analysis of the parareal time parallel time integration method the parareal algorithm is a method to solve time dependent problems parallel in time it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time in this paper the relation of the parareal algorithm to space time multigrid and multiple shooting methods is first briefly discussed the focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals and linear convergence for unbounded intervals

gromov witten invariants of jumping curves given a vector bundle e on a smooth projective variety x we can define subschemes of the kontsevich moduli space of genus zero stable maps m 0 0 x parameterizing maps f p 1 x such that the grothendieck decomposition of f e has a specified splitting type in this paper using a compactification of this locus we define gromov witten invariants of jumping curves associated to the bundle e we compute these invariants for the tautological bundle of grassmannians and the horrocks mumford bundle on p 4 our construction is a generalization of jumping lines for vector bundles on p since for the tautological bundle of the grassmannians the invariants are enumerative we resolve the classical problem of computing the characteristic numbers of unbalanced scrolls

mutations of puzzles and equivariant cohomology of two step flag varieties we introduce a mutation algorithm for puzzles that is a three direction analogue of the classical jeu de taquin algorithm for semistandard tableaux we apply this algorithm to prove our conjectured puzzle formula for the equivariant schubert structure constants of two step flag varieties this formula gives an expression for the structure constants that is positive in the sense of graham thanks to the equivariant version of the quantum equals classical result our formula specializes to a littlewood richardson rule for the equivariant quantum cohomology of grassmannians

stable parareal in time method for first and second order hyperbolic systems the parareal in time algorithm allows one to perform parallel simulations of time dependent problems this algorithm has been implemented on many types of time dependent problems with some success recent contributions have allowed one to extend the domain of application of the parareal in time algorithm so as to handle long time simulations of hamiltonian systems this improvement has managed to avoid the fatally large lack of accuracy of the plain parareal in time algorithm which does not conserve invariant quantities a somewhat similar difficulty occurs for problems where the solution lacks regularity either initially or during the evolution as is the case for hyperbolic systems of conservation laws in this paper we identify the reasons for instabilities of the parareal in time algorithm and propose a simple way to cure them we use the new method to solve a linear wave equation and a nonlinear burgers equation the results illustrate the stability of this variant of the parareal in time algorithm

grid refinement for lattice bgk models lattice boltzmann models proposed at the end of the 1980s as the noise free version of lattice gas models are based on gas kinetic representation of fluid flow their recent modifications the lattice bgk models provide especially simple effective and stable algorithms for the solution of hydrodynamical problems a local second order grid refinement scheme for the lattice bgk model is proposed in this work the refinement scheme and a boundary fitting scheme for complicated geometries are applied to simulate a benchmark problem of flow past a cylinder in a channel with small and moderate reynolds numbers

analysis for parareal algorithms applied to hamiltonian differential equations long time integrations are an important issue in the numerical solution of hamiltonian systems they are time consuming and it is natural to consider the use of parallel architectures for reasons of efficiency in this context the parareal algorithm has been proposed by several authors the present work is a theoretical study of the parareal algorithm when it is applied to hamiltonian differential equations the idea of backward error analysis is employed to get insight into the long time behavior of numerical approximations one of the main results is that convergence of the parareal iterations restricts the length of the time window for nearly integrable systems its length is bounded by the square root of the inverse of the accuracy of the coarse integrator the theoretical bounds are confirmed by numerical experiments

event based parareal a data flow based implementation of parareal parareal is an iterative algorithm that in effect achieves temporal decomposition for a time dependent system of differential or partial differential equations a solution is obtained in a shorter wall clock time but at the expense of increased compute cycles the algorithm combines a fine solver that solves the system to acceptable accuracy with an approximate coarse solver the critical task for the successful implementation of parareal on any system is the development of a coarse solver that leads to convergence in a small number of iterations compared to the number of time slices in the full time interval and is at the same time much faster than the fine solver very fast coarse solvers may not lead to sufficiently rapid convergence and slow coarse solvers may not lead to significant gains even if the number of iterations to convergence is satisfactory we find that the difficulty of meeting these conflicting demands can be substantially eased by using a data driven event based implementation of parareal as a result tasks for one iteration do not wait for the previous iteration to complete but are started when the needed data are available for given convergence properties the event based approach relaxes the speed requirements on the coarse solver by a factor of k where k is the number of iterations required for a converged solution this may for many problems lead to an efficient parareal implementation that would otherwise not be possible or would require substantial coarse solver development in addition the framework used for this implementation executes a task when the data dependencies are satisfied and computational resources are available this leads to improved computational efficiency over previous approaches that pipeline or schedule groups of tasks to a particular processor or group of processors

