a characterization of hybridized mixed methods for second order elliptic problems in this paper we give a new characterization of the approximate solution given by hybridized mixed methods for second order self adjoint elliptic problems we apply this characterization to obtain an explicit formula for the entries of the matrix equation for the lagrange multiplier unknowns resulting from hybridization we also obtain necessary and sufficient conditions under which the multipliers of the raviart thomas and the brezzi douglas marini methods of similar order are identical

a priori and a posteriori analysis of finite volume discretizations of darcy s equations this paper is devoted to the numerical analysis of some finite volume discretizations of darcy s equations we propose two finite volume schemes on unstructured meshes and prove their equivalence with either conforming or nonconforming finite element discrete problems this leads to optimal a priori error estimates in view of mesh adaptivity we exhibit residual type error indicators and prove estimates which allow to compare them with the error in a very accurate way

adjoint based a posteriori analysis of multiscale mortar discretizations with multinumerics in this paper we derive a posteriori error estimates for linear functionals of the solution to an elliptic problem discretized using a multiscale nonoverlapping domain decomposition method the error estimates are based on the solution of an appropriately defined adjoint problem we present a general framework that allows us to consider both primal and mixed formulations of the forward and adjoint problems within each subdomain the primal subdomains are discretized using either an interior penalty discontinuous galerkin method or a continuous galerkin method with weakly imposed dirichlet conditions the mixed subdomains are discretized using raviart thomas mixed finite elements the a posteriori error estimate also accounts for the errors due to adjoint inconsistent subdomain discretizations the coupling between the subdomain discretizations is achieved via a mortar space we show that the numerical discretization error can be broken down into subdomain and mortar components which may be used to drive

explicit and averaging a posteriori error estimates for adaptive finite volume methods local mesh refining algorithms known from adaptive finite element methods are adopted for locally conservative and monotone finite volume discretizations of boundary value problems for steady state convection diffusion reaction equations the paper establishes residual type explicit error estimators and averaging techniques for a posteriori finite volume error control with and without upwind in global h1 and l2 norms reliability and efficiency are verified theoretically and confirmed empirically with experimental support for the superiority of the suggested adaptive mesh refining algorithms over uniform mesh refining a discussion of adaptive computations in the simulation of contaminant concentration in a nonhomogeneous water reservoir concludes the paper

a posteriori error estimations of some cell centered finite volume methods for diffusion convection reaction problems this paper presents an a posteriori residual error estimator for diffusion convection reaction problems approximated by some cell centered finite volume methods on isotropic or aniso tropic meshes in rd d 2 or 3 for that purpose we built a reconstructed approximation which is an appropriate interpolant of the finite volume solution the error is then the difference between the exact solution and this interpolant the residual error estimator is based on the jump of the normal derivative of the interpolant we then prove the equivalence between the energy norm of the error and the residual error estimator some numerical tests confirm our theoretical results

a reference direction approach to multiple objective integer linear programming abstract we propose the use of a reference direction reference point approach to solving multiple objective integer linear programming problems the reference direction reference point is determined by the aspiration levels for the criteria that the decision maker wants to improve within this framework two methods are developed viz a pure integer method that operates entirely with integer solutions and a continuous integer method that works with continuous solutions and finds an integer solution closest to the continuous solution in terms of the achievement scalarizing function obviously the pure integer method is time consuming we therefore propose a decision support system that combines the two methods this way the adbantages of each method can be used to their fullest extent we illustrate the proposed decision support system with a numerical example

ellam for resolving the kinematics of two dimensional resistive magnetohydrodynamic flows we combine the finite element method with the eulerian lagrangian localized adjoint method ellam to solve the convection diffusion equations that describe the kinematics of magnetohydrodynamic flows i e the advection and diffusion of a magnetic field simulations of three two dimensional test problems are presented and in each case we analyze the energy of the magnetic field as it evolves towards its equilibrium state our numerical results highlight the accuracy and efficiency of the ellam approach for convection dominated problems

