small and strong formulations for unions of convex sets from the cayley embedding there is often a significant trade off between formulation strength and size in mixed integer programming when modeling convex disjunctive constraints e g unions of convex sets adding auxiliary continuous variables can sometimes help resolve this trade off however standard formulations that use such auxiliary continuous variables can have a worse than expected computational effectiveness which is often attributed precisely to these auxiliary continuous variables for this reason there has been considerable interest in constructing strong formulations that do not use continuous auxiliary variables we introduce a technique to construct formulations without these detrimental continuous auxiliary variables to develop this technique we introduce a natural non polyhedral generalization of the cayley embedding of a family of polytopes and show it inherits many geometric properties of the original embedding we then show how the associated formulation technique can be used to construct small and strong formulation for a wide range of disjunctive constraints in particular we show it can recover and generalize all known strong formulations without continuous auxiliary variables

discontinuous spectral element methods for time and space fractional advection equations we develop spectral element methods for a time and space fractional advection equation of the form prescript 0 mathcal d tau t u x t theta prescript 0 mathcal d nu x u x t f x t of order tau in 0 1 nu in 0 1 subject to dirichlet initial boundary conditions we present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms instead of employing traditional low order time integration methods to this end we first develop a petrov galerkin in time and discontinuous galerkin in space pg dg method where we carry out the time integration using a single time domain spectral method sm and we perform the spatial discretization using the discontinuous spectral hp element method dsem this scheme also leads to a more efficient time integration when the time derivative is integer order i e tau 1 we develop the sm dsem scheme based on a new spectral theory for fractional sturm liouville problems

on mathematical programming with indicator constraints in this paper we review the relevant literature on mathematical optimization with logical implications i e where constraints can be either active or disabled depending on logical conditions to hold in the case of convex functions the theory of disjunctive programming allows one to formulate these logical implications as convex nonlinear programming problems in a space of variables lifted with respect to its original dimension we concentrate on the attempt of avoiding the issue of dealing with large nlps in particular we review some existing results that allow to work in the original space of variables for two relevant special cases where the disjunctions corresponding to the logical implications have two terms then we significantly extend these special cases in two different directions one involving more general convex sets and the other with disjunctions involving three terms computational experiments comparing disjunctive programming formulations in the original space of variables with straightforward bigm ones show that the former are computationally viable and promising

an efficient implicit fem scheme for fractional in space reaction diffusion equations fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity however the presence of a fractional differential operator causes memory time fractional or nonlocality space fractional issues that impose a number of computational constraints in this paper we develop efficient scalable techniques for solving fractional in space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector our approach is show cased by solving the fractional fisher and fractional allen cahn reaction diffusion equations in two and three spatial dimensions and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying laplacian operator

on stabilized finite element methods for linear systems of convection diffusion reaction equations abstract a stabilized finite element method for solving systems of convection diffusion reaction equations is studied in this paper the method is based on the subgrid scale approach and an algebraic approximation to the subscales after presenting the formulation of the method it is analyzed how it behaves under changes of variables showing that it relies on the law of change of the matrix of stabilization parameters associated to the method an expression for this matrix is proposed for the case of general coupled systems of equations that is an extension of the expression proposed for a one dimensional 1d model problem applications of the stabilization technique to the stokes problem with convection and to the bending of reissner mindlin plates are discussed next the design of the matrix of stabilization parameters is based on the identification of the stability deficiencies of the standard galerkin method applied to these two problems

sufficient and necessary conditions for semidefinite representability of convex hulls and sets a set s subseteq mathbb r n is called semidefinite programming sdp representable or semidefinite representable if s equals the projection of a set in higher dimensional space which is describable by some linear matrix inequality lmi clearly if s is sdp representable then s must be convex and semialgebraic it is describable by conjunctions and disjunctions of polynomial equalities or inequalities this paper proves sufficient conditions and necessary conditions for sdp representability of convex sets and convex hulls by proposing a new approach to constructing sdp representations the contributions of this paper are i for bounded sdp representable sets w 1 dots w m we give an explicit construction of an sdp representation for mathrm conv cup k 1 mw k this provides a technique for building global sdp representations from the local ones ii for the sdp representability of a compact convex semialgebraic set s we prove sufficient the boundary partial s is nonsingular and positively curved while necessary is partial s has nonnegative curvature at each nonsingular point in terms of defining polynomials for s nonsingular boundary amounts to them having nonvanishing gradient at each point on partial s and the curvature condition can be expressed as their strict versus nonstrict quasi concavity at those points on partial s where they vanish the gaps between them are partial s having or not having singular points either of the gradient or of the curvature s positivity a sufficient condition bypassing the gaps is when some defining polynomials of s satisfy an algebraic condition called sos concavity iii for the sdp representability of the convex hull of a compact nonconvex semialgebraic set t we find that the critical object is partial c t the maximum subset of partial t contained in partial mathrm conv t we prove sufficient for sdp representability partial c t is nonsingular and positively curved and necessary is partial c t has nonnegative curvature at nonsingular points the gaps between our sufficient and necessary conditions are similar to case ii the positive definite lagrange hessian pdlh condition which meshes well with constructions is also discussed

