numerical investigation on the stability of singular driven cavity flow by applying the singularity subtraction technique to the unsteady driven cavity problem the stability of the impulsively started flow is investigated without smoothing the corner singularity a second order spectral projection method allows localization of the critical reynolds number for the first hopf bifurcation in the interval 8017 6 8018 8

a new class of cutting planes for the symmetric travelling salesman problem a comprehensive class of cutting planes for the symmetric travelling salesman problem tsp is proposed which contains the known comb inequalities the path inequalities and the 3 star constraints as special cases its relation to the clique tree inequalities is discussed the cutting planes are shown to be valid for a relaxed version of the tsp the travelling salesman problem on a road network and under certain conditions to define facets of the polyhedron associated with this problem

a note on the undirected rural postman problem polytope this note refers to the article by g ghiani and g laporte a branch and cut algorithm for the undirected rural postman problem math program 87 2000 we show that some conditions for the facet defining property of the basic non trivial inequalities are not sufficient and that the rural postman problem polytope is more complex even when focusing on canonical inequalities only

inequalities for the generalized elliptic integrals and modular functions abstract in this paper the authors study monotonicity and convexity of the generalized elliptic integrals and certain combinations of these special functions such as m a r and a r making use of these results the authors obtain some sharp inequalities for the so called ramanujan s generalized modular functions

the symmetric traveling salesman polytope and its graphical relaxation composition of valid inequalities the graphical relaxation of the traveling salesman problem is the relaxation obtained by requiring that the salesman visit each city at least once instead of exactly once this relaxation has already led to a better understanding of the traveling salesman polytope in cornuejols fonlupt and naddef 1985 we show here how one can compose facet inducing inequalities for the graphical traveling salesman polyhedron and obtain other facet inducing inequalities this leads to new valid inequalities for the symmetric traveling salesman polytope this paper is the first of a series of three papers on the symmetric traveling salesman polytope the next one studies the strong relationship between that polytope and its graphical relaxation and the last one applies all the theoretical developments of the two first papers to prove some new facet inducing results

on the cut polytope the cut polytopep c g of a graphg v e is the convex hull of the incidence vectors of all edge sets of cuts ofg we show some classes of facet defining inequalities ofp c g we describe three methods with which new facet defining inequalities ofp c g can be constructed from known ones in particular we show that inequalities associated with chordless cycles define facets of this polytope moreover for these inequalities a polynomial algorithm to solve the separation problem is presented we characterize the facet defining inequalities ofp c g ifg is not contractible tok 5 we give a simple characterization of adjacency inp c g and prove that for complete graphs this polytope has diameter one and thatp c g has the hirsch property a relationship betweenp c g and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied

numerical methods for the stokes and navier stokes equations driven by threshold slip boundary conditions abstract in this article we discuss the numerical solution of the stokes and navier stokes equations completed by nonlinear slip boundary conditions of friction type in two and three dimensions to solve the stokes system we first reduce the related variational inequality into a saddle point point problem for a well chosen augmented lagrangian to solve this saddle point problem we suggest an alternating direction method of multiplier together with finite element approximations the solution of the navier stokes system combines finite element approximations time discretization by operator splitting and augmented lagrangian method numerical experiment results for two and three dimensional flow confirm the interest of these approaches

accurate approximations for the complete elliptic integral of the second kind abstract in this paper we prove that the double inequality s 11 4 7 4 1 r e r s 11 4 7 4 1 r holds for all r 0 1 if and only if 2 1 570796 and 11 7 1 571428 where r 1 r 2 1 2 e r 0 2 1 r 2 sin 2 t d t is the complete elliptic integral of the second kind and s p q a b q a p b p p a q b q 1 p q is the stolarsky mean of a and b