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

quantification of errors for operator split advection diffusion calculations abstract multiphysics simulations frequently are composed from highly optimized solvers for physical subprocesses through the use of temporal operator splitting the subphysics are evolved sequentially passing information between solver components as needed it is often useful yet difficult to determine how the simulation error depends on the temporal splitting method the time step size and the discretization parameters in the solver components this paper proposes a framework to decompose the total error in a quantity of interest for an advection diffusion simulation into two primary contributions that due to the operator splitting as implied by the structure of the code and that due to the discretization errors from the component solvers the method is applied to the advection diffusion equation with boundaries the advection and diffusion operators require separate boundary conditions upon splitting the specification of these impacts the splitting error contribution computational examples demonstrate that the proposed method successfully identifies the splitting error contribution including that induced by a suboptimal imposition of boundary conditions and the decrease in the splitting error contribution in moving to a higher order splitting method the discretization error contribution is decomposed further into contributions attributed to the separate advection and diffusion solvers

high order methods for the approximation of the incompressible navier stokes equations in a moving domain in this paper we address the numerical approximation of the incompressible navier stokes equations in a moving domain by the spectral element method and high order time integrators we present the arbitrary lagrangian eulerian ale formulation of the incompressible navier stokes equations and propose a numerical method based on the following kernels a lagrange basis associated with fekete points in the spectral element method context bdf time integrators an ale map of high degree and an algebraic linear solver in particular the high degree ale map is appropriate to deal with a computational domain whose boundary is described with curved elements finally we apply the proposed strategy to a test case c 2011 elsevier b v all rights reserved

interior penalty discontinuous galerkin method on very general polygonal and polyhedral meshes this paper provides a theoretical foundation for interior penalty discontinuous galerkin methods for second order elliptic equations on very general polygonal or polyhedral meshes the mesh can be composed of any polygons or polyhedra that satisfy certain shape regularity conditions characterized in a recent paper by two of the authors wang and ye 2012 11 the usual h 1 conforming finite element methods on such meshes are either very complicated or impossible to implement in practical computation the interior penalty discontinuous galerkin method provides a simple and effective alternative approach which is efficient and robust results with such general meshes have important application in computational sciences

from electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex the electrostatic interpretation of the jacobi gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex moreover several new estimates based on extensive numerical studies for approximation along the line using jacobi gauss lobatto quadrature points as the nodal sets are presented the electrostatic analogy is extended to the two dimensional case with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented the matrices are evaluated by computing the lebesgue constants and they share the property that the nodes along the edges of the simplex are the gauss lobatto quadrature points of the chebyshev and legendre polynomials respectively this makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h p finite element methods

an interactive procedure for multiple objective integer linear programming problems abstract there are many practical problems where the objective functions and the constraints can be represented by linear functions but the decision variables are restricted to only integer values our objective is to present an interactive procedure for the multiple objective integer linear programming problem that is fast does not place too many demands on the decision maker and can be implemented without requiring any special software we illustrate the proposed procedure with a numerical example and report some computational experience

stable spectral methods on tetrahedral elements a framework for the construction of stable spectral methods on arbitrary domains with unstructured grids is presented although most of the developments are of a general nature an emphasis is placed on schemes for the solution of partial differential equations defined on the tetrahedron in the first part the question of well behaved multivariate polynomial interpolation on the tetrahedron is addressed and it is shown how to extend the electrostatic analogy of the jacobi polynomials to problems beyond the line this allows for the identification of nodal sets suitable for polynomial interpolation within the tetrahedron and subsequently for the formulation of stable spectral schemes on such unstructured nodal sets the second part of this work is devoted to a discussion of weakly imposed boundary conditions and energy stable schemes are formulated for a wide class of problems exemplified by advection problems advection diffusion problems and linear symmetric hyperbolic systems finally in the third part issues related to computational efficiency and implementation of the schemes are discussed the spectral accuracy of the approximation is confirmed through an example and factorization methods for the efficient computation of derivatives on the general nodal sets within the d simplex are developed ensuring that the proposed schemes are competitive with tensor product based methods in this last part we also show that the advective operator results in an cal o n 2 restriction on the time step similar to that of spectral collocation methods employing a tensor product based approximation the performance of the proposed scheme is illustrated by solving a wave problem on a triangulated domain confirming the expected accuracy and stability