on the generalized navier stokes equations in this paper we present a general model of the classical navier stokes equations with the help of laplace fourier sine transforms finite fourier sine transforms and finite hankel transforms an exact solutions for three different special cases have been obtained

multi domain spectral collocation method for variable order nonlinear fractional differential equations abstract spectral and spectral element methods using galerkin type formulations are efficient for solving linear fractional pdes fpdes of constant order but are not efficient in solving nonlinear fpdes and cannot handle fpdes with variable order in this paper we present a multi domain spectral collocation method that addresses these limitations we consider fpdes in the riemann liouville sense and employ jacobi lagrangian interpolants to represent the solution in each element we provide variable order differentiation formulas which can be computed efficiently for the multi domain discretization taking into account the nonlocal interactions we enforce the interface continuity conditions by matching the solution values at the element boundaries via the lagrangian interpolants and in addition we minimize the jump in integer fluxes using a penalty method we analyze numerically the effect of the penalty parameter on the condition number of the global differentiation matrix and on the stability and convergence of the penalty collocation scheme we demonstrate the effectiveness of the new method for the fractional helmholtz equation of constant and variable order using h p refinement for different values of the penalty parameter we also solve the fractional burgers equation with constant and variable order and compare with solutions obtained with a single domain spectral collocation method

embedding formulations and complexity for unions of polyhedra it is well known that selecting a good mixed integer programming mip formulation is crucial for effectively obtaining a solution with state of the art solvers although best practices and guidelines for constructing good formulations abound there is rarely a single systematic construction approach that leads to the best possible formulation here we introduce embedding formulations and complexity as a new mip formulation paradigm for systematically constructing formulations for disjunctive constraints that are optimal with respect to size more specifically they yield the smallest possible ideal formulation i e one whose lp relaxation has integral extreme points among all formulations that only use 0 1 auxiliary variables we use the paradigm to characterize optimal formulations for special ordered sets of type 2 and certain piecewise linear functions of two variables we also show that the resultant formulations can provide a significant computational advantage over all known formulations for pi

numerical methods for the solution of partial differential equations of fractional order anomalous diffusion is a possible mechanism underlying plasma transport in magnetically confined plasmas to model this transport mechanism fractional order space derivative operators can be used here the numerical properties of partial differential equations of fractional order 1 2 are studied two numerical schemes an explicit and a semi implicit one are used in solving these equations two different discretization methods of the fractional derivative operator have also been used the accuracy and stability of these methods are investigated for several standard types of problems involving partial differential equations of fractional order

spectral methods for the time fractional diffusion wave equation in a semi infinite channel in this paper we consider the numerical approximation of the time fractional diffusion wave equation in a semi infinite channel the time fractional derivative is described in caputo sense with order 1 2 a fully discrete spectral scheme based on a finite difference method in the time direction and a laguerre legendre spectral method in the space direction is proposed we also propose an alternating direction implicit adi spectral scheme in order to reduce the amount of computation the stability and convergence of both schemes are rigorously established numerical results are presented to support our theoretical analysis

guaranteed and robust discontinuous galerkin a posteriori error estimates for convection diffusion reaction problems we propose and study a posteriori error estimates for convection diffusion reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior penalty discontinuous galerkin methods our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction we first derive locally computable estimates for the error measured in the energy semi norm these estimates are evaluated using h div w conforming diffusive and convective flux reconstructions thereby extending the previous work on pure diffusion problems the resulting estimates are semi robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local peclet and damkohler numbers fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy semi norm a dual norm of the skew symmetric part of the differential operator and a suitable contribution of the interelement jumps of the discrete solution numerical experiments are presented to illustrate the theoretical results