the general routing problem polyhedron facets from the rpp and gtsp polyhedra in this paper we study the polyhedron associated with the general routing problem grp this problem first introduced by orloff in 1974 is a generalization of both the rural postman problem rpp and the graphical traveling salesman problem gtsp and thus is np hard we describe a formulation of the problem such that from every non trivial facet inducing inequality for the rpp and gtsp polyhedra we obtain facet inducing inequalities for the grp polyhedron we describe a new family of facet inducing inequalities for the grp the honeycomb constraints which seem to be very useful for solving grp and rpp instances finally new classes of facets obtained by composition of facet inducing inequalities are presented

a polyhedral approach to the rural postman problem abstract in this paper we study the polyhedron associated with the rural postman problem rpp because the rpp is np hard we cannot expect to find a complete description of the rural postman polyhedron of a general graph but a partial knowledge of such a description frequently proves to be useful for both theoretical and computational purposes we have tried to characterize the facial structure of this unbounded full dimensional polyhedron sets of valid inequalities inducing facets have been studied as well as their use in a cutting plane algorithm the application of this algorithm to a set of rpp instances taken from the literature and two instances of larger size taken from a real world graph is described all these instances were solved to optimality

odd minimum cut sets and b matchings we show that the determination of a minimum cut set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of gomory and hu for multi terminal networks we connect this problem to the problem of identifying a matching or blossom constraint that chops off a point which is not contained in the convex hull of matchings or proving that no such inequality exists both the b matching problems without and with upper bounds are considered we discuss how the results of this paper can be used in conjunction with commercial lp packages lo solve b matching problems

the maximum benefit chinese postman problem and the maximum benefit traveling salesman problem abstract this paper introduces the maximum benefit chinese postman problem mbcpp and the maximum benefit traveling salesman problem mbtsp for the mbcpp a benefit is realized each time a link of a directed graph is traversed the mbcpp relaxes the constraint of the chinese postman problem that each link be traversed at least once and finds a tour of maximum total net benefit for the mbtsp a benefit is derived when a node is visited and a cost is incurred when a link is traversed the mbtsp relaxes the constraint of the traveling salesman problem that each node be visited exactly or at least once and finds a tour of maximum total net benefit both problems are formulated as linear integer programming problems a solution algorithm is given that repeatedly solves a relaxation of the formulation as a minimum cost flow problem using a branch and bound procedure computational examples are presented

turan type inequalities for generalized complete elliptic integrals in this paper our aim is to establish some turan type inequalities for gaussian hypergeometric functions and for generalized complete elliptic integrals these results complete the earlier result of p turan proved for legendre polynomials moreover we show that there is a close connection between a turan type inequality and a sharp lower bound for the generalized complete elliptic integral of the first kind at the end of this paper we prove a recent conjecture of t sugawa and m vuorinen related to estimates of the hyperbolic distance of the twice punctured plane

the traveling salesman problem on a graph and some related integer polyhedra given a graphg n e and a length functionl e the graphical traveling salesman problem is that of finding a minimum length cycle goingat least once through each node ofg this formulation has advantages over the traditional formulation where each node must be visited exactly once we give some facet inducing inequalities of the convex hull of the solutions to that problem in particular the so called comb inequalities of grotschel and padberg are generalized some related integer polyhedra are also investigated finally an efficient algorithm is given wheng is a series parallel graph

asymptotical bounds for complete elliptic integrals of the second kind abstract in this paper we establish several asymptotical bounds for the complete elliptic integrals of the second kind e r and improve the well known conjecture e r 1 1 r 2 3 4 2 2 3 2 for all r 0 1 proposed by m vuorinen

a cutting plane algorithm for the general routing problem the general routing problem grp is the problem of finding a minimum cost route for a single vehicle subject to the condition that the vehicle visits certain vertices and edges of a network it contains the rural postman problem chinese postman problem and graphical travelling salesman problem as special cases we describe a cutting plane algorithm for the grp based on facet inducing inequalities and show that it is capable of providing very strong lower bounds and in most cases optimal solutions

the rural postman problem with deadline classes vehicle routing problems with general time windows are extremely difficult to solve however the time windows in a particular problem may have a special structure which can be exploited we consider a single vehicle arc routing problem in which the arcs are partitioned into deadline classes it is shown that a cutting plane approach works well for this problem