parallel algorithms for initial value problems for difference and differential equations abstract let yn be a trajectory defined sequentially by a nonlinear vector difference equation yn 1 fn 1 yn with y0 known a steffensen like iterative method is proposed which starts from a guessed sequence un 0 for the approximation of yn and allows certain computations to be performed in parallel the sequence u k 1 n is obtained from u k n by means of a formula of the form u k 1 n 1 v k 1 n 1 k 1 n 1 u k 1 n u k n here the vectors v k 1 n 1 and the matrices k 1 n 1 each require a function evaluation but can be computed in parallel with respect to n in a suitable interval nk mk the length of which depends on the number of processors available therefore for the algorithm to serve effectively the steffensen iteration must converge quickly and the machine used must possess a large number of processors finally note that the theory includes the case that the sequence yn is given by a one step ode solver

dense pgl orbits in products of grassmannians abstract in this paper we find some necessary and sufficient conditions on the dimension vector d d 1 d k n so that the diagonal action of p gl n on i 1 k gr d i n has a dense orbit consequently we obtain some algorithms for finding dense and sparse dimension vectors and classify dense dimension vectors with small length or size we also classify dimension vectors where d i d j 3 for all i j generalizing a theorem of popov 11

a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

quantum and affine schubert calculus and macdonald polynomials abstract we definitively establish that the theory of symmetric macdonald polynomials aligns with quantum and affine schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong bruhat order poset on the type a affine weyl group we construct two one parameter families of functions that respectively transition positively with hall littlewood and macdonald s p functions and specialize to the representatives for schubert classes of homology and cohomology of the affine grassmannian our approach leads us to conjecture that all elements in a defining set of 3 point genus 0 gromov witten invariants for flag manifolds can be formulated as strong covers

parallel in time approximation of the lattice boltzmann method for laminar flows abstract fluid dynamics simulations using grid based methods such as the lattice boltzmann equation can benefit from parallel in space computation however for a fixed size simulation of this type the efficiency of larger processor counts will saturate when the number of grid points per core becomes too small to overcome this fundamental strong scaling limit in space parallel approaches we present a novel time parallel version of the lattice boltzmann method using the parareal algorithm this method is based on a predictor corrector scheme combined with mesh refinement to enable the simulation of larger number of time steps we present results of up to a 32 increase in speed for a model system consisting of a cylinder with conditions for laminar flow the parallel gain obtainable is predicted with strong accuracy providing a quantitative understanding of the potential impact of this method

theory and applications of an alternative lattice boltzmann grid refinement algorithm this contribution proposes an alternative lattice boltzmann grid refinement algorithm that overcomes the drawbacks that plague existing approaches we demonstrate that this algorithm is accurate and applicable for all values of the relaxation time we also show that this algorithm can significantly speed up the flow settlement process by using a hierarchy of grid levels the stationary regime can be approached up to a thousand times faster than with a single grid resolution

gromov witten invariants on grassmannians we prove that any three point genus zero gromov witten invariant on a type a grassmannian is equal to a classical intersection number on a two step flag variety we also give symplectic and orthogonal analogues of this result in these cases the two step flag variety is replaced by a sub maximal isotropic grassmannian our theorems are applied in type a to formulate a conjectural quantum littlewood richardson rule and in the other classical lie types to obtain new proofs of the main structure theorems for the quantum cohomology of lagrangian and orthogonal grassmannians