computable exact bounds for linear outputs from stabilized solutions of the advection diffusion reaction equation summary the paper introduces a methodology to compute strict upper and lower bounds for linear functional outputs of the exact solutions of the advection diffusion reaction equation the bounds are computed using implicit a posteriori error estimators from stabilized finite element approximations of the exact solution the new methodology extends the a posteriori error estimates yielding bounds for the standard galerkin formulation to be able to obtain bounds for stabilized formulations this methodology is combined with both hybrid flux and flux free techniques for error assessment the application to stabilized formulations provides sharper estimates than when applied to galerkin methods the best results are found in combination with the flux free technique copyright 2012 john wiley sons ltd

a simple subgrid scale stabilized method for the advection diffusion reaction equation abstract a subgrid scale method has been developed for the advection diffusion reaction equation with a simple intrinsic time scale parameter the proposed formulation combines very good accuracy and stability in all the physical regimes of the equation that is in the exponential regime for both negative and positive source terms and in the propagation regime numerical examples confirm the robustness and accuracy of the method

stable multi domain spectral penalty methods for fractional partial differential equations we propose stable multi domain spectral penalty methods suitable for solving fractional partial differential equations with fractional derivatives of any order first a high order discretization is proposed to approximate fractional derivatives of any order on any given grids based on orthogonal polynomials the approximation order is analyzed and verified through numerical examples based on the discrete fractional derivative we introduce stable multi domain spectral penalty methods for solving fractional advection and diffusion equations the equations are discretized in each sub domain separately and the global schemes are obtained by weakly imposed boundary and interface conditions through a penalty term stability of the schemes are analyzed and numerical examples based on both uniform and nonuniform grids are considered to highlight the flexibility and high accuracy of the proposed schemes

a spectral tau algorithm based on jacobi operational matrix for numerical solution of time fractional diffusion wave equations in this paper an efficient and accurate spectral numerical method is presented for solving second fourth order fractional diffusion wave equations and fractional wave equations with damping the proposed method is based on jacobi tau spectral procedure together with the jacobi operational matrix for fractional integrals described in the riemann liouville sense the main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought for spectral approximations the validity and effectiveness of the method are demonstrated by solving five numerical examples numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier an efficient numerical scheme for time fractional diffusion wave equations is proposed a time space jacobi tau approximation is developed for such equations several tau spectral methods can be achieved as special cases the validity and applicability of the proposed method are demonstrated accurate numerical results are obtained by selecting limited collocation nodes

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

fractional adams bashforth moulton methods we first formulate a fractional class of explicit adams bashforth a b and implicit adams moulton a m methods of first and second order accuracy for the time integration of d t 0 c u x t g t u 0 1 where d t 0 c denotes the fractional derivative in the caputo sense in this fractional setting and in contrast to the standard adams methods an extra history load term emerges and the associated weight coefficients are dependent however when 1 the developed schemes reduce to the well known a b and a m methods with standard coefficients hence in terms of scientific computing our approach constitutes a minimal modification of the existing adams libraries next we develop an implicit explicit imex splitting scheme for linear and nonlinear fractional pdes of a general advection reaction diffusion type and we apply our scheme to the time space fractional keller segel chemotaxis system in this context we evaluate the nonlinear advection term explicitly employing the fractional a b method in the prediction step and we treat the corresponding diffusion term implicitly in the correction step using the fractional a m scheme moreover we perform the corresponding spatial discretization by employing an efficient and spectrally accurate fractional spectral collocation method our numerical experiments exhibit the efficiency of the proposed imex scheme in solving nonlinear fractional pdes

a parareal method for time fractional differential equations in this paper a parareal method is proposed for the parallel in time integration of time fractional differential equations tfdes it is a generalization of the original parareal method proposed for classic differential equations to match the global feature of fractional derivatives the new method has in the correction step embraced the history part of the solution we provide a convergence analysis under the assumption of lipschitz stability conditions we use a multi domain spectral integrator to build the serial solvers and numerical results demonstrate the feasibility of the new approach and confirm the convergence analysis studies also show that both the coarse resolution and the nature of the differential operators can affect the performance

certain new models of the multi space fractional gardner equation abstract in this paper we study an accurate numerical method for solving the multi space fractional gardner equation msfge with the caputo fabrizio cf and atangana baleanu ab fractional derivatives where the space fractional terms are under the sense of caputo to the best knowledge of the reader we are first to use the spectral collocation method based on the third chebyshev approximations to reduced the multi space fractional gardner equation to a system of ordinary differential equations by using the properties of chebyshev polynomials and then solved them via the finite difference method fdm by computing the absolute errors we present the effectiveness and accuracy of the proposed methods the present paper investigates the dynamics of gardner equation by considering two fractional operators that is the caputo fabrizio and atangana baleanu this is entirely new idea by using the two operators on a gardner equation our results prove that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non local singular kernels

