the element free galerkin method efg this is a meshless method in which the shape functions are constructed by using weight functions of compact support for non convex domains two approaches to the determination of whether a node affects approximation at a particular point are used a contained path criterion and the visibility criterion we show that for non convex domains the visibility criterion leads to discontinuous weight functions and discontinuous shape functions the resulting approximation is no longer conforming and its convergence must be established by inspection of the so called consistency term we show that the variant of the element free galerkin method which uses the discontinuous shape functions is convergent and that in the practically important case of linear shape functions the convergence rate is not affected by the discontinuities the convergence of the discontinuous approximation is first established by the classical and generalized patch test as these tests do not provide an estimate of the convergence rate the rate of convergence in the energy norm is examined for both the continuous and discontinuous efg shape functions and for smooth and non smooth solutions by a direct inspection of the error terms

the p version of the finite element method for problems requiring c 1 continuity in the p version of the finite element method the triangulation of the domain is fixed and the degree p of the local approximating polynomial is increased this is in contrast with the h version in which the degree p is fixed and the maximum diameter h of all triangles is allowed to go to zero we derive a convergence result for the approximating solution to the biharmonic equation when using the p version this result shows that the rate of convergence for the p version is twice that of h version with quasi uniform mesh in the approximation of singular solutions caused by corners this result which applies to problems which require c 1 global continuity is analogous to an earlier result which applied to c 0 problems applications to several benchmark problems in plate bending are presented sample results are examined and compared with theoretical predictions

p version hierarchical three dimensional curved shell element for elastostatics this paper presents a hierarchical three dimensional curved shell finite element formulation based on the p approximation concept the element displacement approximation can be of arbitrary and different polynomial orders in the plane of the shell and the transverse direction the curved shell element approximation functions and the corresponding nodal variables are derived by first constructing the approximation functions of orders p p and p and the corresponding nodal variable operators for each of the three directions and and then taking their products sometimes also known as tensor product this procedure gives the approximation functions and the corresponding nodal variables corresponding to the polynomial orders p p and p both the element displacement functions and the nodal variables are hierarchical therefore the resulting element matrices and the equivalent nodal load vectors are hierarchical also i e the element properties corresponding to the polynomial orders p p and p are a subset of those corresponding to the orders p 1 p 1 and p 1 the formulation guarantees c continuity or smoothness of the displacement field across the interelement boundaries the geometry of the element is described by the co ordinates of the nodes on its middle surface 0 and the nodal vectors describing its bottom 1 and top 1 surfaces the element properties are derived using the principle of virtual work and the hierarchical element approximation the formulation is equally effective for very thin as well as very thick plates and curved shells in fact in many three dimensional applications the element can be used to replace the hierarchical three dimensional solid element without loss of accuracy but significant gain in modelling convenience numerical examples are presented to demonstrate the accuracy efficiency and overall superiority of the present formulation the results obtained from the present formulation are compared with those available in the literature as well as analytical solutions

limit theorems for coupled continuous time random walks scaling limits of continuous time random walks are used in physics to model anomalous diffusion in which a cloud of particles spreads at a different rate than the classical brownian motion governing equations for these limit processes generalize the classical diffusion equation in this article we characterize scaling limits in the case where the particle jump sizes and the waiting time between jumps are dependent this leads to an efficient method of computing the limit and a surprising connection to fractional derivatives

two bracketing theorems characterizing the eigensolution for theh version of the finite element method this paper demonstrates that the classical inclusion principle is in general not valid for the h version of the finite element method whereas the inclusion principle is valid for second order systems discretized by the h version of the finite element method provided linear interpolation functions are used as admissible functions the principle is not valid for fourth order systems to characterize the computed eigenvalues for fourth order systems discretized by the h version of the finite element method this paper formulates two bracketing theorems

compact difference scheme for the fractional sub diffusion equation with neumann boundary conditions an effective finite difference scheme is considered for solving the time fractional sub diffusion equation with neumann boundary conditions a difference scheme combining the compact difference approach the spatial discretization and l 1 approximation for the caputo fractional derivative is proposed and analyzed although the spatial approximation order at the neumann boundary is one order lower than that for interior mesh points the unconditional stability and the global convergence order o t 2 a h 4 in discrete l 2 norm of the compact difference scheme are proved rigorously where t is the temporal grid size and h is the spatial grid size numerical experiments are included to support the theoretical results and comparison with the related works are presented to show the effectiveness of our method

a priori and a posteriori error analyses of a high order unfitted mixed fem for stokes flow abstract we propose and analyze a high order unfitted mixed finite element method for the pseudostress velocity formulation of the stokes problem with dirichlet boundary condition on a fluid domain with curved boundary the method consists of approximating by a polyhedral subdomain d h with boundary h where a galerkin method is applied to approximate the solution and on a transferring technique based on integrating the extrapolated discrete gradient of the velocity to approximate the dirichlet boundary data on the computational boundary h the associated galerkin scheme is defined by raviart thomas elements of order k 0 for the pseudostress and discontinuous polynomials of degree k for the velocity provided suitable hypotheses on the mesh near the boundary we prove well posedness of the galerkin scheme by means of the babuska brezzi theory and establish the corresponding optimal convergence of o h k 1 moreover for the case when h is constructed through a piecewise linear interpolation of we propose a reliable and quasi efficient residual based a posteriori error estimator its definition makes use of a postprocessed velocity with enhanced accuracy to achieve the same rate of convergence of the method when the solution is smooth enough numerical experiments illustrate the performance of the scheme show the behavior of the associated adaptive algorithm and validate the theory