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

driven cavity flows by efficient numerical techniques abstract efficient and reliable numerical techniques of high order accuracy are presented for solving problems of steady viscous incompressible flow in the plane and are used to obtain accurate solutions for the driven cavity a solution is obtained at reynolds number 10 000 on a 180 180 grid the numerical methods combine an efficient linear system solver an adaptive newton like method for nonlinear systems and a continuation procedure for following a branch of solutions over a range of reynolds numbers

on the undirected rural postman problem tight bounds based on a new formulation the rural postman problem rpp is a classic edge routing problem a mathematical programming formulation of the rpp that differs fundamentally from those in the literature was introduced but not tested computationally by garfinkel and webb 1999 a rudimentary algorithm that yields lower bounds via cutting planes and upper bounds via heuristics is developed and tested for a variation of that formulation computational results are encouraging especially in terms of the relatively small number of added inequalities needed to get good lower bounds and the fact that the vast majority of these have efficient exact separation procedures note that a first algorithm based on this new formulation is computationally competitive allowing the possibility of far more efficient and complex future realizations

a differentially interpolated direct forcing immersed boundary method for predicting incompressible navier stokes equations in time varying complex geometries a dispersion relation preserving dual compact scheme developed in cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time varying domains the artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body we develop in this study a differential based interpolation scheme which can be easily extended to three dimensional simulation the results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems the accuracy and fidelity of the ib flow solver developed to predict flows with irregular boundaries are therefore demonstrated

new results on the mixed general routing problem in this paper we deal with the polyhedral description and the resolution of the mixed general routing problem this problem in which the service activity occurs both at some of the nodes and at some of the arcs and edges of a mixed graph contains a large number of important arc and node routing problems as special cases here a large family of facet defining inequalities the honeycomb inequalities is described furthermore a cutting plane algorithm for this problem that incorporates new separation procedures for the k c regular path bridge and honeycomb inequalities is presented branch and bound is invoked when the final solution of the cutting plane procedure is fractional extensive computational experiments over different sets of instances are included

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

solving the prize collecting rural postman problem abstract in this work we present an algorithm for solving the prize collecting rural postman problem this problem was recently defined and is a generalization of other arc routing problems like for instance the rural postman problem the main difference is that there are no required edges instead there is a profit function on the edges that must be taken into account only the first time that an edge is traversed the problem is modeled as a linear integer program where the system has an exponential number of inequalities we propose a solution algorithm where iteratively we solve relaxed models with a small number of inequalities that provide upper bounds and we propose exact separation procedures for generating violated cuts when possible we also propose a simple heuristic to generate feasible solutions that provide lower bounds at each iteration we use a two phase method with different solvers at each phase despite the difficulty of the problem the numerical results from a series of computational experiments with various types of instances assess the good behavior of the algorithm in particular 75 of the instances were optimally solved with the lp relaxation of the model the remaining instances were optimally solved on a second phase most of them in small computation times

concavity of the complete elliptic integrals of the second kind with respect to holder means abstract in this paper we establish a necessary and sufficient condition for the concavity of the complete elliptic integrals of the second kind with respect to holder means

a branch and cut algorithm for the profitable windy rural postman problem abstract in this paper we study the profitable windy rural postman problem this is an arc routing problem with profits defined on a windy graph in which there is a profit associated with some of the edges of the graph consisting of finding a route maximizing the difference between the total profit collected and the total cost this problem generalizes the rural postman problem and other well known arc routing problems and has real life applications mainly in snow removal operations we propose here a formulation for the problem and study its associated polyhedron several families of facet inducing inequalities are described and used in the design of a branch and cut procedure the algorithm has been tested on a large set of benchmark instances and compared with other existing algorithms the results obtained show that the branch and cut algorithm is able to solve large sized instances optimally in reasonable computing times