a posteriori finite element bounds for linear functional outputs of elliptic partial differential equations we present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order coercive linear partial differential equations in two space dimensions the method is based upon the construction of an augmented lagrangian in which the objective is a quadratic energy reformulation of the desired output and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements the bounds on the output for a suitably fine truth mesh discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid flux candidate lagrange multipliers generated by a k element coarser working mesh approximation independent of the form of the original partial differential equation the computation on the truth mesh is reduced to k decoupled subdomain local symmetric neumann problems the technique is illustrated for the convection diffusion and linear elasticity equations

guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains this is the pre peer reviewed version of the following article pares n santos h diez p guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains international journal for numerical methods in engineering 23 marc 2009 vol 79 num 10 p 1203 1244 which has been published in final form at http www3 interscience wiley com journal 122269073 abstract

quadratic spline collocation method for the time fractional subdiffusion equation in this paper exploiting the quadratic spline collocation qsc method we numerically solve the time fractional subdiffusion equation with dirichelt boundary value conditions the coefficient matrix of the discretized linear system is investigated in detail theoretical analyses and numerical examples demonstrate the proposed technique can enjoy the global error bound with o 3 h 3 under the l norm provided that the solution v x t has four order continual derivative with respects to x and t and it can achieve the accuracy of o 4 h 4 at collocation points where h are the step sizes in time and space respectively

fractional ginzburg landau equation for fractal media abstract we derive the fractional generalization of the ginzburg landau equation from the variational euler lagrange equation for fractal media to describe fractal media we use the fractional integrals considered as approximations of integrals on fractals some simple solutions of the ginzburg landau equation for fractal media are considered and different forms of the fractional ginzburg landau equation or nonlinear schrodinger equation with fractional derivatives are presented the agrawal variational principle and its generalization have been applied

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

goal oriented error estimation and adaptivity for the finite element method abstract in this paper we study a new approach in a posteriori error estimation in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm these so called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs we present here the theory with respect to a class of elliptic boundary value problems and in particular show how to obtain accurate estimates as well as upper and lower bounds on the error we also study the new concept of goal oriented adaptivity which embodies mesh adaptation procedures designed to control error in specific quantities numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

strong valid inequalities for orthogonal disjunctions and bilinear covering sets in this paper we derive a closed form characterization of the convex hull of a generic nonlinear set when this convex hull is completely determined by orthogonal restrictions of the original set although the tools used in this construction include disjunctive programming and convex extensions our characterization does not introduce additional variables we develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed we demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed integer polyhedral sets and finding the convex hull of certain mixed pure integer bilinear sets we then extend the utility of the convexification tool to relaxing nonconvex inequalities which are not naturally disjunctive by providing sufficient conditions for establishing the convex extension property over the non negative orthant we illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non negative orthant although we illustrate our results primarily on bilinear covering sets they also apply to more general polynomial covering sets for which they yield new tight relaxations

a fast finite difference method for two dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate complicated dense or full coefficient matrices consequently these numerical methods were traditionally solved by gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two and especially three dimensional space fractional diffusion equations we develop a fast and yet accurate solution method for the implicit finite difference discretization of space fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma

a new finite element formulation for computational fluid dynamics x the compressible euler and navier stokes equations a space time element method is presented for solving the compressible euler and navier stokes equations the proposed formulation includes the variational equation predictor multi corrector algorithms and boundary conditions the variational equation is based on the time discontinuous galerkin method in which the physical entropy variables are employed a least squares operator and a discontinuity capturing operator are added resulting in a high order accurate and unconditionally stable method implicit explicit predictor multi corrector algorithms applicable to steady as well as unsteady problems are presented techniques are developed to enhance their efficiency implementation of boundary conditions is addressed in particular a technique is introduced to satisfy nonlinear essential boundary conditions and a consistent method is presented to calculate boundary fluxes numerical results are presented to demonstrate the performance of the method

bounds of functional outputs for parabolic problems part ii bounds of the exact solution the paper introduces a methodology to compute upper and lower bounds for linear functional outputs of the exact solutions of parabolic problems in this second part the bounds account for the error both in space and time the assumption stating that the error introduced by the time marching scheme is negligible used in the first part is removed here the bounds are computed starting from an approximation of the exact solution associated with a spatial mesh and a time grid nevertheless the bounds are guaranteed with respect to the exact solution with no reference to any mesh or time discretization