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

the biobjective travelling purchaser problem the purpose of this article is to present and solve the biobjective travelling purchaser problem which consists in determining a route through a subset of markets in order to collect a set of products minimizing the travel distance and the purchasing cost simultaneously the most convenient purchase of the product in the visited markets is easily computed once the route has been determined therefore this problem contains a finite set of solutions one for each route and the problem belongs to the field of the biobjective combinatorial optimization it is here formulated as a biobjective mixed integer linear programming model with an exponential number of valid inequalities and this model is used within a cutting plane algorithm to generate the set of all supported and non supported efficient points in the objective space a variant of the algorithm computes only supported efficient points for each efficient point in the objective space exactly one pareto optimal solution in the decision space is computed by solving a single objective problem each of these single objective problems in turn is solved by a specific branch and cut approach a heuristic improvement based on saving previously generated cuts in a common cut pool structure has also been developed with the aim of speeding up the algorithm performance results based on benchmark instances from literature show that the common cut pool heuristic is very useful and that the proposed algorithm manages to solve instances containing up to 100 markets and 200 different products the general procedure can be extended to address other biobjective combinatorial optimization problems whenever a branch and cut algorithm is available to solve a single objective linear combination of these criteria

high order optimal edge elements for pyramids prisms and hexahedra edge elements are a popular method to solve maxwell s equations especially in time harmonic domain however when non affine elements are considered elements of the nedelec s first family 19 are not providing an optimal rate of the convergence of the numerical solution toward the solution of the exact problem in h curl norm we propose new finite element spaces for pyramids prisms and hexahedra in order to recover the optimal convergence in the particular case of pyramids a comparison with other existing elements found in the literature is performed numerical results show the good behavior of these new finite elements

fenchel cutting planes for integer programs a technique for generating cutting planes for integer programs is introduced that is based on the ability to optimize a linear function on a polyhedron rather than explicit knowledge of the underlying polyhedral structure of the integer program the theoretical properties of the cuts and their relationship to lagrangian relaxation are discussed the cut generation procedure is described and computational results are presented

a branch and cut algorithm for the undirected traveling purchaser problem the purpose of this paper is to present a branch and cut algorithm for the undirectedtraveling purchaser problem which consists of determining a minimum cost route through a subset of markets where the cost is the sum of travel and purchase costs the problem is formulated as an integer linear program and several families of valid inequalities are derived to strengthen the linear relaxation the polyhedral structure of the formulation is analyzed and several classes of valid inequalities are proved to be facet defining a branch and cut procedure is developed and tested over four classes of randomly generated instances results show that the proposed algorithm outperforms all previous approaches and can optimally solve instances containing up to 200 markets

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle the main results of this paper are the analysis of the quality of approximation of polynomial interpolation and the computation of the approximate optimal interpolation points in the triangle we introduce various norms for the interpolation operator computational results indicate that for a given polynomial degree the set that minimizes the mean l2 norm of the interpolation operator is close to the smallest lebesgue constant interpolation set in particular for the triangle this set gives the smallest lebesgue constant currently known

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

a posteriori error bounds and global error control for approximation of ordinary differential equations the author analyzes a finite element method for the integration of initial value problems in ordinary differential equations general and contractive problems are treated and quasi optimal a priori and a posteriori error bounds obtained in each case in particular good results are obtained for a class of stiff dissipative problems these results are used to construct a rigorous and robust theory of global error control the author also derives an asymptotic error estimate that is used in a discussion of the behavior of the error in conclusion the properties of the error control are exhibited in a series of numerical experiments

spectral methods on arbitrary grids stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations these new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions specifically we show how to implement legendre galerkin legendre collocation and laguerre galerkin methodology on arbitrary grids