an effective pso inspired algorithm for the team orienteering problem abstract the team orienteering problem top is a particular vehicle routing problem in which the aim is to maximize the profit gained from visiting customers without exceeding a travel cost time limit this paper proposes a new and fast evaluation process for top based on an interval graph model and a particle swarm optimization inspired algorithm psoia to solve the problem experiments conducted on the standard benchmark of top clearly show that our algorithm outperforms the existing solving methods psoia reached a relative error of 0 0005 whereas the best known relative error in the literature is 0 0394 our algorithm detects all but one of the best known solutions moreover a strict improvement was found for one instance of the benchmark and a new set of larger instances was introduced

solving the hierarchical chinese postman problem as a rural postman problem abstract in the undirected hierarchical chinese postman problem hcpp the edges of a graph are partitioned into clusters and must be serviced while respecting a hierarchy or precedence relation between clusters two objectives are considered a hierarchical objective and a makespan objective in this article a transformation of the hcpp into an equivalent rural postman problem rpp is presented the hcpp is solved optimally for both objectives by applying an exact branch and cut rpp algorithm to the transformed problem two heuristics based on the rpp algorithm are also described and assessed computationally

the hierarchical mixed rural postman problem polyhedral analysis and a branch and cut algorithm abstract the hierarchical mixed rural postman problem is defined on a mixed graph where arcs and edges that require a service are divided into clusters that have to be serviced in a hierarchical order the problem generalizes the mixed rural postman problem and thus is np hard in this paper we provide a polyhedral analysis of the problem and propose a branch and cut algorithm for its solution based on the introduced classes of valid inequalities extensive computational experiments are reported on benchmark instances the exact approach allows to find the optimal solutions in less than 1 hour for instances with up to 999 vertices 2678 links and five clusters

generalized hersch pfluger distortion function and complete elliptic integrals abstract in this paper we study the relationship between the generalized hersch pfluger distortion function k a r and complete elliptic integrals of the second kind e r and establish several inequalities for k a r and e r

a least squares finite element method for the numerical solution of the navier stokes cahn hilliard system modeling the motion of the contact line in this article we discuss the numerical solution of the navier stokes cahn hilliard system modeling the motion of the contact line separating two immiscible incompressible viscous fluids near a solid wall the method we employ combines a finite element space approximation with a time discretization by operator splitting to solve the cahn hilliard part of the problem we use a least squares conjugate gradient method we also show that the scheme has the total energy decaying in time property under certain conditions our numerical experiments indicate that the method discussed here is accurate stable and efficient

lower bounds and heuristics for the windy rural postman problem in this paper we present several heuristic algorithms and a cutting plane algorithm for the windy rural postman problem this problem contains several important arc routing problems as special cases and has very interesting real life applications extensive computational experiments over different sets of instances are also presented

the general routing polyhedron a unifying framework it is shown how to transform the general routing problem grp into a variant of the graphical travelling salesman problem gtsp this transformation yields a projective characterisation of the grp polyhedron using this characterisation it is shown how to convert facets of the gtsp polyhedron into valid inequalities for the grp polyhedron the resulting classes of inequalities subsume several previously published classes

inequalities for means abstract a monotone form of l hospital s rule is obtained and applied to derive inequalities between the arithmetic geometric mean of gauss the logarithmic mean and stolarsky s identric mean some related inequalities are given for complete elliptic integrals

a branch and cut algorithm for the undirected rural postman problem the well known undirected rural postman problem is considered and a binary linear problem using new dominance relations is presented polyhedral properties are investigated and a branch and cut algorithm is developed extensive computational results indicate that the algorithm is capable of solving much larger instances than previously reported