representation for multiple right hand sides we are given finitely many polyhedra defined by linear constraints using the same constraint matrix and different right hand sides we consider a simple constraint system and give necessary and sufficient conditions for this system to define the union of the polyhedra we also show that deciding whether the system does define the union is np hard

bounds of functional outputs for parabolic problems part i exact bounds of the discontinuous galerkin time discretization classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest consequently the bounds obtained are only guaranteed asymptotically that is with respect to the reference solution computed with the fine mesh exact bounds that is bounds guaranteed with respect to the exact solution are needed to properly certify the accuracy of the results especially if the meshes are coarse the paper introduces a procedure to compute strict upper and lower bounds of the error in linear functional outputs of parabolic problems in this first part the bounds account for the error associated with the spatial discretization the error coming from the time marching scheme is therefore assumed to be negligible in front of the spatial error the time discretization is performed using the discontinuous galerkin method both for the primal and adjoint problems in the error estimation procedure equilibrated fluxes at interelement edges are calculated using hybridization techniques

strict and effective bounds in goal oriented error estimation applied to fracture mechanics problems solved with xfem in this work we analyze a method that leads to strict and high quality local error bounds in the context of fracture mechanics we investigate in particular the capability of this method to evaluate the discretization error for quantities of interest computed using the extended finite element method xfem the goal oriented error estimation method we are focusing on uses the concept of constitutive relation error along with classical extraction techniques the main innovation in this paper resides in the methodology employed to construct admissible fields in the xfem framework which involves enrichments with singular and level set basis functions we show that this construction can be performed through a generalization of the classical procedure used for the standard finite element method thus the resulting goal oriented error estimation method leads to relevant and very accurate information on quantities of interest that are specific to fracture mechanics such as mixed mode stress intensity factors the technical aspects and the effectiveness of the method are illustrated through two dimensional numerical examples

a multi domain spectral method for time fractional differential equations this paper proposes an approach for high order time integration within a multi domain setting for time fractional differential equations since the kernel is singular or nearly singular two main difficulties arise after the domain decomposition how to properly account for the history memory part and how to perform the integration accurately to address these issues we propose a novel hybrid approach for the numerical integration based on the combination of three term recurrence relations of jacobi polynomials and high order gauss quadrature the different approximations used in the hybrid approach are justified theoretically and through numerical examples based on this we propose a new multi domain spectral method for high order accurate time integrations and study its stability properties by identifying the method as a generalized linear method numerical experiments confirm hp convergence for both time fractional differential equations and time fractional partial differential equations

error estimates of generalized spectral iterative methods with accurate convergence rates for solving systems of fractional two point boundary value problems abstract the main purpose of this study is to provide an efficient spectral iterative method based on fractional interpolants for solving a class of linear and nonlinear fractional two point boundary value problems involving left and right fractional derivatives in order to achieve this goal by composition of iterative method and spectral method in each step the unknown solution of system is expanded by using two classes of generalized jacobi functions to obtain numerically coefficients for this system of fractional differential equations on the interval 1 1 the singularities of the solution are considered of types 1 t and 1 x where 0 l 2 norm finally some numerical results are given to demonstrate the effectiveness and applicability of the proposed method and accuracy of the presented convergence rates

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

a simplification for some disjunctive formulations abstract we give sufficient conditions under which a relatively compact integer formulation can be obtained for some instances of multiple right hand side constraints with the special property that the linear relaxation of this formulation is as small tight as possible among all integer formulations the formulation we obtain is a simplification of the disjunctive formulation and requires exactly as many additional nonnegative variables as alternative r h s with one additional constraint we conclude with several applications of our results

a higher order numerical method for time fractional partial differential equations with nonsmooth data abstract gao et al 11 2014 introduced a numerical scheme to approximate the caputo fractional derivative with the convergence rate o k 3 0 1 by directly approximating the integer order derivative with some finite difference quotients in the definition of the caputo fractional derivative see also lv and xu 20 2016 where k is the time step size under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth lv and xu 20 2016 proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate o k 3 0 1 uniformly with respect to the time variable t however in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate o k 3 0 1 uniformly with respect to the time variable t in this paper we first obtain a similar approximation scheme to the riemann liouville fractional derivative with the convergence rate o k 3 0 1 as in gao et al 11 2014 by approximating the hadamard finite part integral with the piecewise quadratic interpolation polynomials based on this scheme we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using laplace transform methods that the time discretization scheme has the convergence rate o k 3 0 1 for any fixed t n 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases numerical examples are given to show that the theoretical results are consistent with the numerical results