numerical solutions for fractional reaction diffusion equations fractional diffusion equations are useful for applications in which a cloud of particles spreads faster than predicted by the classical equation in a fractional diffusion equation the second derivative in the spatial variable is replaced by a fractional derivative of order less than two the resulting solutions spread faster than the classical solutions and may exhibit asymmetry depending on the fractional derivative used fractional reaction diffusion equations combine the fractional diffusion with a classical reaction term in this paper we develop a practical method for numerical solution of fractional reaction diffusion equations based on operator splitting then we present results of numerical simulations to illustrate the method and investigate properties of numerical solutions we also discuss applications to biology where the reaction term models species growth and the diffusion term accounts for movements

transport of conservative solutes in simulated fracture networks 2 ensemble solute transport and the correspondence to operator stable limit distributions 1 in networks where individual fracture lengths follow a fractal distribution ensemble transport of conservative solute particles at the leading plume edge exhibit characteristics of operator stable densities these densities have as their governing equations of transport either fractional order or integer order advection dispersion equations model selection depends on the identification of either multi gaussian or operator stable transport regimes which in turn depends on the power law exponent of the fracture length distribution low to moderately fractured networks with power law fracture length exponents less than or equal to 1 9 produce solute plumes that exhibit power law leading edge concentration profiles and super fickian plume growth rates for these network types a multiscaling fractional advection dispersion equation mfade provides a model of multidimensional solute transport where different rates of power law particle motion are defined along multiple directions the mfade model is parameterized by a scaling matrix to describe the super fickian growth process in which the eigenvectors correspond to primary fracture group orientations and the eigenvalues code fracture length and transmissivity the approximation of particle clouds by a multi gaussian a subset of the operator stable for densely fractured networks with finite variance fracture length distributions can be ascribed to the classical ade where fickian scaling rates pertain along orthogonal plume growth directions fracture networks show long term particle retention in low velocity fractures so that coupling of the equations of motion with retention models such as continuous time random walk or multirate mobile immobile will increase accuracy near the source particle arrival times at exit boundaries for multi gaussian plumes vary with spatial density generally arrival times are faster in sparsely fractured domains where transport is governed by a few very long fractures

a second order accurate numerical method for the two dimensional fractional diffusion equation spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow finance and others in this paper we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation this numerical method combines the alternating directions implicit adi approach with a crank nicolson discretization and a richardson extrapolation to obtain an unconditionally stable second order accurate finite difference method the stability and the consistency of the method are established numerical solutions for an example super diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method

spectral element methods for elliptic problems in nonsmooth domains engineering applications frequently require the numerical solution of elliptic boundary value problems in irregularly shaped domains for smooth problems spectral element methods have proved very successful since they can accommodate fairly complicated geometries while retaining a rapid rate of convergence geometric singularities however often give rise to singular solutions the accuracy of the spectral element methods is then degraded and they offer no apparent advantage over low order finite element methods in many cases however the singular structure of the solution is known and its form may be exploited by the spectral element method among the various ways of doing so through supplementary basis functions eigenfunction expansions and graded meshes the method of auxiliary mapping proves to be particularly effective for certain simple cases the problem is transformed to a coordinate system in which the solution is analytic and exponential convergence is recovered even when this is not possible the singularity is usually much weaker after mapping so that other treatments are more effective in the new coordinate system in this paper we study different ways of treating singularities and in particular the method of auxiliary mapping coupled with the use of supplementary basis functions error estimates are presented explaining why the combined approach is more effective and these estimates are confirmed through a number of numerical experiments for the laplace poisson and helmholtz equations

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

fractional vector calculus for fractional advection dispersion abstract we develop the basic tools of fractional vector calculus including a fractional derivative version of the gradient divergence and curl and a fractional divergence theorem and stokes theorem these basic tools are then applied to provide a physical explanation for the fractional advection dispersion equation for flow in heterogeneous porous media

the h p and h p versions of the finite element methods in 1 dimension part iii the adaptive h p version abstract the paper is the third and final part in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimension the first part analyzed the p version and the second part concentrated on the h and h p version this paper analyzes a theoretical frame of the adaptive h p version and based on it the authors provide concrete algorithm for the one dimensional problem it is proven that in the case that the solution has x sub alpha type singularity the adaptive algorithm give an exponential rate of convergence very close to the optimal one analyzed in the second part of the paper additional keyword error analysis

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

the post processing approach in the finite element method part 1 calculation of displacements stresses and other higher derivatives of the displacements this is the first in a series of three papers in which we discuss a method for post processing a finite element solution to obtain high accuracy approximations for displacements stresses stress intensity factors etc rather than take the values of these quantities directly from the finite element solution we evaluate certain weighted averages of the solution over the entire region these yield approximations are of the same order of accuracy as the strain energy we obtain error estimates and also present some numerical examples to illustrate the practical effectiveness of the technique in the third paper of this series we address the matters of adaptive mesh selection and a posteriori error estimation

the application of symbolic computing to the hierarchical finite element method one difficulty associated with the hierarchical finite element method is that of accurately determining the integrals necessary to establish the stiffness and mass matrices of arbitrarily high order elements this paper describes how such integrals can be calculated exactly using symbolic computing results are presented which permit the construction of high order hierarchical finite elements applicable to a variety of problems involving two dimensional elasticity and plate and shell bending three dimensional solid or brick elements are not considered

improvements in 3 node triangular shell elements low order triangular finite shell elements are computationally economical and easy to implement but often exhibit very slow convergence two new membrane formulations for triangular shell elements are examined which rectify these drawbacks the first element is based on the marguerre shallow shell theory and a strain projection method that eliminates spurious membrane strain energy resulting expressions are provided in an explicit form for easy implementation of the element the second element is based on a linear membrane field governed by normal rotations and reduced quadrature the difficulties with shell normal rotations are analysed and a method for omitting these rotations while preserving rigid body motion is presented and tested finally a set of test problems are examined which show the importance of mesh patterns and degrees of freedom per node on triangular element performance