on approximating the arithmetic geometric mean and complete elliptic integral of the first kind abstract in the article we prove that the double inequalities 1 6 p 7 r p 5 p 6 r tanh 1 r 2 r k r 1 6 q 7 r q 5 q 6 r tanh 1 r 2 r q a 1 r 5 q 6 g 1 r a 1 r 6 q 7 g 1 r l 1 r a g m 1 r p a 1 r 5 p 6 g 1 r a 1 r 6 p 7 g 1 r l 1 r hold for all r 0 1 if and only if p 2 1 570796 and q 89 69 1 289855 where k r 0 2 1 r 2 sin 2 t 1 2 d t is the complete elliptic integral of the first kind tanh 1 r log 1 r 1 r 2 is the inverse hyperbolic tangent function r 1 r 2 and a 1 r 1 r 2 g 1 r r l 1 r r 1 log r and a g m 1 r are the arithmetic geometric logarithmic and gaussian arithmetic geometric means of 1 and r respectively

a branch and cut algorithm for the maximum benefit chinese postman problem the maximum benefit chinese postman problem mbcpp is an np hard problem that considers several benefits associated with each edge one for each time the edge is traversed with a service the objective is to find a closed walk with maximum benefit we propose an ip formulation for the undirected mbcpp and based on the description of its associated polyhedron we propose a branch and cut algorithm and present computational results on instances with up to 1 000 vertices and 3 000 edges

transformation of facets of the general routing problem polytope we study the class of polytopes associated with the 0 1 variable formulation by g ghiani and g laporte math program 87 2000 pp 467 481 of the rural postman problem and general routing problem we introduce a technique called switching which allows one to obtain new valid or facet defining inequalities from known ones we study the switched forms which can be obtained from two important types of known valid inequalities namely so called path bridge a n letchford eur j oper res 96 1997 pp 317 322 and honeycomb a corberan and j m sanchis eur j oper res 108 1998 pp 538 550 inequalities for the switched path bridge inequalities a generic blueprint for separation routines is developed we give a polynomial time separation algorithm for a special subclass of these inequalities

objectivity tests for navier stokes simulations the revealing of non physical solutions produced by laplace formulations abstract laplace formulations are weak formulations of the navier stokes equations commonly used in computational fluid dynamics in these schemes the viscous terms are given as a function of the laplace diffusion operator only despite their popularity recently it has been proven that they violate a fundamental principle of continuum mechanics the principle of objectivity it is remarkable that such flaw has not being noticed before neither detected in numerical experiments in this work a series of objectivity tests have been designed with the purpose of revealing such problem in real numerical experiments through the tests it is shown how for slip boundaries or free surfaces laplace formulations generate non physical solutions which widely depart from the real fluid dynamics these tests can be easily reproduced not requiring complex simulation tools furthermore they can be used as benchmarks to check consistency of developed or commercial software the article is closed with a discussion of the mathematical aspects involved including the issues of boundary conditions and objectivity

monotonicity theorems and inequalities for the complete elliptic integrals we prove monotonicity properties of certain combinations of complete elliptic integrals of the first and second kind k and e these results lead to sharp symmetrical bounds for k and e which improve recently discovered inequalities

sharp approximations for the complete elliptic integrals of the second kind by one parameter means abstract let e r be the complete elliptic integrals of the second kind and j p a b be the one parameter mean of two distinct positive numbers a and b defined by j p a b p p 1 a p 1 b p 1 a p b p for p 1 0 in this study using the recurrence method with the sign rule of a special power series we prove that the double inequality 7 22 j 2 2 1 r 2 e r j 2 2 1 r holds for r 1 r 2 0 1 where the upper bound is sharp while the lower bound is related to the archimedes approximation 22 7 for using the same method we also easily show that both of the functions r 2 2 e r j 7 4 1 r r 12 and r 2 1 r 10 1 1 2 e r 1 j 7 4 1 r are absolutely monotonic on 0 1 which greatly improves the known results moreover it should be emphasized that our method may be used to explore special functions related to hypergeometric functions

wall driven incompressible viscous flow in a two dimensional semi circular cavity the main goal of this article is to investigate the capability of an operator splitting finite elements based methodology at handling accurately incompressible viscous flow at large reynolds number re in regions with corners and curved boundaries to achieve this goal the authors have selected a wall driven flow in a semi circular cavity on the basis of the numerical experiments reported in this article it seems that the method under investigation has no difficulty at capturing the formation of primary secondary and tertiary vortices as re increases it has also the capability of identifying a hopf bifurcation phenomenon taking place around re 6600

inequalities for zero balanced hypergeometric functions the authors study certain monotoneity and convexity properties of the gaussian hypergeometric function and those of the euler gamma function

exploring variants of 2 opt and 3 opt for the general routing problem the general routing problem grp is the problem of finding a minimum length tour visiting a number of specified vertices and edges in an undirected graph in this paper we describe how the well known 2 opt and 3 opt local search procedures for node routing problems can be adapted to solve arc and general routing problems successfully two forms of the 2 opt and 3 opt approaches are applied to the grp the first version is similar to the conventional approach for the traveling salesman problem the second version includes a dynamic programming procedure and explores a larger neighborhood at the expense of higher running times extensive computational tests including ones on larger instances than previously reported in the arc routing literature are performed with variants of both algorithms in combination with the guided local search metaheuristic and the mechanisms of marking and neighbor lists the procedures systematically detect optimal or high quality solutions within limited computation time

stable loosely coupled type algorithm for fluid structure interaction in blood flow we introduce a novel loosely coupled type algorithm for fluid structure interaction between blood flow and thin vascular walls this algorithm successfully deals with the difficulties associated with the added mass effect which is known to be the cause of numerical instabilities in fluid structure interaction problems involving fluid and structure of comparable densities our algorithm is based on a time discretization via operator splitting which is applied in a novel way to separate the fluid sub problem from the structure elastodynamics sub problem in contrast with traditional loosely coupled schemes no iterations are necessary between the fluid and structure sub problems this is due to the fact that our novel splitting strategy uses the added mass effect to stabilize rather than to destabilize the numerical algorithm this stabilizing effect is obtained by employing the kinematic lateral boundary condition to establish a tight link between the velocities of the fluid and of the structure in each sub problem the stability of the scheme is discussed on a simplified benchmark problem and we use energy arguments to show that the proposed scheme is unconditionally stable due to the crucial role played by the kinematic lateral boundary condition the proposed algorithm is named the kinematically coupled scheme

new results for the directed profitable rural postman problem abstract we study a generalization of the directed rural postman problem where not all arcs requiring a service have to be visited provided that a penalty cost is paid if a service arc is not crossed the problem known as directed profitable rural postman problem looks for a tour visiting the selected set of service arcs while minimizing both traveling and penalty costs we add different valid inequalities to a known mathematical formulation of the problem and develop a branch and cut algorithm that introduces connectivity constraints both in a lazy and in a standard way we also propose a matheuristic followed by an improvement heuristic final refinement the matheuristic exploits information provided by a problem relaxation to select promising service arcs used to solve optimally directed rural postman problems the ex post refinement tries to improve the solution provided by the matheuristic using a branch and cut algorithm the method gets a quick convergence through the introduction of connectivity cuts that are not guaranteed to be valid inequalities and thus may exclude integer feasible solutions all proposed methods have been tested on benchmark instances found in literature and compared to state of the art algorithms results show that heuristic methods are extremely effective outperforming existing algorithms moreover our exact method is able to close in less than one hour all the 22 benchmark instances that have not been solved to optimality yet

remarks on generalized elliptic integrals we study the monotonicity for certain combinations of generalized elliptic integrals thus generalizing analogous well known results for classical complete elliptic integrals and prove a conjecture put forward by heikkala vamanamurthy and vuorinen

finite element approximation of the nonstationary navier stokes problem part ii stability of solutions and error estimates uniform in time in this paper assumptions about the stability of a solution are introduced in the numerical analysis of the nonstationary navier stokes problem for the purpose of extending local a priori error estimates and local a posteriori error estimates globally in time