a posteriori error estimations of some cell centered finite volume methods this paper presents the natural framework to residual based a posteriori error estimation of some cell centered finite volume methods for the laplace equation in r d d 2 or 3 for that purpose we associate with the finite volume solution a reconstructed approximation which is a kind of morley interpolant the error is then the difference between the exact solution and this morley interpolant the residual error estimator is based on the jump of normal and tangential derivatives of the morley interpolant we then prove the equivalence between the discrete h1 seminorm of the error and the residual error estimator numerical tests confirm our theoretical results

an overview of techniques for solving multiobjective mathematical programs multiobjective mathematical programming has been one of the fastest growing areas of or ms during the last 15 years this paper presents some reasons for the rapidly growing increase in interest in multiobjective mathematical programming a discussion of the advantages and disadvantages of the three general approaches articulation of the decision maker s preference structure over the multiple objectives prior to during or after the optimization towards multiobjective mathematical programming a nontechnical overview of many of the specific solution techniques for multiobjective mathematical programming and a discussion of important areas for further research the overview concentrates on those techniques which require an articulation of the decision maker s preference structure either during or after the optimization since these are the areas where most of the recent research has been conducted it differs from previous overviews in that in addition to the timing of the elicited preference informa

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

a fast algorithm for spectral differentiation a simple algorithm is presented for matrix multiplication in just over half the number of operations entailed by the conventional algorithm in cases where the matrix possesses the degree of symmetry widely exhibited by derivative matrices the algorithm is used to multiply the chebyshev derivative matrix by a vector for the larger values of n the ratio of execution times approached the expected value of 2

the role of the penalty in the local discontinuous galerkin method for maxwell s eigenvalue problem abstract the local discontinuous galerkin ldg method is a promising emerging technology for discretizing a wide range of differential operators it is well known that the robustness of this method is contingent upon a stabilization term we analyze the role of this term in the ldg discretization of the maxwell curl curl operator demonstrating that the stabilization parameter can be used to split the spectrum into two sections the first set of eigenvalues approximate the spectrum of the curl conforming finite element curl curl operator while the second set are spurious eigenvalues that can be made arbitrarily large by increasing the stabilization parameter we confirm this analysis with a range of computational experiments that compare the spectra obtained from ldg and curl conforming discretizations

a branch and bound algorithm for the traveling purchaser problem an important generalization of the traveling salesman problem called the traveling purchaser problem is considered a branch and bound algorithm which solves a related simple plant location problem for calculating the bounds is proposed for this problem computational experiments with this algorithm provide the evidence that moderate size problems of upto 25 cities and 100 commodities can be solved in reasonable computation time the problem is shown to have applications in the areas of scheduling warehousing and routing

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

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

a posteriori error estimator for finite volume methods in this paper first we discuss a technique to compare finite volume method and some well known finite element methods namely the dual mixed methods and nonconforming primal methods for elliptic equations these both equivalences are exploited to give us a posteriori error estimator for finite volume methods this estimator is explicitly given easy to compute and asymptotically exact without any regularity of the solution in unstructured grids

residual flux based a posteriori error estimates for finite volume and related locally conservative methods we derive in this paper a posteriori error estimates for discretizations of convection diffusion reaction equations in two or three space dimensions our estimates are valid for any cell centered finite volume scheme and in a larger sense for any locally conservative method such as the mimetic finite difference covolume and other we consider meshes consisting of simplices or rectangular parallelepipeds and also provide extensions to nonconvex cells and nonmatching interfaces we allow for the cases of inhomogeneous and anisotropic diffusion dispersion tensors and of convection dominance the estimates are established in the energy semi norm for a locally postprocessed approximate solution preserving the conservative fluxes and are of residual type they are fully computable all occurring constants are evaluated explicitly and locally efficient give a local lower bound on the error times an efficiency constant so that they can serve both as indicators for adaptive refinement and for the actual control of the error they are semi robust in the sense that the local efficiency constant only depends on local variations in the coefficients and becomes optimal as the local peclet number gets sufficiently small numerical experiments confirm their accuracy