k schur expansions of catalan functions abstract we make a broad conjecture about the k schur positivity of catalan functions symmetric functions which generalize the parabolic hall littlewood polynomials we resolve the conjecture with positive combinatorial formulas in cases which address the k schur expansion of 1 hall littlewood polynomials proving the q 0 case of the strengthened macdonald positivity conjecture from 24 2 the product of a schur function and a k schur function when the indexing partitions concatenate to a partition describing a class of gromov witten invariants for the quantum cohomology of complete flag varieties 3 k split polynomials solving a substantial special case of a problem of broer and shimozono weyman on parabolic hall littlewood polynomials 37 in addition we prove the conjecture that the k schur functions defined via k split polynomials 25 agree with those defined in terms of strong tableaux 21

an algorithm with polylog parallel complexity for solving parabolic partial differential equations the standard numerical algorithms for solving parabolic partial differential equations are inherently sequential in the time direction this paper describes an algorithm for the time accurate solution of certain classes of parabolic partial differential equations that can be parallelized in both time and space it has a serial complexity that is proportional to the serial complexities of the best known algorithms the algorithm is a variant of the multigrid waveform relaxation method where the scalar ordinary differential equations that make up the kernel of computation are solved using a cyclic reduction type algorithm experimental results obtained on a massively parallel multiprocessor are presented

a theoretical framework for proving accuracy results for deferred corrections general techniques are described for proving accuracy results for deferred correction solutions to differential equations these techniques apply also to computational estimates of the local discretization error the proofs avoid the necessity of demonstrating the existence of asymptotic expansions of the global error in powers of some meshsize parameter

communication aware adaptive parareal with application to a nonlinear hyperbolic system of partial differential equations in the strong scaling limit the performance of conventional spatial domain decomposition techniques for the parallel solution of pdes saturates when sub domains become small halo communication and other overheard come to dominate a potential path beyond this scaling limit is to introduce domain decomposition in time with one such popular approach being the parareal algorithm which has received a lot of attention due to its generality and potential scalability low efficiency particularly on convection dominated problems has however limited the adoption of the method in this paper we introduce a new strategy communication aware adaptive parareal caap to overcome some of the challenges with caap we choose time subdomains short enough that convergence of the parareal algorithm is quick yet long enough that the overheard of communicating time subdomain interfaces does not induce a new limit to parallel speed up furthermore we propose an adaptive work scheduling algorithm that overlaps consecutive parareal cycles and decouples the number of time subdomains and number of active node groups in an efficient manner to allow for comparatively high parallel eciency we demonstrate the viability of caap trough the parallel in time integration of a hyperbolic system of pdes in the form of the two dimensional nonlinear shallow water wave equation solved using a 3rd order accurate weno rk discretization for the computational cheap approximate operator needed as a preconditioner in the parareal corrections we use a lower order roe type discretization time parallel integration of purely hyperbolic type evolution problems is traditionally considered impractical trough large scale numerical experiments we demonstrate that with caap it is possible not only to obtain time parallel speedup on this class of evolution problems but also that we may obtain parallel acceleration beyond what is possible using conventional spatial domain decomposition techniques alone the approach is widely applicable for parallel in time integration over long time domains regardless of the class of evolution problem

root theoretic young diagrams and schubert calculus planarity and the adjoint varieties abstract we study root theoretic young diagrams to investigate the existence of a lie type uniform and nonnegative combinatorial rule for schubert calculus we provide formulas for co adjoint varieties of classical lie type this is a simplest case after the co minuscule family where a rule has been proved by h thomas and the second author using work of r proctor our results build on earlier pieri type rules of p pragacz j ratajski and of a buch a kresch h tamvakis specifically our formula for og 2 2 n is the first complete rule for a case where diagrams are non planar yet the formulas possess both uniform and non uniform features using these classical type rules as well as results of p e chaput n perrin in the exceptional types we suggest a connection between polytopality of the set of nonzero schubert structure constants and planarity of the diagrams this is an addition to work of a klyachko and a knutson t tao on the grassmannian and of k purbhoo f sottile on cominuscule varieties where the diagrams are always planar