multiscaling fractional advection dispersion equations and their solutions 1 the multiscaling fractional advection dispersion equation ade is a multidimensional model of solute transport that encompasses linear advection fickian dispersion and super fickian dispersion the super fickian term in these equations has a fractional derivative of matrix order that describes unique plume scaling rates in different directions the directions need not be orthogonal so the model can be applied to irregular noncontinuum fracture networks the statistical model underlying multiscaling fractional dispersion is a continuous time random walk ctrw in which particles have arbitrary jump length distributions and finite mean waiting time distributions the meaning of the parameters in a compound poisson process a subset of ctrws is used to develop a physical interpretation of the equation variables the green s function solutions are the densities of operator stable probability distributions the limit distributions of normalized sums of independent and identically distributed random vectors these densities can be skewed heavy tailed and scale nonlinearly resembling solute plumes in granular aquifers they can also have fingers in any direction resembling transport along discrete pathways such as fractures

preconditioned iterative methods for space time fractional advection diffusion equations in this paper we propose practical numerical methods for solving a class of initial boundary value problems of space time fractional advection diffusion equations first we propose an implicit method based on two sided grunwald formulae and discuss its stability and consistency then we develop the preconditioned generalized minimal residual preconditioned gmres method and preconditioned conjugate gradient normal residual preconditioned cgnr method with easily constructed preconditioners importantly because resulting systems are toeplitz like fast fourier transform can be applied to significantly reduce the computational cost we perform numerical experiments to demonstrate the efficiency of our preconditioners even in cases with variable coefficients

toward a universal h p adaptive finite element strategy part 2 a posteriori error estimation abstract in this second paper of the series on h p adaptive finite element methods we present five different techniques for a posteriori error estimation of h p methods for linear elliptic boundary value problems the techniques are referred to as the residual estimation method the duality method the subdomain residual method a method based on interpolation theory and a post processing method the present study examines the theory and behavior of such techniques applied specifically to h p adaptive finite element methods several numerical examples illustrate the efficiency of the methods

a box type scheme for fractional sub diffusion equation with neumann boundary conditions combining order reduction approach and l1 discretization a box type scheme is presented for solving a class of fractional sub diffusion equation with neumann boundary conditions a new inner product and corresponding norm with a sobolev embedding inequality are introduced a novel technique is applied in the proof of both stability and convergence the global convergence order in maximum norm is o t 2 a h 2 the accuracy and efficiency of the scheme are checked by two numerical tests

the approximation of solutions of elliptic boundary value problems via the p version of the finite element method the approximation theory developed in 11 is used to determine the piecewise polynomial approximability of solutions of elliptic problems on polygonal domains in mathbb r 2 and polyhedra in mathbb r 3 from these estimates convergence orders for the p version of the finite element method applied to such problems are readily obtained the critical issue is the approximation of the singularities which occur at the nonsmooth parts of the domain boundaries it is seen that the estimates of 11 involving the weighted sobolev spaces z l s are well suited for treating such singular functions yielding directly the optimal approximation degree up to an arbitrarily small varepsilon numerical results for two problems from two dimensional linear elasticity are also presented the computations show that the predicted order of convergence is achieved even for low values of p moreover in contrast to the usual h version of the finite element method the point at which the p version enters the asym

computational error estimates and adaptive processes for some nonlinear structural problems abstract in earlier papers the authors introduced and analyzed the calculation of reliable a posteriori error estimates for finite element solutions and discussed the design and effectivity of adaptive procedures based upon them while most of this work concerned linear problems this paper is intended to show that the same approaches remain also highly effective in the nonlinear case as a model problem a planar elastic rod is considered involving both material as well as geometrical nonlinearities but as far as possible the discussion is kept general for nonlinear problems of this type interest centers on an analysis of the shape and features of the equilibrium surface after some general remarks about such surfaces and the related computational problems a general continuation process is summarized and some of its extensions for determining limit points and tracing stability paths are discussed then estimators are introduced for the error along the solution paths and of the computed critical values experimental results for the model problem show the effectivity of these estimators and of the adaptive procedures based upon them then some aspects relating to the occurrence and identification of spurious solutions are discussed in order to illustrate desirable generalizations of these results the paper ends with an outline of some results for linear problems of the type that should be achievable also in the nonlinear case

stability of the positive mass theorem for rotationally symmetric riemannian manifolds we study the stability of the positive mass theorem using the intrinsic flat distance in particular we consider the class of complete asymptotically flat rotationally symmetric riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal hypersurfaces or are empty we prove that a sequence of these manifolds whose adm masses converge to zero must converge to euclidean space in the pointed intrinsic flat sense in fact we provide explicit bounds on the intrinsic flat distance between annular regions in the manifold and annular regions in euclidean space by constructing an explicit filling manifold and estimating its volume in addition we include a variety of propositions that can be used to estimate the intrinsic flat distance between riemannian manifolds without rotationally symmetry conjectures regarding the intrinsic flat stability of the positive mass theorem in the general case are proposed in the final section lee is partially supported by a psc cuny research grant and nsf dms 0903467 sormani is partially supported by a psc cuny research grant and nsf dms 1006059 1 ar x iv 1 10 4 26 57 v1 m at h d g 1 4 a pr 2 01 1 2 dan a lee and christina sormani

on the rates of convergence of the finite element method abstract the rate of convergence of the finite element method is a function of the strategy by which the number of degrees of freedom are increased alternative strategies are examined in the light of recent theoretical results and computational experience author