goal oriented a posteriori error estimation for finite volume methods a general framework for goal oriented a posteriori error estimation for finite volume methods is developed the framework does not rely on recasting finite volume methods as special cases of finite element methods but instead directly determines error estimators from the discretized finite volume equations thus the framework can be applied to arbitrary finite volume methods it also provides the proper functional settings to address well posedness issues for the primal and adjoint problems numerical results are presented to illustrate the validity and effectiveness of the a posteriori error estimates and their applicability to adaptive mesh refinement

a pseudo spectral scheme for the incompressible navier stokes equations using unstructured nodal elements a pseudo spectral scheme for solving the incompressible navier stokes equations using unstructured nodal triangles is proposed efficient algorithms are developed with numerical evidence that indicates optimal rates of convergence can be achieved navier stokes simulations of kovasznay shear layer roll up and row past a cylinder are included to show comparisons between the different nodal sets considered and an alternative modal approach c 2000 academic press

a posteriori error analysis of a cell centered finite volume method for semilinear elliptic problems in this paper we conduct a goal oriented a posteriori analysis for the error in a quantity of interest computed from a cell centered finite volume scheme for a semilinear elliptic problem the a posteriori error analysis is based on variational analysis residual errors and the adjoint problem to carry out the analysis we use an equivalence between the cell centered finite volume scheme and a mixed finite element method with special choice of quadrature

a note on a decision support system for multiobjective integer and mixed integer programming problems abstract this note presents the main characteristics of a decision support system dss dealing with multiobjective integer and mixed integer programming problems the dss is based on interactive reference point approaches developed by the authors for this kind of problems it is implemented for windows platforms and aims at providing an open communication protocol for interaction with the decision maker s

an interactive algorithm for solving multiple objective integer linear programming problems abstract we propose a reference direction based interactive algorithm to solve multiple objective integer linear programming moilp problems at each iteration of the solution procedure the algorithm finds weak nondominated solutions to the relaxed moilp problem only at certain iterations if the dm so desires an additional mixed integer programming problem is solved to find an integer weak nondominated solution which is close to the current continuous weak nondominated solution to the relaxed moilp problem in the proposed algorithm dm has to provide only the reference point at each iteration no special software is required to implement the proposed algorithm the algorithm is illustrated with an example

an exact e constraint method for bi objective combinatorial optimization problems application to the traveling salesman problem with profits abstract this paper describes an exact constraint method for bi objective combinatorial optimization problems with integer objective values this method tackles multi objective optimization problems by solving a series of single objective subproblems where all but one objectives are transformed into constraints we show in this paper that the pareto front of bi objective problems can be efficiently generated with the constraint method furthermore we describe heuristics based on information gathered from previous subproblems that significantly speed up the method this approach is used to find the exact pareto front of the traveling salesman problem with profits a variant of the traveling salesman problem in which a profit or prize value is associated with each vertex the goal here is to visit a subset of vertices while addressing two conflicting objectives maximize the collected prize and minimize the travel costs we report the first exact results for this problem on instances derived from classical vehicle routing and traveling salesman problem instances with up to 150 vertices results on approximations of the pareto front obtained from a variant of our exact algorithm are also reported

an adaptive discontinuous finite volume method for elliptic problems an adaptive discontinuous finite volume method is developed and analyzed in this paper we prove that the adaptive procedure achieves guaranteed error reduction in a mesh dependent energy norm and has a linear convergence rate numerical results are also presented to illustrate the theoretical analysis

stable spectral methods for conservation laws on triangles with unstructured grids this paper presents an asymptotically stable scheme for the spectral approximation of linear conservation laws defined on a triangle lagrange interpolation on a general two dimensional nodal set is employed and by imposing the boundary conditions weakly through a penalty term the scheme is proven stable in l 2 this result is established for a general unstructured grid in the triangle a special case for which the nodes along the edges of the triangle are chosen as the legendre gauss lobatto quadrature points is discussed in detail the eigenvalue spectrum of the approximation to the advective operator is computed and is shown to result in an o n 2 restriction on the time step when considering explicit time stepping c 1999 elsevier science s a all rights reserved

