stability of large amplitude ekman hartmann boundary layers in mhd the case of ill prepared data in this paper we study an incompressible highly rotating fluid submitted to a high magnetic field between two planes with a dirichlet boundary condition we investigate the nonlinear stability of ekman hartmann boundary layers under a spectral assumption for general initial data this means that the data can be chosen as an arbitrary but smooth enough three dimensional divergence free vector field independent of the small parameter

efficient inversion of the galerkin matrix of general second order elliptic operators with nonsmooth coefficients this article deals with the efficient approximate inversion of finite element stiffness matrices of general second order elliptic operators with l coefficients it will be shown that the inverse stiffness matrix can be approximated by hierarchical matrices h matrices furthermore numerical results will demonstrate that it is possible to compute an approximate inverse with almost linear complexity

accurate multiscale finite element methods for two phase flow simulations in this paper we propose a modified multiscale finite element method for two phase flow simulations in heterogeneous porous media the main idea of the method is to use the global fine scale solution at initial time to determine the boundary conditions of the basis functions this method provides a significant improvement in two phase flow simulations in porous media where the long range effects are important this is typical for some recent benchmark tests such as the spe comparative solution project m christie m blunt tenth spe comparative solution project a comparison of upscaling techniques spe reser eval eng 4 2001 308 317 where porous media have a channelized structure the use of global information allows us to capture the long range effects more accurately compared to the multiscale finite element methods that use only local information to construct the basis functions we present some analysis of the proposed method to illustrate that the method can indeed capture the long range effect in channelized media

computing non stationary s s policies using mixed integer linear programming abstract this paper addresses the single item single stocking location non stationary stochastic lot sizing problem under the s s control policy we first present a mixed integer non linear programming minlp formulation for determining near optimal s s policy parameters to tackle larger instances we then combine the previously introduced minlp model and a binary search approach these models can be reformulated as mixed integer linear programming milp models which can be easily implemented and solved by using off the shelf optimization software computational experiments demonstrate that optimality gaps of these models are less than 0 3 of the optimal policy cost and computational times are reasonable

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

a sigmoid approximation of the standard normal integral abstract this paper presents a simple sigmoid function for approximation of cumulative standard normal probabilities the approximation has an error of 0 000043 for 8 z 8

efficient implementation of the localized orthogonal decomposition method abstract in this paper we present algorithms for an efficient implementation of the localized orthogonal decomposition method lod the lod is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales we show how the method can be implemented in a fairly standard finite element framework and discuss its realization for different types of problems such as linear elliptic problems with rough coefficients and linear eigenvalue problems

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

some remarks on a multigrid preconditioner this paper is devoted to a class of multilevel preconditioners developed in math comp 55 1990 pp 1 22 by bramble pasciak and xu and in the ph d thesis of xu at cornell university ithaca ny 1989 a simple proof is given for optimal estimation of the conditioning a derivation of matrix representations and a description of efficient implementation techniques are also given some modified preconditioners are proposed combining the hierarchical basis method numerical examples are also given that compare various preconditioners discussed in this paper

the method of auxiliary mapping for the finite element solutions of elasticity problems containing singularities babuska and oh have introduced a new approach called the method of auxiliary mapping mam to deal with elliptic boundary value problems with singularities they showed that for the laplace equation with corner singularities in the context of the p version of the finite element method mam yielded an exponential rate of convergence at virtually no extra cost in this paper those results are extended by showing the exponential convergence of mam for homogeneous laplace equations with boundary singularities and helmholtz equations with both corner and boundary singularities in addition a convergence result is developed for mam as applied to the h p version of the finite element method to clarify the power of mam a series of benchmark runs are made for three examples using our implementation mapfem comparisons are made with two h version finite element codes pltmg6 and fesop both of which use adaptive meshes the finite difference code ellpack and the recent singular element code isbfm the examples include the well known motz problem and both homogeneous and nonhomogeneous helmholtz equations

change of sign of the corrector s determinant for homogenization in three dimensional conductivity in this paper we study the positivity of the determinant of the local electric field in a conducting composite we know by 1 that the positivity holds true in two dimensions for any periodic structure using a different approach from 11 we prove that is also the case for a laminate microstructure in any dimension however and this is the main result of the paper we provide an example of a two phase three dimensional periodic composite for which the determinant changes sign

double precision rational approximation algorithms for the standard normal first and second order loss functions abstract we present double precision algorithms based upon piecewise rational approximations for the standard normal first and second order loss functions these functions are used frequently in inventory management no direct approximation or closed formulation exists for the standard first and second order loss functions current state of the art algorithms require intermediate computations of the cumulative normal distribution or tabulations and they do not compute to full double precision we deal with both these issues and present direct double precision accurate algorithms which are valid in the full range of double precision floating point numbers