boundary conditions for two sided fractional diffusion abstract this paper develops appropriate boundary conditions for the two sided fractional diffusion equation where the usual second derivative in space is replaced by a weighted average of positive left and negative right fractional derivatives mass preserving reflecting boundary conditions for two sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary stable consistent explicit and implicit euler methods are detailed and steady state solutions are derived steady state solutions for two sided fractional diffusion equations using both riemann liouville and caputo flux are computed for riemann liouville flux and reflecting boundary conditions the steady state solution is singular at one or both of the end points for caputo flux and reflecting boundary conditions the steady state solution is a constant function numerical experiments illustrate the convergence of these numerical methods finally the influence of the reflecting boundary on the steady state behavior subject to both the riemann liouville and caputo fluxes is discussed

fractional diffusion with two time scales abstract moving particles that rest along their trajectory lead to time fractional diffusion equations for the scaling limit distributions for power law waiting times with infinite mean the equation contains a fractional time derivative of order between 0 and 1 for finite mean waiting times the most revealing approach is to employ two time scales one for the mean and another for deviations from the mean for finite mean power law waiting times the resulting equation contains a first derivative as well as a derivative of order between 1 and 2 finite variance waiting times lead to a second order partial differential equation in time in this article we investigate the various solutions with regard to moment growth and scaling properties and show that even infinite mean waiting times do not necessarily induce subdiffusion but can lead to super diffusion if the jump distribution has non zero mean

modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit lagrangian scheme this study develops an explicit two step lagrangian scheme based on the renewal reward process to capture transient anomalous diffusion with mixed retention and early arrivals in multidimensional media the resulting 3 d anomalous transport simulator provides a flexible platform for modeling transport the first step explicitly models retention due to mass exchange between one mobile zone and any number of parallel immobile zones the mobile component of the renewal process can be calculated as either an exponential random variable or a preassigned time step and the subsequent random immobile time follows a hyper exponential distribution for finite immobile zones or a tempered stable distribution for infinite immobile zones with an exponentially tempered power law memory function the second step describes well documented early arrivals which can follow streamlines due to mechanical dispersion using the method of subordination to regional flow applicability and implementation of the lagrangian solver are further checked against transport observed in various media results show that although the time nonlocal model parameters are predictable for transport with retention in alluvial settings the standard time nonlocal model cannot capture early arrivals retention and early arrivals observed in porous and fractured media can be efficiently modeled by our lagrangian solver allowing anomalous transport to be incorporated into 2 d 3 d models with irregular flow fields extensions of the particle tracking approach are also discussed for transport with parameters conditioned on local aquifer properties as required by transient flow and nonstationary media

aquifer operator scaling and the effect on solute mixing and dispersion 1 since aquifer parameters may have statistical dependence structures that are present across a huge range of scales the concepts of fractional brownian motion fbm have been used in both analytic and numerical stochastic settings most previous models have used isotropic scaling characterized by a single scalar hurst coefficient any real world anisotropy has been handled by an elliptical stretching random k field we define a d dimensional extension of fbm in which the fractional order integration may take on different orders in the d primary possibly nonorthogonal scaling directions and the degree of connectivity and long range dependence is freely assigned via a probability measure on the unit sphere this approach accounts for the different scaling found in the vertical versus horizontal directions in sedimentary aquifers and allows very general degrees of continuity of k in certain directions it also allows for the representation of fracture networks in a continuum setting the eigenvectors of the scaling matrix describe the primary fracture scaling directions and discrete weights of fractional integration represent fracture continuity that may be limited to a small number of directions in a numerical experiment the motion of solutes through 2 d operator fractional gaussian fields depends very much on transverse hurst coefficients transverse scaling in the range of fractional gaussian noise engenders greater plume mixing and a transition to a fickian regime higher orders of integration in the range of fractional brownian motion are associated with thicker layering of aquifer sediments and more preferential unmixed transport therefore direct representation of the unique directional scaling properties of an aquifer is important for realistic simulation of transport

an o n log2n alternating direction finite difference method for two dimensional fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices consequently the numerical solution of these methods often require computational work of o n 3 per time step and memory of o n 2 for where n is the number of grid points in this paper we develop a fast alternating direction implicit finite difference method for space fractional diffusion equations in two space dimensions the method only requires computational work of o n log 2n per time step and memory of o n while retaining the same accuracy and approximation property as the regular finite difference method with gaussian elimination our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations to achieve the same accuracy the new method has a significant reduction of the cpu time from more than 2 months and 1 week consumed by a traditional finite difference method to 1 5h using less than one thousandth of memory the standard method does this demonstrates the utility of the method

curved shell elements based on hierarchical p approximation in the thickness direction for linear static analysis of laminated composites this paper presents a new curved shell finite element formulation for linear static analysis of laminated composite plates and shells where the displacement approximation in the direction of the shell thickness can be of an arbitrary polynomial order p thereby permitting strains of at least p 1 order this is accomplished by introducing additional nodal variables in the element displacement approximation corresponding to the lagrange interpolating polynomials in the element thickness direction the resulting element displacement approximation has an important hierarchical property i e the approximation functions and the generalized nodal variables corresponding to an approximation order p are a subset of those corresponding to an approximation order p 1 the element formulation ensures c0 continuity or smoothness of displacements across the interelement boundaries the element properties stiffness matrix and equivalent load vectors are derived using the principle of virtual work and the hierarchical element approximation the formulation is extended for generally orthotropic material behaviour where the material directions are not necessarily parallel to the global axes further extension of this formulation for laminated composites is accomplished by incorporating the material properties of each lamina by numerically integrating the element stiffness matrix for each lamina the formulation has no restriction on either the number of laminas or the layup pattern of the laminas each lamina can be generally orthotropic and the material directions and the lamina thicknesses may vary from point to point within each lamina the geometry of the laminated shell element is described by the co ordinates of the nodes lying on the middle surface of the element and the lamina thicknesses at each node the formulation permits any desired order displacement or strain approximation in the shell thickness direction without remodelling numerical examples are presented to demonstrate the accuracy efficiency and overall superiority of the present formulation the results obtained from the present formulation are compared with those available in the literature as well as the analytical solutions