a review of interactive methods for multiobjective integer and mixed integer programming this paper makes a review of interactive methods devoted to multiobjective integer and mixed integer programming moip momip problems the basic concepts concerning the characterization of the non dominated solution set are first introduced followed by a remark about non interactive methods vs interactive methods then we focus on interactive moip momip methods including their characterization according to the type of preference information required from the decision maker the computing process used to determine non dominated solutions and the interactive protocol used to communicate with the decision maker we try to draw out some contrasts and similarities of the different types of methods

solving 0 1 integer programming problems arising from large scale planning models we present methods that are useful in solving some large scale hierarchical planning models involving 0 1 variables these 0 1 programming problems initially could not be solved with any standard techniques we employed several approaches to take advantage of the hierarchical structure of variables ordered by importance and other structures present in the models critical but not sufficient for success was a strong linear programming formulation we describe methods for strengthening the linear programs as well as other techniques necessary for a commercial branch and bound code to be successful in solving these problems

an interactive weighted tchebycheff procedure for multiple objective programming the procedure samples the efficient set by computing the nondominated criterion vector that is closest to an ideal criterion vector according to a randomly weighted tchebycheff metric using filtering techniques maximally dispersed representatives of smaller and smaller subsets of the set of nondominated criterion vectors are presented at each iteration the procedure has the advantage that it can converge to non extreme final solutions especially suitable for multiple objective linear programming the procedure is also applicable to integer and nonlinear multiple objective programs

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

use of reference points for molp problems analysis abstract two scalar problems intended for analysis of multiple objective linear programming molp problems are considered in the paper these problems use reference points and it is shown how to change the reference point in order to improve the attained value of one criterion

chva tal closures for mixed integer programming problems chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality this viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems the basic ingredient in these cutting plane proofs is that for a polyhedronp and integral vectorw if max wx x p wx integer t thenwx t is valid for all integral vectors inp we consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that bar wx be integer for some other integral vector bar w the cutting plane proofs thus obtained may be seen either as an abstraction of gomory s mixed integer cutting plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by balas and jeroslow our main result is that for a given polyhedronp the set of vectors that satisfy every cutting plane forp with respect to a specified subset of integer variables is again a polyhedron this allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron analogous to the process of repeatedly taking chvatal closures in the integer programming case these results are illustrated with a number of examples from combinatorial optimization our work can be seen as a continuation of that of nemhauser and wolsey on mixed integer cutting planes

using cutting planes in an interactive reference point approach for multiobjective integer linear programming problems we propose an interactive approach for multiple objective integer linear programming moilp problems that combines the use of the tchebycheff metric with cutting plane techniques at each interaction the method computes the nondominated solution for the moilp problem that is closest to a reference point according to the tchebycheff metric the information provided by the decision maker in each dialogue phase is used to adjust the next reference point through a sensitivity analysis stage cutting plane techniques enable the method to take advantage of computations performed at previous iterations to solve the next scalarizing integer program we address both theoretical issues and the computational implementation

a posteriori error analysis for a cut cell finite volume method we study the solution of a diffusive process in a domain where the diffusion coefficient changes discontinuously across a curved interface we consider discretizations that use regularly shaped meshes so that the interface cuts through the cells elements or volumes without respecting the regular geometry of the mesh consequently the discontinuity in the diffusion coefficients has a strong impact on the accuracy and convergence of the numerical method this motivates the derivation of computational error estimates that yield accurate estimates for specified quantities of interest for this purpose we adapt the well known adjoint based a posteriori error analysis technique used for finite element methods in order to employ this method we describe a systematic approach to discretizing a cut cell problem that handles complex geometry in the interface in a natural fashion yet reduces to the well known ghost fluid method in simple cases we test the accuracy of the estimates in a series of examples