acceleration of lattice bgk schemes with grid refinement abstract the lbgk method with local grid refinement has been shown to be an efficient and accurate tool for the simulation of incompressible viscous flows over complex geometries in the present study further improvement of this concept is proposed enabling the use of smaller amount of time steps on refined grids without impairing the spatial or temporal accuracy this extension of the lbgk method has been proved by analytical and numerical investigations the gain in computational time was found to be significant

a combinatorial rule for co minuscule schubert calculus we prove a root system uniform concise combinatorial rule for schubert calculus of minuscule and cominuscule flag manifolds g p the latter are also known as compact hermitian symmetric spaces we connect this geometry to the poset combinatorics of proctor thereby giving a generalization of schutzenberger s jeu de taquin formulation of the littlewood richardson rule that computes the intersection numbers of grassmannian schubert varieties our proof introduces cominuscule recursions a general technique to relate the numbers for different lie types

a parareal waveform relaxation algorithm for semi linear parabolic partial differential equations we report a new parallel iterative algorithm for semi linear parabolic partial differential equations pdes by combining a kind of waveform relaxation wr techniques into the classical parareal algorithm the parallelism can be simultaneously exploited by wr and parareal in different directions we provide sharp error estimations for the new algorithm on bounded time domain and on unbounded time domain respectively the iterations of the parareal and the wr are balanced to optimize the performance of the algorithm furthermore the speedup and the parallel efficiency of the new approach are analyzed numerical experiments are carried out to verify the effectiveness of the theoretic work

quantum cohomology of the odd symplectic grassmannian of lines abstract odd symplectic grassmannians are a generalization of symplectic grassmannians to odd dimensional spaces here we compute the classical and quantum cohomology of the odd symplectic grassmannian of lines although these varieties are not homogeneous we obtain pieri and giambelli formulas that are very similar to the symplectic case we notice that their quantum cohomology is semisimple which enables us to check dubrovin s conjecture for this case

a space time multigrid method for parabolic partial differential equations we consider the solution of parabolic partial differential equations pdes in standard time stepping techniques multigrid can be used as an iterative solver for the elliptic equations arising at each discrete time step by contrast the method presented in this paper treats the whole of the space time problem simultaneously thus the multigrid operations of smoothing and coarse grid correction are defined on all of the space time variables of a given grid level the method is characterized by a coarsening strategy with prolongation and restriction operators which depend at each grid level on the degree of anisotropy of the discretization stencil numerical results for the one and two dimensional heat equations are presented and are shown to agree closely with predictions from fourier mode analysis

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

a two dimensional shear deformable beam element based on the absolute nodal coordinate formulation in this study a new two dimensional shear deformable beam element is proposed for large deformation problems the kinematics of the beam are defined using an exact displacement field where the rotation angles of the cross section caused by bending and shear deformations are described separately cubic interpolation is used for determining the curvature of the beam due to bending while linear interpolation polynomials are used for the shear strain the absolute nodal coordinate formulation in which global displacements and slopes are used as the nodal coordinates is employed for the finite element discretization of the beam the capability of the element to predict static deformation is studied using numerical examples the results imply that the element is free of a phenomenon called shear locking the capability of the element to model highly nonlinear behaviour is established using a bending test where the cantilever is bent into a full circle using only four elements a flexible pendulum and a spin up manoeuvre are modelled in order to study the behaviour of the element in dynamical problems the proposed element is also compared with an existing shear deformable beam element based on the absolute nodal coordinate formulation finally the simple linearization of the beam curvature based on the assumption of small strain will be discussed