operator levy motion and multiscaling anomalous diffusion the long term limit motions of individual heavy tailed power law particle jumps that characterize anomalous diffusion may have different scaling rates in different directions operator stable motions y t tg 0 are limits of d dimensional random jumps that are scale invariant according to c h y t y mathrm ct where h is a d ifmmode times else texttimes fi d matrix the eigenvalues of the matrix have real parts 1 ensuremath alpha j with each positive ensuremath alpha j l 2 in each of the j principle directions the random motion has a different fickian or super fickian diffusion dispersion rate proportional to t 1 ensuremath alpha j these motions have a governing equation with a spatial dispersion operator that is a mixture of fractional derivatives of different order in different directions subsets of the generalized fractional operator include i a fractional laplacian with a single order ensuremath alpha and a general directional mixing measure m ensuremath theta and ii a fractional laplacian with uniform mixing measure the riesz potential the motivation for the generalized dispersion is the observation that tracers in natural aquifers scale at different super fickian rates in the directions parallel and perpendicular to mean flow

solution of elasticity problems by a self adaptive mesh refinement technique for boundary element computation a self adaptive mesh refinement technique is developed for boundary element solutions of the two dimensional navier s equation the method is based on error reduction and applied on the element and global level to estimate the residual error associated with each mesh results for several problems employing constant two dimensional isotropic elastostatic boundary elements are presented

an isoparametric spectral element method for solution of the navier stokes equations in complex geometry abstract an isoparametric spectral element technique is presented for solution of the two dimensional navier stokes equations in arbitrary curvy geometries the spatial discretization is described and the method is illustrated by solution of poisson s equation in a cylindrical annulus a fractional step navier stokes solver based on a discretely consistent poisson equation for the pressure is then presented and exponential convergence in space is demonstrated for the case of separated flow between eccentric rotating cylinders the technique is made efficient by the use of explicit collocation rather than galerkin operators for the convective terms and use of static condensation to solve the elliptic equations resulting from the stokes problem the algorithm is amenable to a high degree of parallelism in both multiprocessor and vector pipeline environments

a new triangular and tetrahedral basis for high order hp finite element methods in this paper we describe the foundations of a new hierarchical modal basis suitable for high order hp finite element discretizations on unstructured meshes it is based on a generalized tensor product of mixed weight jacobi polynomials the generalized tensor product property leads to a low operation count with the use of sum factorization techniques variable p order expansions in each element are readily implemented which is a crucial property for efficient adaptive discretizations numerical examples demonstrate the exponential convergence for smooth solutions and the ability of this formulation to handle easily very complex two and three dmensional computational domains employing standard meshes

adaptive hp finite element for transient analysis abstract even though a tremendous effort has been devoted to the area of transient analysis many of these models fall short of achieving the overall desired objective due to oversights and rigid simplifying assumptions some of those shortfalls are summarized as follows 1 i the grid used is too fine or too coarse and is usually fixed in size 2 ii no error estimates are being used to trigger a grid size change 3 iii a fixed polynomial degree is being used to interpolate the approximate solution of the differential equation at hand finite element approach 4 iv an approximate differential equation is used to model the physical phenomena rather than the exact one finite difference approach in our study a self adaptive grid refinement technique coupled with the p version of the finite element method is investigated this type of approach is called the hp version of the finite element in the modeling process the approximate solution to the exact differential equation achieves convergence by applying two distinct solution enrichment strategies one strategy is to solve the problem using a very coarse grid and to enrich the quality of the approximation by increasing the degree p of the interpolating polynomial shape functions for those elements where it is needed the other strategy involves local grid h refinements for those elements where it is needed both strategies would interact synergistically to produce an optimal computational model that produces an accurate simulation of the transient phenomena addressed with minimal computational effort the triggering for the increase or decrease in polynomial degree p and placement or removal of local mesh h refinements is based on the calculation of a reliable a posteriori error estimate over each element at the end of each time step acase study of particular interest is the simulation of wave propagation in a semi infinite media

hydraulic conductivity velocity and the order of the fractional dispersion derivative in a highly heterogeneous system 1 a one dimensional fractional order advection dispersion equation accurately models the movement of the core of the tritium plume at the highly heterogeneous made site an a priori estimate of the parameters in that equation including the order of the fractional dispersion derivative was based on the assumption that the observed power law heavy tail of the hydraulic conductivity k field would create a similarly distributed velocity field monte carlo simulations were performed to test this hypothesis results from the monte carlo analysis show that heavy tailed k fields do give rise to heavy tailed velocity fields however the exponent of the power law the tail parameter describing these two distributions is not necessarily the same the tail parameter that characterizes a velocity distribution is not solely dependent on the tail parameter that characterizes the k distribution the k field must also have long range dependence so that water may flow through relatively continuous high k channels

an extended equivalent domain integral method for mixed mode fracture problems by thep version of fem a new path independent contour integral formula is presented to estimate the crack tip integral parameter j value for two dimensional cracked elastic bodies which may quantify the severity of the crack tip stress fields the conventional j integral method based on a line integral has been converted to an equivalent area or domain integral edi by the divergence theorem it is noted that the edi method is very attractive because all the quantities necessary for computation of domain integrals are readily available in a finite element analysis the details and its implementation are extended to the p version fe model with hierarchic elements using integrals of legendre polynomials by decomposing the displacement field obtained from the p version finite element analysis into symmetric and antisymmetric displacement fields with respect to the crack line the mode i and mode ii non dimensional stress intensity factors can be determined by using the decomposition method the example problems for validating the proposed techniques are centrally oblique cracked plates under tensile loading the numerical results associated with the variation of oblique angles show very good agreement with the existing solutions also the selective distribution of polynomial orders and the corner elements for automatic mesh generation are applied to improve the numerical solution in this paper 1998 john wiley sons ltd