computing bounds for linear functionals of exact weak solutions to poisson s equation we present a method for poisson s equation that computes guaranteed upper and lower bounds for the values of piecewise polynomial linear functional outputs of the exact weak solution of the infinite dimensional continuum problem with piecewise polynomial forcing the method results from exploiting the lagrangian saddle point property engendered by recasting the output problem as a constrained minimization problem localization is achieved by lagrangian relaxation and the bounds are computed by appeal to a local dual problem the proposed method computes approximate lagrange multipliers using traditional finite element approximations to calculate a primal and an adjoint solution along with well known hybridization techniques to calculate interelement continuity multipliers the computed bounds hold uniformly for any level of refinement and in the asymptotic convergence regime of the finite element method the bound gap decreases at twice the rate of the energy norm measure of the error in the finite element solution given a finite element solution and its output adjoint solution the method can be used to provide a certificate of precision for the output with an asymptotic complexity that is linear in the number of elements in the finite element discretization the elemental contributions to the bound gap are always positive and hence lend themselves to be used as adaptive indicators as we demonstrate with a numerical example

convex programming for disjunctive convex optimization given a finite number of closed convex sets whose algebraic representation is known we study the problem of finding the minimum of a convex function on the closure of the convex hull of the union of those sets we derive an algebraic characterization of the feasible region in a higher dimensional space and propose a solution procedure akin to the interior point approach for convex programming

perspective reformulations of mixed integer nonlinear programs with indicator variables we study mixed integer nonlinear programs minlp s that are driven by a collection of indicator variables where each indicator variable controls a subset of the decision variables an indicator variable when it is turned off forces some of the decision variables to assume fixed values and when it is turned on forces them to belong to a convex set many practical minlps contain integer variables of this type we first study a mixed integer set defined by a single separable quadratic constraint and a collection of variable upper and lower bound constraints and a convex hull description of this set is derived we then extend this result to produce an explicit characterization of the convex hull of the union of a point and a bounded convex set defined by analytic functions further we show that for many classes of problems the convex hull can be expressed via conic quadratic constraints and thus relaxations can be solved via second order cone programming our work is closely related with the earlier work of ceria and soares math program 86 595 614 1999 as well as recent work by frangioni and gentile math program 106 225 236 2006 and akturk et al oper res lett 37 187 191 2009 finally we apply our results to develop tight formulations of mixed integer nonlinear programs in which the nonlinear functions are separable and convex and in which indicator variables play an important role in particular we present computational results for three applications quadratic facility location network design with congestion and portfolio optimization with buy in thresholds that show the power of the reformulation technique

the computation of bounds for linear functional outputs of weak solutions to the two dimensional elasticity equations we present a method for the computation of upper and lower bounds for linear functional outputs of the exact solutions to the two dimensional elasticity equations the method can be regarded as a generalization of the well known complementary energy principle the desired output is cast as the supremum of a quadratic linear convex functional over an infinite dimensional domain using duality the computation of an upper bound for the output of interest is reduced to a feasibility problem for the complementary or dual problem in order to make the problem tractable from a computational perspective two additional relaxations that preserve the bounding properties are introduced first the domain is triangulated and a domain decomposition strategy is used to generate a sequence of independent problems to be solved over each triangle the lagrange multipliers enforcing continuity are approximated using piecewise linear functions over the edges of the triangulation second the solution of the adjoint problem is approximated over the triangulation using a standard galerkin finite element approach a lower bound for the output of interest is computed by repeating the process for the negative of the output reversing the sign of the computed upper bound for the negative of the output yields a lower bound for the actual output the method can be easily generalized to three dimensions however a constructive proof for the existence of feasible solutions for the outputs of interest is only given in two dimensions the computed bound gaps are found to converge optimally that is at the same rate as the finite element approximation an attractive feature of the proposed approach is that it allows for a data set to be generated that can be used to certify and document the computed bounds using this data set and a simple algorithm the correctness of the computed bounds can be established without recourse to the original code used to compute them in the present paper only computational domains whose boundary is made up of straight sided segments and polynomially varying loads are considered two examples are given to illustrate the proposed methodology

a branch and cut method for 0 1 mixed convex programming we generalize the disjunctive approach of balas ceria and cornuejols 2 and devevlop a branch and cut method for solving 0 1 convex programming problems we show that cuts can be generated by solving a single convex program we show how to construct regions similar to those of sherali and adams 20 and lovasz and schrijver 12 for the convex case finally we give some preliminary computational results for our method

calculation of strict error bounds for finite element approximations of non linear pointwise quantities of interest this paper deals with the verification of simulations performed using the finite element method more specifically it addresses the calculation of strict bounds on the discretization errors affecting pointwise outputs of interest which may be non linear with respect to the displacement field the method is based on classical tools such as the constitutive relation error and extraction techniques associated with the solution of an adjoint problem however it uses two specific and innovative techniques the enrichment of the adjoint solution using a partition of unity method which enables one to consider truly pointwise quantities of interest and the decomposition of the non linear quantities of interest by means of projection properties in order to take into account higher order terms in establishing the bounds thus no linearization is performed and the property that the local error bounds are guaranteed is preserved the effectiveness of the approach and the quality of the bounds are illustrated with two dimensional applications in the context of elastic fatigue problems

exponentially accurate spectral and spectral element methods for fractional odes current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy here we present different methods for fractional odes that lead to exponentially fast decay of the error first we develop a petrov galerkin pg spectral method for fractional initial value problems fivps of the form d t 0 u t f t and fractional final value problems ffvps d t t u t g t where 0 1 subject to dirichlet initial final conditions these schemes are developed based on a new spectral theory for fractional sturm liouville problems fslps which has been recently developed in 1 specifically we obtain solutions to fivps and ffvps in terms of the new fractional non polynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of first kind fslp i correspondingly we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of second kind fslp ii subsequently we develop a discontinuous spectral method dsm of petrov galerkin sense for the aforementioned fivps and ffvps where the basis functions do not satisfy the initial final conditions finally we extend the dsm scheme to a discontinuous spectral element method dsem for efficient longer time integration and adaptive refinement in these discontinuous schemes we employ the asymptotic eigensolutions to fslp i ii which are of jacobi polynomial forms as basis and test functions our numerical tests confirm the exponential algebraic convergence respectively in p and h refinements for various test cases with integer and fractional order solutions

upper bounds of the error in local quantities using equilibrated and compatible finite element solutions for linear elastic problems abstract when local quantities are computed using the principle of virtual work dual analysis which provides an upper bound of the global error may also be applied to the virtual problem greenberg and washizu proposed alternative approaches to combine the global error bounds of the real and virtual problems providing upper bounds of the local error it is shown in this paper that optimising greenberg s approach corresponds to using washizu s approach which in turn may be further improved these approaches are used to provide finite element error indicators for adaptive refinement

subdomain based flux free a posteriori error estimators a new residual type flux free error estimator is presented it estimates upper and lower bounds of the error in energy norm the proposed approach precludes the main drawbacks of standard residual type estimators circumvents the need of flux equilibration and results in a simple implementation that uses standard resources available in finite element codes this is specially interesting for 3d applications where the implementation of this technique is as simple as in 2d recall that on the contrary the complexity of the flux equilibration techniques increases drastically in the 3d case the bounds for the energy norm of the error are used to produce upper and lower bounds of linear functional outputs representing quantities of engineering interest the presented estimators demonstrate their efficiency in numerical tests producing sharp estimates both for the energy and the quantities of interest

an analysis of the modified l1 scheme for time fractional partial differential equations with nonsmooth data we introduce a modified l1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous c

exact bounds for linear outputs of the advection diffusion reaction equation using flux free error estimates the paper introduces a methodology to compute strict upper and lower bounds for linear functional outputs of the exact solutions of the advection reaction diffusion equation the proposed approach is an alternative to the standard residual type estimators hybrid flux circumventing the need of flux equilibration following a flux free error estimation strategy the presented estimator provides sharper estimates than the ones provided by both the standard hybrid flux techniques and other flux free techniques

a new equilibrated residual method improving accuracy and efficiency of flux free error estimates this paper presents a new methodology to compute guaranteed upper bounds for the energy norm of the error in the context of linear finite element approximations of the reaction diffusion equation the new approach revisits the ideas in pares et al 2009 6 4 with the goal of substantially reducing the computational cost of the flux free method while retaining the good quality of the bounds the new methodology provides also a technique to compute equilibrated boundary tractions improving the quality of standard equilibration strategies the zeroth order equilibration conditions are imposed using an alternative less restrictive form of the first order equilibration conditions along with a new efficient minimization criterion this new equilibration strategy provides much more accurate upper bounds for the energy and requires only doubling the dimension of the local linear systems of equations to be solved

modeling disjunctive constraints with a logarithmic number of binary variables and constraints many combinatorial constraints over continuous variables such as sos1 and sos2 constraints can be interpreted as disjunctive constraints that restrict the variables to lie in the union of a finite number of specially structured polyhedra known mixed integer binary formulations for these constraints have a number of binary variables and extra constraints linear in the number of polyhedra we give sufficient conditions for constructing formulations for these constraints with a number of binary variables and extra constraints logarithmic in the number of polyhedra using these conditions we introduce mixed integer binary formulations for sos1 and sos2 constraints that have a number of binary variables and extra constraints logarithmic in the number of continuous variables we also introduce the first mixed integer binary formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints logarithmic in the number of linear pieces of the functions we prove that the new formulations for piecewise linear functions have favorable tightness properties and present computational results showing that they can significantly outperform other mixed integer binary formulations

a posteriori error estimators for stationary convection diffusion problems a computational comparison abstract we consider the scalar convection diffusion problem in 2 d present standard and stabilized finite element methods and give a survey of different a posteriori error estimators these estimators are either based on the evaluation of local residuals or on the solution of discrete local neumann problems besides that a heuristic indicator based on local averaging techniques for the solution gradient is presented and compared with the estimators we give an overview of another aspect of adaptive mesh refinement namely the decision which elements have to be marked for refinement several marking strategies are described and compared

a unified spectral method for fpdes with two sided derivatives part ii stability and error analysis abstract we present the stability and error analysis of the unified petrov galerkin spectral method developed in 1 for linear fractional partial differential equations with two sided derivatives and constant coefficients in any 1 d dimensional space time hypercube d 1 2 3 subject to homogeneous dirichlet initial boundary conditions specifically we prove the existence and uniqueness of the weak form and perform the corresponding stability and error analysis of the proposed method finally we perform several numerical simulations to compare the theoretical and computational rates of convergence

mixed integer linear programming formulation techniques a wide range of problems can be modeled as mixed integer linear programming mip problems using standard formulation techniques however in some cases the resulting mip can be either too weak or

superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations we consider an initial boundary value problem for partial tu partial t alpha nabla 2u f t that is for a fractional diffusion 1 alpha 0 or wave 0 alpha 1 equation a numerical solution is found by applying a piecewise linear discontinuous galerkin dg method in time combined with a piecewise linear conforming finite element method in space the time mesh is graded appropriately near t 0 but the spatial mesh is quasi uniform previously we proved that the error measured in the spatial l 2 norm is of order k 2 alpha h 2 ell k uniformly in t where k is the maximum time step h is the maximum diameter of the spatial finite elements alpha min alpha 0 le0 and ell k max 1 log k here we prove convergence of order k 3 2 alpha ell k h 2 at each time level t n for 1 alpha 1 thus if 1 2 alpha 1 then the dg solution is superconvergent which generalizes a known result for the classical heat equation i e the case alpha 0 a

discontinuous galerkin method for fractional convection diffusion equations we propose a discontinuous galerkin method for fractional convection diffusion equations with a superdiffusion operator of order alpha 1 alpha 2 defined through the fractional laplacian the fractional operator of order alpha is expressed as a composite of first order derivatives and a fractional integral of order 2 alpha the fractional convection diffusion problem is expressed as a system of low order differential integral equations and a local discontinuous galerkin method scheme is proposed for the equations we prove stability and optimal order of convergence cal o h k 1 for the fractional diffusion problem and an order of convergence of cal o h k frac 1 2 is established for the general fractional convection diffusion problem the analysis is confirmed by numerical examples

computing upper and lower bounds for the j integral in two dimensional linear elasticity we present an a posteriori method for computing rigorous upper and lower bounds for the j integral in two dimensional linear elasticity the j integral which is typically expressed as a contour integral is recast as a quadratic continuous functional of the displacement involving only area integration by expanding the quadratic output about an approximate finite element solution the output is expressed as a known computable quantity plus linear and quadratic functionals of the solution error the quadratic component is bounded by the energy norm of the error scaled by a continuity constant which is determined explicitly the linear component is expressed as an inner product of the errors in the displacement and in a computed adjoint solution and bounded by an appropriate combination of the energy norms of the error in the displacement and the adjoint upper bounds for the energy norm of the error are obtained by using a complementary energy approach requiring the computation of equilibrated stress fields the method is illustrated with two fracture problems in plane strain elasticity an important feature of the method presented is that the computed bounds are rigorous with respect to the exact weak solution of the elasticity equations