a method to calculate numerical errors using adjoint error estimation for linear advection this paper is concerned with the computation of numerical discretization error for uncertainty quantification an a posteriori error formula is described for a functional measurement of the solution to a scalar advection equation that is estimated by finite volume approximations an exact error formula and computable error estimate are derived based on an abstractly defined approximation of the adjoint solution the adjoint problem is divorced from the finite volume method used to approximate the forward solution variables and may be approximated using a low order finite volume method the accuracy of the computable error estimate provably satisfies an a priori error bound for sufficiently smooth solutions of the forward and adjoint problems computational examples are provided that show support of the theory for smooth solutions the application to problems with discontinuities is also investigated computationally

a stable penalty method for the compressible navier stokes equations i open boundary conditions the purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible navier stokes equations in three spatial dimensions the treatment uses the conservation form of the navier stokes equations and utilizes linearization and localization at the boundaries based on these variables the proposed boundary conditions are applied through a penalty procedure thus ensuring correct behavior of the scheme as the reynolds number tends to infinity the versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder

cutting planes for integer programs with general integer variables we investigate the use of cutting planes for integer programs with general integer variables we show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0 1 variables we also explore the use of gomory s mixed integer cuts we address both theoretical and computational issues and show how to embed these cutting planes in a branch and bound framework we compare results obtained by using our cut generation routines in two existing systems with a commercially available branch and bound code on a range of test problems arising from practical applications 1998 the mathematical programming society inc published by elsevier science b v

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

overlapping schwarz methods for fekete and gauss lobatto spectral elements the classical overlapping schwarz algorithm is here extended to the triangular tetrahedral spectral element tsem discretization of elliptic problems this discretization based on fekete nodes is a generalization to nontensorial elements of the tensorial gauss lobatto legendre quadrilateral spectral elements qsem the overlapping schwarz preconditioners are based on partitioning the domain of the problem into overlapping subdomains solving local problems on these subdomains and solving an additional coarse problem associated with either the subdomain mesh or the spectral element mesh the overlap size is generous i e one element wide in the tsem case while it is minimal or variable in the qsem case the results of several numerical experiments show that the convergence rate of the proposed preconditioning algorithm is independent of the number of subdomains n and the spectral degree p in case of generous overlap otherwise it depends inversely on the overlap size the proposed preconditioners are also robust with respect to arbitrary jumps of the coefficients of the elliptic operator across subdomains

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

a high order discontinuous galerkin method for the unsteady incompressible navier stokes equations we present a high order discontinuous galerkin discretization of the unsteady incompressible navier stokes equations in convection dominated flows using triangular and tetrahedral meshes the scheme is based on a semi explicit temporal discretization with explicit treatment of the nonlinear term and implicit treatment of the stokes operator the nonlinear term is discretized in divergence form by using the local lax friedrichs fluxes thus local conservativity is inherent spatial discretization of the stokes operator has employed both equal order p k p k and mixed order p k p k 1 velocity and pressure approximations a second order approximate algebraic splitting is used to decouple the velocity and pressure calculations leading to an algebraic helmholtz equation for each component of the velocity and a consistent poisson equation for the pressure the consistent poisson operator is replaced by an equivalent in stability and convergence operator namely that arising from the interior penalty discretization of the standard poisson operator with appropriate boundary conditions this yields a simpler and more efficient method characterized by a compact stencil size we show the temporal and spatial behavior of the method by solving some popular benchmarking tests for an unsteady stokes problem second order temporal convergence is obtained while for the taylor vortex test problem on both semi structured and fully unstructured triangular meshes spectral convergence with respect to the polynomial degree k is obtained by studying the orr sommerfeld stability problem we demonstrate that the p k p k method yields a stable solution while the p k p k 1 formulation leads to unphysical instability the good performance of the method is further shown by simulating vortex shedding in flow past a square cylinder we conclude that the proposed discontinuous galerkin method with the p k p k formulation is an efficient scheme for accurate and stable solution of the unsteady navier stokes equations in convection dominated flows