predictor corrector p and hp versions of the finite element method for poisson s equation in polygonal domains abstract we consider boundary value problems for the poisson equation on polygonal domains with general nonhomogeneous mixed boundary conditions and derive on the one hand explicit extraction formulas for the coefficients of the singularities on the other hand the formulas are used to construct efficient adaptations for the h p and h p versions of the finite element method for the numerical treatment a priori error estimates show that the h version of the finite element algorithm exhibits the same rate of convergence as it is known for problems with smooth solutions however the principal results of the present work are the robust exponential convergence results for the p and h p versions of the finite element method on quasiuniform meshes in fact it is shown that if the input data source term and boundary data are piecewise analytic then with appropriate choices of conforming finite element subspaces v n of dimension n n the p and h p versions of the finite element algorithms on quasiuniform meshes yield approximate solutions u n to the exact solution u that satisfy the estimates u u n h 1 c 1 e b 1 n 2 3 and u u n h 1 c 2 e b 2 n 1 3 respectively several numerical experiments are included to illustrate the practical effectiveness of the proposed algorithms the results show that the theoretical error analyses are attained within the range of engineering accuracy

on upscaling operator stable levy motions in fractal porous media the dynamics of motile particles such as microbes in random porous media are modeled with a hierarchical set of stochastic differential equations which correspond to micro meso and macro scales on the microscale the motile particle is modeled as an operator stable levy process with stationary ergodic markov drift the micro to meso and meso to macro scale homogenization is handled with generalized central limit theorems on the mesoscale the lagrangian drift or the lagrangian acceleration is assumed levy to account for the fractal character of many natural porous systems diffusion on the mesoscale is a result of the microscale asymptotics while diffusion on the macroscale results from the mesoscale asymptotics renormalized fokker planck equations with time dependent dispersion tensors and fractional derivatives are presented at the macro scale

analysis of singular cylindrical shells by p version of fem abstract a new hierarchic p version cylindrical shell element based on exact mapping is presented its rigid body modes round off error and convergence characteristics are investigated its performance in the presence of singularities is demonstrated with the help of three test problems namely pinched cylinder problem lockheed test problem 2 and cracked cylinder problem

on the use of lagrange multipliers in domain decomposition for solving elliptic problems the primal hybrid method for solving second order elliptic equa tions is extended from finite element approximations to general bases varia tional techniques are used to show convergence of approximations to the so lution of the homogeneous dirichlet problem for selfadjoint equations error estimates are obtained and examples are given

relating notions of convergence in geometric analysis abstract we relate l p convergence of metric tensors or volume convergence to a given smooth metric to intrinsic flat and gromov hausdorff convergence for sequences of riemannian manifolds we present many examples of sequences of conformal metrics which demonstrate that these notions of convergence do not agree in general even when the sequence is conformal g j f j 2 g 0 to a fixed manifold we then prove a theorem demonstrating that when sequences of metric tensors on a fixed manifold m are bounded 1 1 j g 0 g j k g 0 and either the volumes converge vol j m vol 0 m or the metric tensors converge in the l p sense then the riemannian manifolds m g j converge in the measured gromov hausdorff and volume preserving intrinsic flat sense to m g 0

on using the cell discretization algorithm for mixed boundary value problems and domain decomposition the cell discretization algorithm is used to approximate solutions to self adjoint elliptic equations with general nonhomogeneous dirichlet or neumann or mixed boundary values error estimates are obtained showing general convergence this provides the framework for a nonoverlapping iterative algorithm for domain decomposition the domain of a dirichlet problem is partitioned into two subdomains solutions on the two subdomains can be patched together to form a solution to the original problem provided the solutions agree across the common interface of the subdomains and the normal derivatives there have equal absolute values and are opposite in sign we generate such a solution by alternating between imposing neumann and dirichlet conditions on the interface with boundary data adapted from the results of the previous approximation a posteriori error estimates show that we have global convergence provided computed interface errors are sufficiently small

the p and hp finite element method for problems on thin domains abstract the p and hp versions of the finite element method allow the user to change the polynomial degree to increase accuracy we survey these methods and show how this flexibility can be exploited to counter four difficulties that occur in the approximation of problems over thin domains such as plates beams and shells these difficulties are 1 control of modeling error 2 approximation of corner singularities 3 resolution of boundary layers and 4 control of locking our guidelines enable the efficient resolution of these difficulties when a p hp code is available

the h p and h p versions of the finite element method in 1 dimension this paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting the main emphasis is placed on the analysis when the exact solution has singularity of x type the first part analyzes thep version the second theh version and generalh p version and the final third part addresses the problems of the adaptiveh p version

advection and dispersion in time and space abstract previous work showed how moving particles that rest along their trajectory lead to time nonlocal advection dispersion equations if the waiting times have infinite mean the model equation contains a fractional time derivative of order between 0 and 1 in this article we develop a new advection dispersion equation with an additional fractional time derivative of order between 1 and 2 solutions to the equation are obtained by subordination the form of the time derivative is related to the probability distribution of particle waiting times and the subordinator is given as the first passage time density of the waiting time process which is computed explicitly

fractal mobile immobile solute transport 1 a fractal mobile immobile model for solute transport assumes power law waiting times in the immobile zone leading to a fractional time derivative in the model equations the equations are equivalent to previous models of mobile immobile transport with power law memory functions and are the limiting equations that govern continuous time random walks with heavy tailed random waiting times the solution is gained by performing an integral transform on the solution of any boundary value problem for transport in the absence of an immobile phase in this regard the output from a multidimensional numerical model can be transformed to include the effect of a fractal immobile phase the solutions capture the anomalous behavior of tracer plumes in heterogeneous aquifers including power law breakthrough curves at late time and power law decline in the measured mobile mass the made site mobile tritium mass decline is consistent with a fractional time derivative of order 0 33 while haggerty et al s 2002 stream tracer test is well modeled by a fractional time derivative of order 0 3