application of the absolute nodal co ordinate formulation to multibody system dynamics abstract the floating frame of reference formulation is currently the most widely used approach in flexible multibody simulations the use of this approach however has been limited to small deformation problems in this investigation the computer implementation of the newabsolute nodal co ordinate formulationand its use in the small and large deformation analysis of flexible multibody systems that consist of interconnected bodies are discussed while in the floating frame of reference formulation a mixed set of absolute reference and local elastic co ordinates are used in the absolute nodal co ordinate formulation only absolute co ordinates are used in the absolute nodal co ordinate formulation new interpretation of the nodal co ordinates of the finite elements is used no infinitesimal or finite rotations are used as nodal co ordinates from beams and plates instead global slopes are used to define the element nodal co ordinates using this interpretation of the element co ordinates beams and plates can be considered as isoparametric elements and as a result exact modelling of the rigid body dynamics can be obtained using the element shape function and the absolute nodal co ordinates unlike the floating frame of reference approach no co ordinate transformation is required in order to determine the element inertia the mass matrix of the finite elements is a constant matrix and therefore the centrifugal and coriolis forces are equal to zero when the absolute nodal co ordinate formulation is used another advantage of using the absolute nodal co ordinate formulation in the dynamic simulation of multibody systems is its simplicity in imposing some of the joint constraints and also its simplicity in formulating the generalized forces due to spring damper elements the results obtained in this investigation show an excellent agreement with the results obtained using the floating frame of reference formulation when large rotation small deformation problems are considered

convergent iterative schemes for time parallelization parallel methods are usually not applied to the time domain because of the inherit sequentialness of time evolution but for many evolutionary problems computer simulation can benefit substantially from time parallelization methods in this paper we present several such algorithms that actually exploit the sequential nature of time evolution through a predictor corrector procedure this sequentialness ensures convergence of a parallel predictor corrector scheme within a fixed number of iterations the performance of these novel algorithms which are derived from the classical alternating schwarz method are illustrated through several numerical examples using the reservoir simulator athena

a geometric littlewood richardson rule we describe a geometric littlewood richardson rule interpreted as deforming the intersection of two schubert varieties into the union of schubert varieties there are no restrictions on the base eld and all multiplicities arising are 1 this is important for applications this rule should be seen as a generalization of pieri s rule to arbitrary schubert classes by way of explicit homotopies it has straightforward bijections to other littlewood richardson rules such as tableaux and knutson and tao s puzzles this gives the rst geometric proof and interpretation of the littlewood richardson rule geometric consequences are described here and in v2 kv1 kv2 v3 for example the rule also has an interpretation in k theory suggested by buch which gives an extension of puzzles to k theory

the quantum cohomology of flag varieties and the periodicity of the schubert structure constants we give conditions on a curve class that guarantee the vanishing of the structure constants of the small quantum cohomology of partial flag varieties f k1 kr n for that class we show that many of the structure constants of the quantum cohomology of flag varieties can be computed from the image of the evaluation morphism in fact we show that a certain class of these structure constants are equal to the ordinary intersection of schubert cycles in a related flag variety we obtain a positive geometric rule for computing these invariants see coskun in a littlewood richardson rule for partial flag varieties preprint our study also reveals a remarkable periodicity property of the ordinary schubert structure constants of partial flag varieties

development of simple models for the elastic forces in the absolute nodal co ordinate formulation abstract the objective of this study is to develop simple and accurate elastic force models that can be used in the absolute nodal co ordinate formulation for the analysis of two dimensional beams these force models which account for the coupling between bending and axial deformations are derived using a continuum mechanics approach without the need for introducing a local element co ordinate system four new different force models that include different degrees of complexity are presented it is shown that the vector of the elastic forces can be significantly simplified as compared to the elastic force model developed for the absolute nodal co ordinate formulation using a local element frame 1 despite the simplicity of the new models they account for elastic non linearity in the strain displacement relationship therefore they lead to more accurate results as compared to the more complex models developed using the local frame method which does not account for the non linearities in the strain displacement relationships numerical results are presented in order to demonstrate the use of the new models and test their performances in the analysis of large deformations of flexible multibody systems

littlewood richardson coefficients for reflection groups abstract in this paper we explicitly compute all littlewood richardson coefficients for semisimple and kac moody groups g that is the structure constants also known as the schubert structure constants of the cohomology algebra h g p c where p is a parabolic subgroup of g these coefficients are of importance in enumerative geometry algebraic combinatorics and representation theory our formula for the littlewood richardson coefficients is purely combinatorial and is given in terms of the cartan matrix and the weyl group of g however if some off diagonal entries of the cartan matrix are 0 or 1 the formula may contain negative summands on the other hand if the cartan matrix satisfies a i j a j i 4 for all i j then each summand in our formula is nonnegative that implies nonnegativity of all littlewood richardson coefficients we extend this and other results to the structure coefficients of the t equivariant cohomology of flag varieties g p and bott samelson varieties i g

multiscale lattice boltzmann schemes with turbulence modeling abstract the viability of multiscale lattice boltzmann schemes for the numerical simulation of turbulent flows is discussed and numerically demonstrated for turboaxial machine applications the extension of boundary fitting formulas based on wall functions is proposed which enables the efficient computation of turbulent flows in complex curvilinear geometry using a simple cartesian grid examples of two dimensional turbulent flows in an axial compressor cascade are presented

efficient kinetic method for fluid simulation beyond the navier stokes equation we present a further theoretical extension to the kinetic theory based formulation of the lattice boltzmann method of shan j fluid mech 550 413 2006 in addition to the higher order projection of the equilibrium distribution function and a sufficiently accurate gauss hermite quadrature in the original formulation a regularization procedure is introduced in this paper this procedure ensures a consistent order of accuracy control over the nonequilibrium contributions in the galerkin sense using this formulation we construct a specific lattice boltzmann model that accurately incorporates up to third order hydrodynamic moments numerical evidence demonstrates that the extended model overcomes some major defects existing in conventionally known lattice boltzmann models so that fluid flows at finite knudsen number kn can be more quantitatively simulated results from force driven poiseuille flow simulations predict the knudsen s minimum and the asymptotic behavior of flow flux at large kn

reference motion in deformable bodies under rigid body motion and vibration part i theory abstract this paper examines the motions of reference systems linked to deformable bodies under simultaneously vibration and large translations and rotations these motions depend on the particular type of linkage between the moving reference system and the deformable body which is defined by the so called reference conditions when using the rayleigh ritz method the reference conditions also dictate the boundary conditions to be fulfilled by the shape functions used to describe the body s elasticity this paper analyses three different types of reference conditions namely free linkage rigid linkage and two point linkage it is shown that moving reference frames only evolve at a constant velocity in the absence of external forces when the free linkage is used the reference velocities for systems with a free linkage are designated rigid body equivalent velocities for the deformable body here such velocities can also be calculated under other types of reference conditions and are usually functions of the elastic and reference co ordinates and also of their derivatives rigid body equivalent velocities are useful for purposes such as estimating the trajectory of deformable bodies moving freely in space without the need to examine the deformations they undergo also their calculation is required with a view to determining the kinematic restitution coefficient for deformable body collisions which is dealt within part ii of this series

flexible translational joint analysis by meshless method element free galerkin method efgm is used to analyze a flexible translational joint a moving constraint condition is a major concern for the translational joint analysis it is shown that the efgm is well suited for an analysis of flexible translational joint original shape function in the efgm is modified so that essential boundary conditions are imposed by the same way as that of finite element method fem the modified shape function possesses physical values at both ends of the element the completeness of the modified shape function is discussed employing the present modified shape function makes easy to implement the moving constraint condition numerical examples for a static cantilever beam are presented for a comparison between the penalty method the lagrange multiplier method and the present one as second example a large deformation problem is solved for a comparison between the fem and the efgm finally a simulation of flexible translational joint is shown

restriction varieties and geometric branching rules abstract this paper develops a new method for studying the cohomology of orthogonal flag varieties restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to not necessarily isotropic flags they interpolate between schubert varieties in orthogonal flag varieties and the restrictions of general schubert varieties in ordinary flag varieties we give a positive geometric rule for calculating their cohomology classes obtaining a branching rule for schubert calculus for the inclusion of the orthogonal flag varieties in type a flag varieties our rule in addition to being an essential step in finding a littlewood richardson rule has applications to computing the moment polytopes of the inclusion of so n in su n the asymptotic of the restrictions of representations of sl n to so n and the classes of the moduli spaces of rank two vector bundles with fixed odd determinant on hyperelliptic curves furthermore for odd orthogonal flag varieties we obtain an algorithm for expressing a schubert cycle in terms of restrictions of schubert cycles of type a flag varieties thereby giving a geometric though not positive algorithm for multiplying any two schubert cycles

coupling methods for heat transfer and heat flow operator splitting and the parareal algorithm abstract we propose an operator splitting method for coupling heat transfer and heat flow equations this work is motivated by the need to couple independent industrial heat transfer solvers e g the aura fluid software package and heat flow solvers e g openfoam such packages are often used to simulate the influence of solar heat in car bodies and are coupled by a b splitting techniques the main goal of this work is the acceleration of the coupled software system by iterative operator splitting methods and additional time parallelism using the parareal algorithm we present these new splitting techniques along with some novel convergence results and test the splitting parareal combination on various numerical problems

on the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach abstract in finite element methods that are based on position and slope coordinates a representation of axial and bending deformation by means of an elastic line approach has become popular such beam and plate formulations based on the so called absolute nodal coordinate formulation have not yet been verified sufficiently enough with respect to analytical results or classical nonlinear rod theories examining the existing planar absolute nodal coordinate element which uses a curvature proportional bending strain expression it turns out that the deformation does not fully agree with the solution of the geometrically exact theory and even more serious the normal force is incorrect a correction based on the classical ideas of the extensible elastica and geometrically exact theories is applied and a consistent strain energy and bending moment relations are derived the strain energy of the solid finite element formulation of the absolute nodal coordinate beam is based on the st venant kirchhoff material therefore the strain energy is derived for the latter case and compared to classical nonlinear rod theories the error in the original absolute nodal coordinate formulation is documented by numerical examples the numerical example of a large deformation cantilever beam shows that the normal force is incorrect when using the previous approach while a perfect agreement between the absolute nodal coordinate formulation and the extensible elastica can be gained when applying the proposed modifications the numerical examples show a very good agreement of reference analytical and numerical solutions with the solutions of the proposed beam formulation for the case of large deformation pre curved static and dynamic problems including buckling and eigenvalue analysis the resulting beam formulation does not employ rotational degrees of freedom and therefore has advantages compared to classical beam elements regarding energy momentum conservation

nonlinear magnetohydrodynamics simulation using high order finite elements a conforming representation composed of 2d finite elements and finite fourier series is applied to 3d nonlinear non ideal magnetohydrodynamics using a semi implicit time advance the self adjoint semi implicit operator and variational approach to spatial discretization are synergistic and enable simulation in the extremely stiff conditions found in high temperature plasmas without sacrificing the geometric flexibility needed for modeling laboratory experiments growth rates for resistive tearing modes with experimentally relevant lundquist number are computed accurately with time steps that are large with respect to the global alfven time and moderate spatial resolution when the finite elements have basis functions of polynomial degree p two or larger an error diffusion method controls the generation of magnetic divergence error convergence studies show that this approach is effective for continuous basis functions with p 2 where the number of test functions for the divergence control terms is less than the number of degrees of freedom in the expansion for vector fields anisotropic thermal conduction at realistic ratios of parallel to perpendicular conductivity is computed accurately with p 3 without mesh alignment a simulation of tearingmode evolution for a shaped toroidal tokamak equilibrium demonstrates the effectiveness of the algorithm in nonlinear conditions and its results are used to verify the accuracy of the numerical anisotropic thermal conduction in 3d magnetic topologies

a littlewood richardson rule for two step flag varieties this paper studies the geometry of one parameter specializations of subvarieties of grassmannians and two step flag varieties as a consequence we obtain a positive geometric rule for expressing the structure constants of the cohomology of two step flag varieties in terms of their schubert basis a corollary is a positive geometric rule for computing the structure constants of the small quantum cohomology of grassmannians we also obtain a positive geometric rule for computing the classes of subvarieties of grassmannians that arise as the projection of the intersection of two schubert varieties in a partial flag variety these rules have numerous applications to geometry representation theory and the theory of symmetric functions