k homology asymptotic representations and unsuspended e theory connes and higson defined a bivariant homology theory e a b for separable c algebras the elements of e a b are taken to be homotopy classes of asymptotic momorphisms from sa k to sb k in symbols e a b sa sb k if a is k nuclear then e theory agrees with kasparov s bivariant k theory we show that in many cases one need not take suspensions to calculate the e theory group e a b for many a we show e a b a b k for all b among the a for which this is true are c0 x pt for x with the homotopy type of a finite connected cw complex this gives a concrete realization of k homology related to the brown douglas fillmore description for example k0 x arises as asymptotic representations k0 x c0 x k other a for which our isomorphic holds include the nonunital dimension drop intervals in this case there is no distinction between homomorphisms and asymptotic morphisms so we have succeeded in classifying all homomorphisms from a dimension drop interval to a stable c algebra this was subsequently used by elliott j reine angew math 443 1993 179 219 in the classification of certain c algebras the dimension drop interval may also be used to describe k x z n in terms of paths of asymptotic representations of c0 x

robust and efficient discontinuous galerkin methods for under resolved turbulent incompressible flows abstract we present a robust and accurate discretization approach for incompressible turbulent flows based on high order discontinuous galerkin methods the dg discretization of the incompressible navier stokes equations uses the local lax friedrichs flux for the convective term the symmetric interior penalty method for the viscous term and central fluxes for the velocity pressure coupling terms stability of the discretization approach for under resolved turbulent flow problems is realized by a purely numerical stabilization approach consistent penalty terms that enforce the incompressibility constraint as well as inter element continuity of the velocity field in a weak sense render the numerical method a robust discretization scheme in the under resolved regime the penalty parameters are derived by means of dimensional analysis using penalty factors of order 1 applying these penalty terms in a postprocessing step leads to an efficient solution algorithm for turbulent flows the proposed approach is applicable independently of the solution strategy used to solve the incompressible navier stokes equations i e it can be used for both projection type solution methods as well as monolithic solution approaches since our approach is based on consistent penalty terms it is by definition generic and provides optimal rates of convergence when applied to laminar flow problems robustness and accuracy are verified for the orr sommerfeld stability problem the taylor green vortex problem and turbulent channel flow moreover the accuracy of high order discretizations as compared to low order discretizations is investigated for these flow problems a comparison to state of the art computational approaches for large eddy simulation indicates that the proposed methods are highly attractive components for turbulent flow solvers

efficient random walks in the presence of complex two dimensional geometries this paper details an efficient algorithm for particles undergoing random walks in the presence of complex two dimensional solid boundaries the scheme is developed for the simulation of vortex diffusion using the random vortex method both vortex blobs and sheets are handled the relevant modifications for using the algorithm with the vorticity redistribution technique are also discussed the algorithm is designed to be used in the framework of an existing fast multipole implementation a measure for the geometric complexity of a body is introduced and the algorithm s efficiency is studied as various parameters are changed for bodies of varying complexity

a high order semi explicit discontinuous galerkin solver for 3d incompressible flow with application to dns and les of turbulent channel flow abstract we present an efficient discontinuous galerkin scheme for simulation of the incompressible navier stokes equations including laminar and turbulent flow we consider a semi explicit high order velocity correction method for time integration as well as nodal equal order discretizations for velocity and pressure the non linear convective term is treated explicitly while a linear system is solved for the pressure poisson equation and the viscous term the key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under resolved high reynolds number flows as well as small time steps this penalty method is similar to the grad div stabilization widely used in continuous finite elements we further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations the solver is specifically designed for large scale computations through matrix free linear solvers including efficient preconditioning strategies and tensor product elements which have allowed us to scale this code up to 34 4 billion degrees of freedom and 147 456 cpu cores we validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large eddy simulation of turbulent channel flow at r e 180 as well as 590

classification ofc algebras of real rank zero and unsuspendede equivalence types abstract in this article examples are given to prove that the graded scaled ordered k group is not the complete invariant for a c algebra in the class of unital separable nuclear c algebras of real rank zero and stable rank one even for a c algebra in the subclass which consists of those real rank zero stable rank one c algebras being expressed as inductive limits of k n i 1 m n i c x n i where x n i are two dimensional finite cw complexes and n i are positive integers in the case of simple such c algebras it has been proved that the above invariant is the complete invariant by george elliott and the author these examples prove that the classification conjecture of elliott for the case of non simple real rank zero c algebras should be revised one needs extra invariants the obstruction preventing two such c algebras with the same graded scaled ordered k group from being isomorphic is that they have different unsuspended e equivalence types a refinement of kk equivalence type of c algebras due to connes and higson in this article it is proved that for the above class of inductive limit c algebras the obstruction of unsuspended e equivalence type is the only obstruction i e if two c algebras in the class are unsuspended e equivalence then they are isomorphic it is a surprise that in the case of simple such c algebras or even the case of c algebras with finitely many ideals the obstruction will disappear see section 4

a fast two dimensional panel method a new two dimensional panel technique has been developed to solve laplacian flows which eliminates the edge effect present in traditional panel methods such a method is very useful for applications where the velocity induced by the panels is required at arbitrary locations particle based flow solvers are a prime example the method however requires considerably more computational effort in this paper the method is modified to improve computational efficiency by adapting the fast multipole algorithm for the panel method significant improvement in computational efficiency is obtained while ensuring that the edge effects are eliminated

new transformations of cauchy matrices and trummer s problem abstract we show some new expressions for a cauchy matrix which enable us to simplify the solution of trummer s problem both in the general case and in the case where the input cauchy matrix is fixed for the problem whereas the input vector varies

approximate projection methods part i inviscid analysis the use of approximate projection methods for modeling low mach number flows avoids many of the numerical complications associated with exact projection methods but introduces additional design choices in developing a robust algorithm in this paper we first explore these design choices in the setting of inviscid incompressible flow using several computational examples we then develop a framework for analyzing the behavior of the different design variations and use that analysis to explain the features observed in the computations as part of this work we introduce a new variation of the approximate projection algorithm that combines the advantages of several of the previous versions

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

using hierarchical cartesian grids with multigrid acceleration this paper describes the use of a multigrid strategy on hierarchical cartesian grids for the simulation of small amplitude inviscid waves in a tank quadtree grid generation by recursive subdivision is described and the grid storage system discussed along with the associated grid operations such as neighbour finding and identification of empty cells a multigrid strategy using a novel global approach is discussed with respect to its application to quadtree grids a comparative study is presented of the performance of multigrid parameters such as number of iterations at each level grid visiting schedules and different prolongation operators it is shown that quadtree grids are ideally suited to a multigrid strategy when the solution of the elliptic laplace s equation is sought with savings of the order of 90 in cpu time with respect to a non multigrid solution copyright 2006 john wiley sons ltd

inductive limits of semiprojective c algebras abstract we prove closure properties for the class of c algebras that are inductive limits of semiprojective c algebras most importantly we show that this class is closed under shape domination and so in particular under shape and homotopy equivalence it follows that the considered class is quite large it contains for instance the stable suspension of any nuclear c algebra satisfying the uct and with torsion free k 0 group in particular the stabilized c algebra of continuous functions on the pointed sphere is isomorphic to an inductive limit of semiprojectives

electrostatics and heat conduction in high contrast composite materials we present a robust integral equation method for the calculation of the electrostatic and thermal properties of systems made of piecewise homogeneous high contrast materials by high contrast we mean that the electrical or thermal conductivity ratios are high our approach involves a modification of the standard integral representation using in electrostatic terms the polarization charge as an unknown it is related to the perturbation approach proposed by tausch and white j tausch j white capacitance extraction of 3 d conductor systems in dielectric media with high permittivity ratios ieee trans microwave theory tech 47 1999 18 26 but based on a simple form of projection rather than on the solution of auxiliary integral equations numerical implementation is straightforward in the multiple dielectric case

a high order discontinuous galerkin method for 2d incompressible flows in this paper we introduce a high order discontinuous galerkin method for two dimensional incompressible flow in the vorticity stream function formulation the momentum equation is treated explicitly utilizing the efficiency of the discontinuous galerkin method the stream function is obtained by a standard poisson solver using continuous finite elements there is a natural matching between these two finite element spaces since the normal component of the velocity field is continuous across element boundaries this allows for a correct upwinding gluing in the discontinuous galerkin framework while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability the method is efficient for inviscid or high reynolds number flows optimal error estimates are proved and verified by numerical experiments

a vorticity based method for incompressible unsteady viscous flows a novel approach is presented based on the integral form of the vorticity formulation in which the vorticity transport equation is solved by using the cell centred finite volume method while the velocities needed at the centre of each control volume are calculated by a modified biot savart formula in conjunction with a fast summation algorithm the vorticity and mass conservation in the flow are guaranteed during the calculation by virtue of the finite volume approach and the method of implementing the boundary conditions at the body surface as an example both the early stage development and long term evolution of the flow around an impulsively started circular cylinder are computed using the method the present results are compared with other numerical and experimental results for the same flow problem and show good agreement

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

a spectral multidomain penalty method model for the simulation of high reynolds number localized incompressible stratified turbulence a spectral multidomain penalty method model has been developed for the simulation of high reynolds number localized stratified turbulence this is the first time that a penalty method with a particular focus on subdomain interface treatment has been used with a multidomain scheme to simulate incompressible flows the temporal discretization ensures maximum temporal accuracy by combining third order stiffly stable and backward differentiation schemes with a high order boundary condition for the pressure in the non periodic vertical direction a spectral multidomain discretization is used and its stability for under resolved simulations at high reynolds numbers is ensured through use of penalty techniques spectral filtering and strong adaptive interfacial averaging the penalty method is implemented in different formulations for both the explicit non linear term advancement and the implicit treatment of the viscous terms the multidomain model is validated by comparing results of simulations of the mid to late time stratified turbulent wake with non zero net momentum to the corresponding laboratory data for a towed sphere the model replicates correctly the characteristic vorticity and internal wave structure of the stratified wake and exhibits robust agreement with experiments in terms of the temporal power laws in the evolution of mean profile characteristic velocity and lengthscales

a new diffusion procedure for vortex methods a new method is proposed for simulating diffusion in vortex methods for two dimensional incompressible flows the method resolves length scales up to the spacing of the vortices the grid free nature of vortex methods is fully retained and the distribution of the vortices can be irregular it is shown for the stokes equations that in principle the method can have any order of accuracy it also conserves circulation linear and angular momentum the method is based on exchanging a conserved quantity between arbitrary computational points this suggests that extensions to more general flows may be possible for the two dimensional incompressible flows studied circulation is exchanged between vortices to simulate diffusion the amounts of circulation exchanged must satisfy a linear system of equations based on stability considerations the exchanged amounts should further be positive a procedure to find a solution to this problem is formulated using linear programming techniques to test the method the decay processes of a single point vortex and of a counterrotating pair of point vortices are computed current limitations of the method are discussed

on the stability of projection methods for the incompressible navier stokes equations based on high order discontinuous galerkin discretizations abstract the present paper deals with the numerical solution of the incompressible navier stokes equations using high order discontinuous galerkin dg methods for discretization in space for dg methods applied to the dual splitting projection method instabilities have recently been reported that occur for small time step sizes since the critical time step size depends on the viscosity and the spatial resolution these instabilities limit the robustness of the navier stokes solver in case of complex engineering applications characterized by coarse spatial resolutions and small viscosities by means of numerical investigation we give evidence that these instabilities are related to the discontinuous galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high order accuracy with respect to the temporal discretization this new formulation is stable in the limit of small time steps for both equal order and mixed order polynomial approximations although the dual splitting scheme itself includes inf sup stabilizing contributions we demonstrate that spurious pressure oscillations appear for equal order polynomials and small time steps highlighting the necessity to consider inf sup stability explicitly

a space time hybridizable discontinuous galerkin method for incompressible flows on deforming domains we present the first space time hybridizable discontinuous galerkin hdg finite element method for the incompressible navier stokes and oseen equations major advantages of a space time formulation are its excellent capabilities of dealing with moving and deforming domains and grids and its ability to achieve higher order accurate approximations in both time and space by simply increasing the order of polynomial approximation in the space time elements our formulation is related to the hdg formulation for incompressible flows introduced recently in e g n c nguyen j peraire b cockburn a hybridizable discontinuous galerkin method for stokes flow comput methods appl mech eng 199 2010 582 597 however ours is inspired in typical dg formulations for compressible flows which allow for a more straightforward implementation another difference is the use of polynomials of fixed total degree with space time hexahedral and quadrilateral elements instead of simplicial elements we present numerical experiments in order to assess the quality of the performance of the methods on deforming domains and to experimentally investigate the behavior of the convergence rates of each component of the solution with respect to the polynomial degree of the approximations in both space and time

convergence adaptive refinement and scaling in 1d peridynamics we introduce here adaptive refinement algorithms for the non local method peridynamics which was proposed in j mech phys solids 2000 48 175 209 as a reformulation of classical elasticity for discontinuities and long range forces we use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected we discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1d in the limit of the horizon going to zero continuous micromoduli lead to optimal rates of convergence independent of the grid used while discontinuous micromoduli produce optimal rates of convergence only for uniform grids examples for static and dynamic elasticity problems in 1d are shown the relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement for the same number of nodes copyright 2008 john wiley sons ltd

an artificial compressibility flux for the discontinuous galerkin solution of the incompressible navier stokes equations discontinuous galerkin dg methods have proved to be well suited for the construction of robust high order numerical schemes on unstructured and possibly nonconforming grids for a variety of problems their application to the incompressible navier stokes ins equations has also been recently considered although the subject is far from being fully explored in this work we propose a new approach for the dg numerical solution of the ins equations written in conservation form the inviscid numerical fluxes both in the continuity and in the momentum equation are computed using the values of velocity and pressure provided by the exact solution of the riemann problem associated with a local artificial compressibility perturbation of the equations unlike in most of the existing methods artificial compressibility is here introduced only at the interface flux level therefore resulting in a consistent discretization of the ins equations irrespectively of the amount of artificial compressibility introduced the discretization of the viscous term follows the well established dg scheme named br2 the performance and the accuracy of the method are demonstrated by computing the kovasznay flow and the two dimensional lid driven cavity flow for a wide range of reynolds numbers and for various degrees of polynomial approximation

fast adaptive summation of point forces in the two dimensional poisson equation direct summation of the velocity field introduced by point vortices tends to be time consuming since the velocity of each vortex is found as a sum over all other vortices the resulting number of numerical operations is proportional to the square of the number of vortices here a relatively simple procedure is outlined which significantly reduces the number of operations by replacing selected partial sums by asymptotic series the resulting number of operations appears to vary roughly in proportion to the number of unknowns corresponding to a fast solver

convergence characteristics and computational cost of two algebraic kernels in vortex methods with a tree code algorithm we study the convergence characteristics of two algebraic kernels used in vortex calculations the rosenhead moore kernel which is a low order kernel and the winckelmans leonard kernel which is a high order kernel to facilitate the study a method of evaluating particle cluster interactions is introduced for the winckelmans leonard kernel the method is based on taylor series expansion in cartesian coordinates as initially proposed by lindsay and krasny j comput phys 172 2001 pp 879 907 for the rosenhead moore kernel a recurrence relation for the taylor coefficients of the winckelmans leonard kernel is derived by separating the kernel into two parts and an error estimate is obtained to ensure adaptive error control the recurrence relation is incorporated into a tree code to evaluate vorticity induced velocity next comparison of convergence is made while utilizing the tree code both algebraic kernels lead to convergence but the winckelmans leonard kernel exhibits a superior convergence rate the combined desingularization and discretization error from the winckelmans leonard kernel is an order of magnitude smaller than that from the rosenhead moore kernel at a typical resolution simulations of vortex rings are performed using the two algebraic kernels in order to compare their performance in a practical setting in particular numerical simulations of the side by side collision of two identical vortex rings suggest that the three dimensional evolution of vorticity at finite resolution can be greatly affected by the choice of the kernel we find that the winckelmans leonard kernel is able to perform the same task with a much smaller number of vortex elements than the rosenhead moore kernel greatly reducing the overall computational cost

an adaptive multigrid technique for evaluating long range forces in biomolecular simulations we propose an adaptive multigrid method for evaluating the nonlocal long range interactions in biomolecular models the method provides o h 3 accuracy for the forces where h is the mesh size of the finest cubic grid even though the stencil size is minimal two new algorithmic components are developed an adaptive technique for spreading a three dimensional charge distribution onto a space mesh and a procedure for performing coarse to fine grid interpolation of the potential force components and their derivatives grid to particle interpolation at the finest level in a consistent fashion the adaptive mesh used in the spreading procedure contains two types of nodes regular positioned at nodes cubic grid and flexible one per cubic cell thus the charges located in each grid cell are spread to these nine nodes the method conserves charge dipole and quadrupole moments of cells the spreading and interpolation techniques are economical in computation time and storage due to the overlap design in the stencils implementation is also straightforward our initial tests demonstrate the favorable performance for low accuracy with respect to the fast multipole technique

a kernel independent adaptive fast multipole algorithm in two and three dimensions we present a new fast multipole method for particle simulations the main feature of our algorithm is that it does not require the implementation of multipole expansions of the underlying kernel and it is based only on kernel evaluations instead of using analytic expansions to represent the potential generated by sources inside a box of the hierarchical fmm tree we use a continuous distribution of an equivalent density on a surface enclosing the box to find this equivalent density we match its potential to the potential of the original sources at a surface in the far field by solving local dirichlet type boundary value problems the far field evaluations are sparsified with singular value decomposition in 2d or fast fourier transforms in 3d we have tested the new method on the single and double layer operators for the laplacian the modified laplacian the stokes the modified stokes the navier and the modified navier operators in two and three dimensions our numerical results indicate that our method compares very well with the best known implementations of the analytic fmm method for both the laplacian and modified laplacian kernels its advantage is the relative simplicity of the implementation and its immediate extension to more general kernels

a stable and high order accurate discontinuous galerkin based splitting method for the incompressible navier stokes equations abstract in this paper we consider discontinuous galerkin dg methods for the incompressible navier stokes equations in the framework of projection methods in particular we employ symmetric interior penalty dg methods within the second order rotational incremental pressure correction scheme the major focus of the paper is threefold i we propose a modified upwind scheme based on the vijayasundaram numerical flux that has favourable properties in the context of dg ii we present a novel postprocessing technique in the helmholtz projection step based on h div reconstruction of the pressure correction that is computed locally is a projection in the discrete setting and ensures that the projected velocity satisfies the discrete continuity equation exactly as a consequence it also provides local mass conservation of the projected velocity iii numerical results demonstrate the properties of the scheme for different polynomial degrees applied to two dimensional problems with known solution as well as large scale three dimensional problems in particular we address second order convergence in time of the splitting scheme as well as its long time stability

short note a short note on the discontinuous galerkin discretization of the pressure projection operator in incompressible flow this note reports on the issue of spurious compressibility artifacts that can arise when the popular pressure projection pp method is used for unsteady simulations of incompressible flow using the symmetric interior penalty discontinuous galerkin sip dg method through a spectral analysis of the projection operator s sip dg discretization we demonstrate that the eigenfunctions of the operator do not form a basis that allows for the correct enforcement of the incompressibility constraint this short coming can cause numerical instabilities for inviscid advection dominated and density stratified flow simulations especially for long time integrations and or under resolved situations to remedy this problem we propose a local post processing projection that enforces incompressibility exactly to allow for stable and robust long time integrations

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

a multipole expansion technique in solving boundary integral equations abstract the boundary integral equation method reduces the spatial dimension of an elliptic problem by converting the original n dimensional partial differential equation to an n 1 dimensional boundary integral equation defined on the boundary of the domain of the original problem at the same time the discretisation of the problem is also remarkably simplified the price of these advantages however is that the structure as well as the algebraic properties of the resulting boundary element matrices are somewhat unpleasant since they are neither sparse nor self adjoint in general even if the original problem is self adjoint consequently the computational cost of the boundary integral equation method seems to be unnecessarily high to make the method more economic from a computational point of view we present a numerical technique based on the multipole expansion method which reduces the computational cost of the appearing matrix vector multiplications i e the evaluations of the discretized boundary integrals by a remarkable amount the method is applicable also to the reconstruction problem when the inner solution is to be reconstructed from the boundary solutions

deformations of nilpotent groups and homotopy symmetric c algebras the homotopy symmetric c algebras are those separable c algebras for which one can unsuspend in e theory we find a new simple condition that characterizes homotopy symmetric nuclear c algebras and use it to show that the property of being homotopy symmetric passes to nuclear c subalgebras and it has a number of other significant permanence properties as an application we show that if i g is the kernel of the trivial representation iota c g rightarrow mathbb c for a countable discrete torsion free nilpotent group g then i g is homotopy symmetric and hence the kasparov group kk i g b can be realized as the homotopy classes of asymptotic morphisms i g b otimes mathcal k for any separable c algebra b

a fast adaptive vortex method in three dimensions the method of local corrections mlc developed by anderson for two spatial dimensions is a particle particle particle mesh method in which the calculation of the velocity field induced by a collection of vortices is split into two parts i a finite difference velocity field calculation using a fast poisson solver the results of which are used to represent the velocity field induced by vortices far from the evaluation point and ii an n body calculation to compute the velocity field at a vortex induced by nearby vortices we present a fast vortex method for incompressible flow in three dimensions based on the extension of the mlc algorithm from two to three spatial dimensions and the use of adaptive mesh refinement in the finite difference calculation of the mlc calculations with a vortex ring in three dimensions show that the break even point between the mlc with amr and the direct method is at n 3000 on a cray y mp for n 64 000 mlc with amr can be 12 times faster than the direct method results from calculations of two colliding inviscid vortex rings demonstrate the increased resolution which can be obtained using fast methods

lbb stability of a mixed galerkin finite element pair for fluid flow simulations we introduce a new mixed finite element for solving the 2 and 3 dimensional wave equations and equations of incompressible flow the element which we refer to as p1 d p2 uses discontinuous piecewise linear functions for velocity and continuous piecewise quadratic functions for pressure the aim of introducing the mixed formulation is to produce a new flexible element choice for triangular and tetrahedral meshes which satisfies the lbb stability condition and hence has no spurious zero energy modes the advantage of this particular element choice is that the mass matrix for velocity is block diagonal so it can be trivially inverted it also allows the order of the pressure to be increased to quadratic whilst maintaining lbb stability which has benefits in geophysical applications with coriolis forces we give a normal mode analysis of the semi discrete wave equation in one dimension which shows that the element pair is stable and demonstrate that the element is stable with numerical integrations of the wave equation in two dimensions an analysis of the resultant discrete laplace operator in two and three dimensions on various meshes which shows that the element pair does not have any spurious modes we provide convergence tests for the element pair which confirm that the element is stable since the convergence rate of the numerical solution is quadratic

on the asymptotic homotopy type of inductive limit c algebras let x y be compact connected metrisable spaces with base points xo yo and let denote the compact operators it is shown that co x xo is asymptotically homotopic or shape equivalent to co y yo yi r if and only if x and y have isomorphic k groups similar results are obtained for certain inductive limits of nuclear c algebras let r denote the category whose objects are all the separable c algebras and whose set of morphisms from a to b denoted a b consists of homotopy classes of asymptotic morphisms the construction of this category is due to connes and higson ch who defined a bivariant homology theory e a b lisa sb ff and have shown how to define the intersection product for arbitrary extensions of separable c algebras if a is k nuclear then e theory agrees with kasparov s bivariant k theory k on the other hand the asymptotic homotopy category d appears to be the right framework for the homotopy theory of separable c algebras this point of view is supported by results in h ch d chi ck d1 for instance we have shown in d1 that asymptotic homotopy is equivalent to a strong shape theory and hence is intimately related to the shape theories of ek1 and b which were intended as homotopy theories for noncommutative singular spaces in particular it turned out that two separable c algebras are shape equivalent if and only if they are asymptotically homotopic i e isomorphic in r the isomorphism class in sr of a separable c algebra a is called the asymptotic homotopy type of a in this note we exhibit large classes of projeetionless stable nuclear c algebras whose asymptotic homotopy type is determined by k theoretical data theorem 6 this is done via a suspension isomorphism