space fractional advection dispersion equations with variable parameters diverse formulas numerical solutions and application to the macrodispersion experiment site data 1 to model the observed local variation of transport speed an extension of the homogeneous space fractional advection dispersion equation fade to more general cases with space dependent coefficients drift velocity v and dispersion coefficient d has been suggested to provide a rigorous evaluation of this extension we explore the underlying physical meanings of two proposed and one other possible form of the fade by using the generalized mass balance law proposed by meerschaert et al 2006 when the classical fick s law is replaced by its generalized form in the first order mass conversation law the original fade with constant parameters extends to the advection dispersion equation ade with fractional flux denoted as ff ade when the net inflow of dispersive flux is from nonlocal concentration gradients following a fractional divergence we get the ade with fractional divergence fd ade when the total net inflow from both nonlocal advection and nonlocal concentration gradients follows a fractional form the fade contains fully fractional divergence ffd ade these three fades with constant parameters can also be obtained by proper choice of the two memory kernels in the nonlocal dispersive constitutive theory proposed by cushman et al 1994 while the space variable fades correspond to the conditional nonlocal theory proposed by neuman 1993 after specifying the general levy type functional form of the random distribution the corresponding langevin markov models can be found in many cases where the lagrangian stochastic processes can be conditioned directly on local aquifer properties at any practical measurable level and resolution the resulting lagrangian random walk particle tracking methods along with previous numerical solutions using implicit euler finite differences distinguish and elucidate the plume behavior described by these fades the fractional models are applied to fit the tritium plumes measured at the macrodispersion experiment test site when the local parameters gleaned from the hydraulic conductivity k distribution are varied even slightly i e a two zone model allowing the use of observed hydraulic conditions at the site the model fits are well within the variability of the data the extended fades describe the fast and space dependent leading edges of measured plumes in the regional scale alluvial system which was underestimated and could not be fully captured by the original fade with constant parameters applications also favor the fd ade model because of the ease of implementation and consistency with previous analysis of the k statistics

an hp galerkin method with fast solution for linear peridynamic models in one dimension the computational work and memory requirement are bottlenecks for galerkin finite element methods for peridynamic models because of their non locality in this paper fast galerkin and hp galerkin finite element methods are introduced and analyzed to solve a steady state peridynamic model we present a fast solution technique to accelerate non square toeplitz matrixvector multiplications arising from piecewise linear piecewise quadratic and piecewise cubic galerkin methods this fast solution technique is based on a fast fourier transform and depends on the special structure of coefficient matrices and it helps to reduce the computational work from o n3 required by traditional methods to o nlog2n and the memory requirement from o n2 to o n without using any lossy compression where n is the number of unknowns the peridynamic model admits solutions having jump discontinuities for problems with discontinuous solutions we therefore introduce a piecewise constant galerkin method and give an h and p refinement algorithm then we develop fast hp galerkin methods based on hybrid piecewise constant piecewise linear and piecewise constant piecewise quadratic finite element approximations the new method reduces the computational work from o n3 required by the traditional methods to o nlog2n and the memory requirement from o n2 to o n

on the inclusion principle for the hierarchical finite element method the inclusion principle provides a qualitative characterization of the eigenvalues of a matrix the principle has been shown to apply to systems described by a single hermitian matrix the most important of which being the real symmetric matrix self adjoint distributed systems when discretized by either the classical rayleigh ritz method or by the finite element method lead to algebraic eigenvalue problems described in terms of two real symmetric matrices the algebraic eigenvalues problem derived by the classical rayleigh ritz method possesses the embedding feature required by the inclusion principle but that derived by the finite element method in general does not this paper demonstrates that the inclusion principle can be extended to discretized systems derived by the hierarchical finite element method

a posteriori error estimation with finite element methods of lines for one dimensional parabolic systems consider the solution of one dimensional linear initial boundary value problems by a finite element method of lines using a piecewisep th degree polynomial basis a posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degreep 1 that vanish at element ends error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are furthermore applicable in situations that are not supported by the analysis

a stable explicitly solvable numerical method for the riesz fractional advection dispersion equations abstract in this paper we present a finite difference scheme for solving the riesz fractional advection dispersion equations rfades the scheme is obtained by using asymmetric discretization technique and modify the shifted grunwald approximation to fractional derivative by calculating the unknowns in differential nodal point sequences at the odd and even time levels the discrete solution of the scheme can be obtained explicitly the computational cost for the scheme at each time step can be o klog k by using the fast matrix vector multiplication with the help of toeplitz structure where k is the number of unknowns we prove that the scheme is solvable and unconditionally stable we derive the error estimates in discrete l2 norm which is optimal in some cases numerical examples are presented to verify our theoretical results

effective and practical h p version adaptive analysis procedures for the finite element method two practical and effective h p type finite element adaptive procedures are presented the procedures allow not only the final global energy norm error to be well estimated using hierarchic p refinement but in addition give a nearly optimal mesh the design of this is guided by the local information computed on the previous mesh the desired accuracy can always be obtained within one or at most two h p refinements the rate of convergence of the adaptive h p version analysis procedures has been tested for some examples and found to be very strong the presented procedures can easily be incorporated into existing p or h type code structures