tensor product gauss lobatto points are fekete points for the cube tensor products of gauss lobatto quadrature points are frequently used as collocation points in spectral element methods unfortunately it is not known if gauss lobatto points exist in non tensor product domains like the simplex in this work we show that the n dimensional tensor product of gauss lobatto quadrature points are also fekete points this suggests a way to generalize spectral methods based on gauss lobatto points to non tensor product domains since fekete points are known to exist and have been computed in the triangle and tetrahedron in one dimension this result was proved by fejer in 1932 but the extension to higher dimensions in non trivial

nurbs enhanced finite element method nefem the development of nurbs enhanced finite element method nefem is revisited this technique allows a seamless integration of the cad boundary representation of the domain and the finite element method fem the importance of the geometrical model in finite element simulations is addressed and the benefits and potential of nefem are discussed and compared with respect to other curved finite element techniques

a posteriori error analysis of imex multi step time integration methods for advection diffusion reaction equations abstract implicit explicit imex schemes are an important and widely used class of time integration methods for both parabolic and hyperbolic partial differential equations we develop accurate a posteriori error estimates for a user defined quantity of interest for two classes of multi step imex schemes for advection diffusion reaction problems the analysis proceeds by recasting the imex schemes into a variational form suitable for a posteriori error analysis employing adjoint problems and computable residuals the a posteriori estimates quantify distinct contributions from various aspects of the spatial and temporal discretizations and can be used to evaluate discretization choices numerical results are presented that demonstrate the accuracy of the estimates for a representative set of problems

recursive parameter free explicitly defined interpolation nodes for simplices a rule for constructing interpolation nodes for n th degree polynomials on the simplex is presented these nodes are simple to define recursively from families of 1d node sets such as the lobatto

multi objective vehicle routing problems routing problems such as the traveling salesman problem and the vehicle routing problem are widely studied both because of their classic academic appeal and their numerous real life applications similarly the field of multi objective optimization is attracting more and more attention notably because it offers new opportunities for defining problems this article surveys the existing research related to multi objective optimization in routing problems it examines routing problems in terms of their definitions their objectives and the multi objective algorithms proposed for solving them

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

a finite volume method for one dimensional reaction diffusion problems abstract a truly conservative finite volume method for one dimensional reaction diffusion equations is presented the method is based on piecewise second degree polynomials approximations that result in linear fluxes a symmetric mass matrix and a semi discrete system of nonlinear ordinary differential equations in time it is shown that the semi discrete method is linearly stable provided that the problem is mathematically well posed it is also shown that implicit discretizations of the time variable are stable whereas the explicit one is conditionally stable and has a smaller domain of linear stability than the standard explicit finite difference method it is also illustrated that upon lumping the mass matrix the method coincides with a finite difference discretization of the original problem a detailed formulation of the dirichlet neumann and robin boundary conditions is presented such a formulation makes use of control volumes which are here referred to as incomplete ones although the finite volume technique presented here uses a node centered formulation it can be easily extended to cell centered ones the technique is also developed for multidimensional problems

a stable penalty method for the compressible navier stokes equations iii multidimensional domain decomposition schemes this paper concluding the trilogy develops schemes for the stable solution of wave dominated unsteady problems in general three dimensional domains the schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established the complex computational domains are constructed by using nonoverlapping quadrilaterals in the two dimensional case and hexahedrals in the three dimensional space to illustrate the ideas underlying the multidomain method a stable scheme for the solution of the three dimensional linear advection diffusion equation in general curvilinear coordinates is developed the analysis suggests a novel yet simple stable treatment of geometric singularities like edges and vertices the theoretical results are supported by a two dimensional implementation of the scheme the main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible navier stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume with special attention given to a hexahedral domain the treatment includes a stable procedure for dealing with boundary conditions at a solid wall the efficacy of the scheme for the compressible navier stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions the results are found to agree with the solution of the compressible boundary layer equations