homogenization of parabolic equations with a continuum of space and time scales this paper addresses the issue of the homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available namely we consider divergence form linear parabolic operators in omega subset mathbb r n with l infty omega times 0 t coefficients it appears that the inverse operator maps the unit ball of l 2 omega times 0 t into a space of functions which at small time and space scales are close in h 1 norm to a functional space of dimension n it follows that once one has solved these equations at least n times it is possible to homogenize them both in space and in time reducing the number of operation counts necessary to obtain further solutions in practice we show under a cordes type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to caloric coordinates are in l 2 instead of h 1 with euclidean coordinates if the medium is time independent then it is sufficient to solve n times the associated elliptic equation in order to homogenize the parabolic equation

convergence of a nonconforming multiscale finite element method the multiscale finite element method msfem t y hou x h wu and z cai math comp 1998 to appear t y hou and x h wu j comput phys 134 1997 pp 169 189 has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients this is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator our previous study reveals that the leading order error in this approach is caused by the resonant sampling which leads to large error when the mesh size is close to the small scale of the continuous problem similar difficulty also arises in numerical upscaling methods an oversampling technique has been introduced to alleviate this difficulty t y hou and x h wu j comput phys 134 1997 pp 169 189 a consequence of the oversampling method is that the resulting finite element method is no longer conforming here we give a detailed analysis of the nonconforming error our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale we show that the cell resonance error is of lower order our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice

localized orthogonal decomposition techniques for boundary value problems in this paper we propose a local orthogonal decomposition method lod for elliptic partial differential equations with inhomogeneous dirichlet and neumann boundary conditions for this purpose we present new boundary correctors which preserve the common convergence rates of the lod even if the boundary condition has a rapidly oscillating fine scale structure we prove a corresponding a priori error estimate and present numerical experiments we also demonstrate numerically that the method is reliable with respect to thin conductivity channels in the diffusion matrix accurate results are obtained without resolving these channels by the coarse grid and without using patches that contain the channels

stability of large ekman boundary layers in rotating fluids the aim of this paper is to investigate the stability of ekman boundary layers for rotating fluids when the ekman number and the rossby number go to zero more precisely we prove that spectral stability implies linear and nonlinear stabilities of approximate solutions in particular we replace the smallness condition obtained with energy methods in 5 by a weaker spectral condition which is sharp

a uniform approximation to the right normal tail integral this paper presents two simple formulas for approximation of the standard normal right tail probabilities and indicates the method of computing approximations of higher order one of the approximations relies on two simple numerical constants has absolute error of 0 00071 and relative error 0 023 the other uses four numerical constants has absolute error of 0 000019 the relative error is 0 005 and gives at least two significant digits for probabilities over the entire range 0

multiscale homogenization with bounded ratios and anomalous slow diffusion we show that the effective diffusivity matrix d v n for the heat operator t 1 2 rv n r in a periodic potential v n d p nd0 uk x rk obtained as a superposition of hlder continuous periodic potentials uk of period t d vd r d z d d 2 n uk 0 d 0 decays exponentially fast with the number of scales when the scale ratios rkc1 rk are bounded above and below from this we de duce the anomalous slow behavior for a brownian motion in a potential obtained as a superposition of an innite number of scales dyt d d t rv 1 yt dt

multiscale techniques for parabolic equations we use the local orthogonal decomposition technique introduced in malqvist and peterseim math comput 83 290 2583 2603 2014 to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale coefficients we consider nonsmooth initial data and a backward euler scheme for the temporal discretization optimal order convergence rate depending only on the contrast but not on the variations of the coefficients is proven in the l infty l 2 norm we present numerical examples which confirm our theoretical findings

ekman layers of rotating fluids the case of general initial data in this paper we describe the weak solutions of the navier stokes equations with a large coriolis term as the rossby and the ekman numbers go to zero in a special domain with various boundary conditions this work extends the results in the case of well prepared initial data in 19 30 2000 john wiley sons inc

edge residuals dominate a posteriori error estimates for low order finite element methods we prove that up to higher order perturbation terms edge residuals yield global upper and local lower bounds on the error of linear finite element methods both in h1 and l2 norms we present two proofs one uses the standard l2 projection and the other relies on a new weighted clement type interpolation operator

boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems abstract in this paper we study viscous perturbations of quasilinear hyperbolic systems in several dimensions as the viscosity goes to zero the boundary is noncharacteristic for the hyperbolic system we in particular describe the boundary layer which arises near the boundary and give a sufficient condition for the convergence of the solution to the solution of some mixed hyperbolic problem with some nonlinear maximal dissipative boundary conditions a counterexample is given when this condition is not satisfied and the solution blows up as the viscosity goes to 0

piecewise linear lower and upper bounds for the standard normal first order loss function the first order loss function and its complementary function are extensively used in practical settings when the random variable of interest is normally distributed the first order loss function can be easily expressed in terms of the standard normal cumulative distribution and probability density function however the standard normal cumulative distribution does not admit a closed form solution and cannot be easily linearised several works in the literature discuss approximations for either the standard normal cumulative distribution or the first order loss function and their inverse however a comprehensive study on piecewise linear upper and lower bounds for the first order loss function is still missing in this work we initially summarise a number of distribution independent results for the first order loss function and its complementary function we then extend this discussion by focusing first on random variables featuring a symmetric distribution and then on normally distributed random variables for the latter we develop effective piecewise linear upper and lower bounds that can be immediately embedded in milp models these linearisations rely on constant parameters that are independent of the mean and standard deviation of the normal distribution of interest we finally discuss how to compute optimal linearisation parameters that minimise the maximum approximation error

nonlocal effects in two dimensional conductivity the paper deals with the asymptotic behaviour as e 0 of a two dimensional conduction problem whose matrix valued conductivity ae is e periodic and not uniformly bounded with respect to e we prove that only under the assumptions of equi coerciveness and l1 boundedness of the sequence ae the limit problem is a conduction problem of same nature this new result points out a fundamental difference between the two dimensional conductivity and the three dimensional one indeed under the same assumptions of periodicity equi coerciveness and l1 boundedness it is known that the high conductivity regions can induce nonlocal effects in three or greater dimensions

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

homogenization in a periodic and time dependent potential this paper contains a study of the long time behavior of a diffusion process in a periodic potential the first goal is to determine a suitable rescaling of time and space so that the diffusion process converges to some homogeneous limit the issue of interest is to characterize the effective evolution equation the main result is that in some cases large drifts must be removed in order to get a diffusive asymptotic behavior this is applied to the homogenization of parabolic differential equations

multiscale enrichment based on partition of unity a new multiscale enrichment method based on the partition of unity mepu method is presented it is a synthesis of mathematical homogenization theory and the partition of unity method its primary objective is to extend the range of applicability of mathematical homogenization theory to problems where scale separation may not be possible mepu is perfectly suited for enriching the coarse scale continuum descriptions pdes with fine scale features and the quasi continuum formulations with relevant atomistic data numerical results show that it provides a considerable improvement over classical mathematical homogenization theory and quasi continuum formulations copyright 2005 john wiley sons ltd

existence of ℋ matrix approximants to the inverse fe matrix of elliptic operators with l coefficients this article deals with the existence of blockwise low rank approximants so called matrices to inverses of fem matrices in the case of uniformly elliptic operators with l coefficients unlike operators arising from boundary element methods for which the matrix theory has been extensively developed the inverses of these operators do not benefit from the smoothness of the kernel function however it will be shown that the corresponding green functions can be approximated by degenerate functions giving rise to the existence of blockwise low rank approximants of fem inverses numerical examples confirm the correctness of our estimates as a side product we analyse the matrix property of the inverse of the fe mass matrix

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

a new approach to variational problems with multiple scales we introduce a new concept the young measure on micropatterns to study singularly perturbed variational problems that lead to multiple small scales depending on a small parameter e this allows one to extract in the limit e 0 the relevant information at the macroscopic scale as well as the coarsest microscopic scale say e and to eliminate all finer scales to achieve this we consider rescaled functions rx t x s e t viewed as maps of the macroscopic variable s with values in a suitable function space the limiting problem can then be formulated as a variational problem on the young measures generated by rex as an illustration we study a one dimensional model that describes the competition between formation of microstructure and highest gradient regularization we show that the unique minimizer of the limit problem is a young measure supported on sawtooth functions with a given period 2001 john wiley sons inc

double precision rational approximation algorithm for the inverse standard normal second order loss function abstract we present a double precision algorithm based upon rational approximations for the inverse standard normal second order loss function this function is used frequently in inventory management no direct approximation or closed formulation exists for the inverse standard normal second order loss function calculations are currently based on root finding methods and intermediate computations of the cumulative normal distribution or tabulations results then depend on the accuracy and valid range of that underlying function we deal with these issues and present a direct double precision accurate algorithm valid in the full range of double precision floating point numbers

an energy minimizing interpolation for robust multigrid methods we propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation it can handle pde coefficients of various types on structured or unstructured grids under one framework the formulation is general it can be applied to any dimension we demonstrate numerically the effectiveness of the multigrid method in two dimensions by applying it to a discontinuous coefficient problem an oscillatory coefficient problem and an anisotropic problem empirically the convergence rate is independent of the coefficients of the underlying pde in addition to being independent of the mesh size the proposed method is primarily designed for second order elliptic pdes with possible extensions to other classes of problems such as integral equations

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

metric based upscaling we consider divergence form elliptic operators in dimension n ge 2 with l coefficients although solutions of these operators are only holder continuous we show that they are differentiable c1 with respect to harmonic coordinates it follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales this new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged homogenized quantities from subgrid scales into computational scales error bounds can be given and this method can also be used as a compression tool for differential operators 2006 wiley periodicals inc

the discontinuous enrichment method for multiscale analysis computation naturally separates scales of a problem according to the mesh size a variety of improved numerical methods are described by multiscale considerations differing in the treatment of the unresolved fine scales the discontinuous enrichment method provides a unique multiscale approach to computation by employing fine scales that contain solutions of the homogeneous partial differential equation in a discontinuous framework the method thus combines relative ease of implementation with improved numerical performance these properties are demonstrated for both multiscale wave and transport problems pointing to the potential of considerable savings in computational resources

localization of elliptic multiscale problems this paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients the basis functions are solutions of local problems on vertex patches the error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches hence on a uniform mesh of size h patches of diameter h log 1 h are sufficient to preserve a linear rate of convergence in h without pre asymptotic or resonance effects the analysis does not rely on regularity of the solution or scale separation in the coefficient this result motivates new and justifies old classes of variational multiscale methods see more at http www ams org journals mcom 2014 83 290 s0025 5718 2014 02868 8 sthash z2ccfxig dpuf

adaptive variational multiscale methods based on a posteriori error estimation energy norm estimates for elliptic problems we develop a new adaptive multiscale finite element method using the variational multiscale framework together with a systematic technique for approximation of the fine scale part of the solution the fine scale is approximated by a sum of solutions to decoupled localized problems which are solved numerically on a fine grid partition of a patch of coarse grid elements the sizes of the patches of elements may be increased to control the error caused by localization we derive an a posteriori error estimate in the energy norm which captures the dependency of the crucial discretization parameters the coarse grid mesh size the fine grid mesh size and the sizes of the patches based on the a posteriori error estimate we present an adaptive algorithm that automatically tunes these parameters finally we show how the method works in practice by presenting various numerical examples

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

multi scale finite volume method for elliptic problems in subsurface flow simulation in this paper we present a multi scale finite volume msfv method to solve elliptic problems with many spatial scales arising from flow in porous media the method efficiently captures the effects of small scales on a coarse grid is conservative and treats tensor permeabilities correctly the underlying idea is to construct transmissibilities that capture the local properties of the differential operator this leads to a multi point discretization scheme for the finite volume solution algorithm transmissibilities for the msfv have to be constructed only once as a preprocessing step and can be computed locally therefore this step is perfectly suited for massively parallel computers furthermore a conservative fine scale velocity field can be constructed from the coarse scale pressure solution two sets of locally computed basis functions are employed the first set of basis functions captures the small scale heterogeneity of the underlying permeability field and it is computed in order to construct the effective coarse scale transmissibilities a second set of basis functions is required to construct a conservative fine scale velocity field the accuracy and efficiency of our method is demonstrated by various numerical experiments

modelling and computing rn sn policies for inventory systems with non stationary stochastic demand abstract this paper addresses the single item non stationary stochastic demand inventory control problem under the non stationary r s policy in non stationary r s policies two sets of control parameters the review intervals which are not necessarily equal and the order up to levels for replenishment periods are fixed at the beginning of the planning horizon to minimize the expected total cost it is assumed that the total cost is comprised of fixed ordering costs and proportional direct item inventory holding and shortage costs with the common assumption that the actual demand per period is a normally distributed random variable about some forecast value a certainty equivalent mixed integer linear programming model is developed for computing policy parameters the model is obtained by means of a piecewise linear approximation to the non linear terms in the cost function numerical examples are provided

a multiscale finite element method for the helmholtz equation it is well known that when the standard galerkin method is applied to the helmholtz equation it exhibits an error in the wavenumber and the solution does not therefore preserve the phase characteristics of the exact solution improvements on the galerkin method including galerkin least squares gls methods have been proposed however these approaches rely on a dispersion analysis of the underlying difference stencils in order to reduce error in the solution in this paper we propose a multiscale finite element for the helmholtz equation the method employs a multiscale variational formulation which leads to a subgrid model in which subgrid scales are incorporated analytically through appropriate green s functions it is shown that entirely new and accurate methods emerge and that gls methods can be obtained as special cases of the more general subgrid model

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

on the h 1 h1 stability of the l_2 l2 projection onto finite element spaces we study the stability in the h 1 h 1 seminorm of the l 2 l 2 projection onto finite element spaces in the case of nonuniform but shape regular meshes in two and three dimensions and prove in particular stability for conforming triangular elements up to order twelve and conforming tetrahedral elements up to order seven