on using random walks to solve the space fractional advection dispersion equations the solution of space fractional advection dispersion equations fade by random walks depends on the analogy between the fade and the forward equation for the associated markov process the forward equation which provides a lagrangian description of particles moving under specific markov processes is derived here by the adjoint method the fade however provides an eulerian description of solute fluxes there are two forms of the fade based on fractional flux ff ade and fractional divergence fd ade the ff ade is derived by taking the integer order mass conservation of non local diffusive flux while the fd ade is derived by taking the fractional order mass conservation of local diffusive flux the analogy between the fade and the forward equation depends on which form of the fade is used and on the spatial variability of the dispersion coefficient d in the fade if d does not vary in space then the fades can be solved by tracking particles following a markov process with a simple drift and an stable levy noise with index that corresponds to the fractional order of the fade if d varies smoothly in space and the solute concentration at the upstream boundary remains zero the fd ade can be solved by simulating a markov process with a simple drift an stable levy noise and an additional term with the dispersion gradient and an additional levy noise of order 1 however a non markov process might be needed to solve the ff ade with a space dependent d except for specific d such as a linear function of space

a comparative study of finite element and spectral element methods in seismic wavefield modeling the computational accuracy and efficiency of finite element method and spectral element method sem are investigated thoroughly in time domain elastic wavefield modeling the diagonal mass matrices of the fem and sem free from matrix inversion are compared comprehensively by making full use of the mass lumped technique and quadrature rules we investigate the fem and sem based respectively on quadrilateral with the polynomials of degrees one and two and on triangular grids with the polynomials of degrees one and three generally the numerical solutions based on quadrilateral grids have a higher precision than those computed on triangular grids when the same order of polynomials is used the fem has a comparable accuracy to the sem with the same number of interpolant points in view of the triangular and quadrilateral sems the former suffers from larger computational costs and relatively lower accuracy compared with the latter furthermore the convergence study proves that the triangular sem produces consistently larger errors than the quadrilateral sem for any order and element sizes however the triangular sem can adapt to arbitrary complex geometries effectively in terms of efficiency the fem has an efficiency comparable with the sem on condition that the order of interpolation polynomials is identical in addition a perfectly matched layer pml boundary condition in variational form is deduced by introducing four intermediate variables in frequency domain the pml avoids convolution calculation and obtains an exact solution through inverse fourier transform in time domain the numerical examples verify the validity and effectiveness of the pml

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

fractional dynamics of systems with long range space interaction and temporal memory abstract field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power law memory function and long range interaction in systems the method is applied to obtain a fractional generalization of the ginzburg landau and nonlinear schrodinger equations as another example dynamical equations for particle chains with power law interaction and memory are considered in the continuous limit the obtained fractional equations can be applied to complex media with without random parameters or processes

on the stability and convergence of higher order mixed finite element methods for second order elliptic problems we investigate the use of higher order mixed methods for second order elliptic problems by establishing refined stability and convergence estimates which take into account both the mesh size h and polynomial degree p our estimates yield asymptotic convergence rates for the p and h p versions of the finite element method they also describe more accurately than previously proved estimates the increased rate of convergence expected when the h version is used with higher order polynomials

sobolev bounds and convergence of riemannian manifolds abstract we consider sequences of compact riemannian manifolds with uniform sobolev bounds on their metric tensors and prove that their distance functions are uniformly bounded in the holder sense this is done by establishing a general trace inequality on riemannian manifolds which is an interesting result on its own we provide examples demonstrating how each of our hypotheses are necessary in the appendix by the first author with christina sormani we prove that sequences of compact integral current spaces without boundary including riemannian manifolds that have uniform holder bounds on their distance functions have subsequences converging in the gromov hausdorff gh sense if in addition they have a uniform upper bound on mass volume then they converge in the sormani wenger intrinsic flat swif sense to a limit whose metric completion is the gh limit we provide an example of a sequence developing a cusp demonstrating why the swif and gh limits may not agree

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material

mixing driven equilibrium reactions in multidimensional fractional advection dispersion systems we study instantaneous mixing driven bimolecular equilibrium reactions in a system where transport is governed by a multidimensional space fractional dispersion equation the superdiffusive nonlocal nature of the system causes the location and magnitude of reactions that take place to change significantly from a classical fickian diffusion model in particular regions where reaction rates would be zero for the fickian case become regions where the maximum reaction rate occurs when anomalous dispersion operates we also study a global metric of mixing in the system the scalar dissipation rate and compute its asymptotic scaling rates analytically the scalar dissipation rate scales asymptotically as t d where d is the number of spatial dimensions and is the fractional derivative exponent

proof of the riemannian penrose inequality using the positive mass theorem we prove the riemannian penrose conjecture an important case of a con jecture 41 made by roger penrose in 1973 by defining a new flow of metrics this flow of metrics stays inside the class of asymptotically flat riemannian 3 manifolds with nonnegative scalar curvature which contain minimal spheres in particular if we consider a riemannian 3 manifold as a totally geodesic submanifold of a space time in the context of general relativity then outermost minimal spheres with total area a correspond to apparent horizons of black holes contributing a massa 16 scalar cur vature corresponds to local energy density at each point and the rate at which the metric becomes flat at infinity corresponds to total mass also called the adm mass the riemannian penrose conjecture then states that the total mass of an asymptotically flat 3 manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes the flow of metrics we define continuously evolves the original 3 metric to a schwarzschild 3 metric which represents a spherically symmetric black hole in vacuum we define the flow such that the area of the minimal spheres which flow outward and hence the mass contributed by the black holes in each of the metrics in the flow is constant and then use the positive mass theorem to show that the total mass of the metrics is nonincreasing then since the total mass equals the mass of the black hole in a schwarzschild metric the riemannian penrose conjecture follows we also refer the reader to the beautiful work of huisken and ilmanen 30 who used inverse mean curvature flows of surfaces to prove that the total mass is at least the mass contributed by the largest black hole in sections 1 and 2 we motivate the problem discuss important quantities like total mass and horizons of black holes and state the positive mass theorem and the penrose conjecture for riemannian 3 manifolds in section 3 we give the proof of the riemannian penrose

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis