accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

a new class of truly consistent splitting schemes for incompressible flows a new class of splitting schemes for incompressible flows is introduced the new schemes are based on a weak form of the pressure poisson equation and at each time step only require to solve a set of helmholtz type equations for the velocity and a poisson equation in the weak form for the pressure just as pressure correction and velocity correction schemes however unlike pressure correction and velocity correction schemes the new splitting schemes are free of splitting errors and deliver full accuracy on the vorticity and the pressure

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

a splitting method for incompressible flows with variable density based on a pressure poisson equation a new fractional time stepping technique for solving incompressible flows with variable density is proposed the main feature of this method is that as opposed to other known algorithms the pressure is determined by just solving one poisson equation per time step which greatly reduces the computational cost the stability of the method is proved and the performance of the method is numerically illustrated

the efficient rotational pressure correction schemes for the coupling stokes darcy problem abstract in this paper the rotational pressure correction methods for the stokes darcy system are developed and analyzed the central advantage of these methods is a time dependent version of domain decomposition these methods have first order second order accuracy without the incompressibility constraint of the stokes darcy system their main feature is the implementation efficiency in that we only solve one vector valued elliptic equation and one scalar valued poisson equation for the stokes equations per time step the unconditional stability and long time stability are established and numerical experiments are also presented to show their performance

the accuracy of the fractional step method we analyze the accuracy of the fractional step method of kim and moin j comput phys 59 1985 pp 308 323 for the incompressible navier stokes equations we show that the boundary conditions cannot be exactly satisfied in the projection step and that this limits the accuracy of the method we also show that the pressure in any projection method can be at best first order accurate our analysis is simpler and more direct than the previous analyses of this method we also show that there is no numerical boundary layer for velocity or pressure but there is one for the auxiliary pressure variable

on error estimates of the projection methods for the navier stokes equations second order schemes we present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible navier stokes equations the error analysis is accomplished by interpreting the respective projection schemes as second order time discretizations of a perturbed system which approximates the navier stokes equations numerical results in agreement with the error analysis are also presented

subgrid stabilized projection method for 2d unsteady flows at high reynolds numbers a subgrid stabilization technique is developed for solving the two dimensional incompressible navier stokes equations at high reynolds numbers the time marching algorithm is based on a well established fractional step pressure correction projection method the advection diffusion step is enriched by an implicit subgrid stabilizing term and by an explicit dissipative shock capturing term the former is calculated by means of a hierarchical finite element setting the latter is included to avoid gibbs phenomenon in the boundary layer convergence tests on prototypical two dimensional examples are reported and the method is used to simulate the viscous incompressible flows around the airfoil naca0012 at zero incidence and reynolds numbers ranging from 105 to 106

nonassociative tori and applications to t duality in this paper we initiate the study of c algebras open image in new window endowed with a twisted action of a locally compact abelian lie group open image in new window and we construct a twisted crossed product open image in new window which is in general a nonassociative noncommutative algebra the duality properties of this twisted crossed product algebra are studied in detail and are applied to t duality in type ii string theory to obtain the t dual of a general principal torus bundle with general h flux which we will argue to be a bundle of noncommutative nonassociative tori nonassociativity is interpreted in the context of monoidal categories of modules we also show that this construction of the t dual includes the other special cases already analysed in a series of papers

a pressure correction scheme for generalized form of energy stable open boundary conditions for incompressible flows we present a generalized form of open boundary conditions and an associated numerical algorithm for simulating incompressible flows involving open or outflow boundaries the generalized form represents a family of open boundary conditions which all ensure the energy stability of the system even in situations where strong vortices or backflows occur at the open outflow boundaries our numerical algorithm for treating these open boundary conditions is based on a rotational pressure correction type strategy with a formulation suitable for c 0 spectral element spatial discretizations we have introduced a discrete equation and associated boundary conditions for an auxiliary variable the algorithm contains constructions that prevent a numerical locking at the open outflow boundary in addition we have developed a scheme with a provable unconditional stability for a sub class of the open boundary conditions extensive numerical experiments have been presented to demonstrate the performance of our method for several flow problems involving open outflow boundaries we compare simulation results with the experimental data to demonstrate the accuracy of our algorithm long time simulations have been performed for a range of reynolds numbers at which strong vortices or backflows occur at the open outflow boundaries we show that the open boundary conditions and the numerical algorithm developed herein produce stable simulations in such situations

an incremental pressure correction finite element method for the time dependent oldroyd flows abstract in this paper we present an incremental pressure correction finite element method for the time dependent oldroyd flows this method is a fully discrete projection method as we all know most projection methods have been studied without space discretization then the ensuing analysis may not extend to this case we also give the stability analysis and the optimal error analysis the analysis is based on a time discrete error and a spatial discrete error in order to show the effectiveness of the method we also present some numerical results the numerical results confirm our analysis and show clearly the stability and optimal convergence of the incremental pressure correction finite element method for the time dependent oldroyd flows

numerical approximations to a new phase field model for two phase flows of complex fluids abstract we derive a new phase field theory for immiscible mixtures of nematic liquid crystals and viscous fluids using the variational principle coupled with the generalized onsager principle a novel phase transition mechanism is implemented to couple the nematic liquid crystal phase with the viscous fluid phase to arrive at the dissipative hydrodynamic model for incompressible fluid mixtures through a delicate explicit implicit numerical discretization we develop a decoupled linear scheme for a simplified version of the phase field model as well as a coupled nonlinear scheme for the full model both schemes are shown as unconditionally energy stable with consistent discrete dissipative energy laws several numerical examples are presented to show the effectiveness of the new model and the new numerical schemes developed for it

particulate flows with the subspace projection method a novel finite element method for the 3d simulation of many particles in a newtonian carrier liquid is presented the methods features the celebrated one domain approach a subspace projection method to account for the rigid body motion within the particles and operator splitting combined with local mesh refinement the method results in a fast and accurate though conceptually simple to implement algorithm validation is achieved using the sedimentation of one particle and comparing the resulting drag coefficient with theoretical and experimental results furthermore a viscometer is considered where the effective viscosity of a particle laden fluid is compared with analytic results

algebraic fractional step schemes with spectral methods for the incompressible navier stokes equations the numerical investigation of a recent family of algebraic fractional step methods for the solution of the incompressible time dependent navier stokes equations is presented these methods are improved versions of the yosida method proposed in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations comput methods appl mech engrg 188 1 3 2000 505 526 a quarteroni f saleri a veneziani j math pures appl 9 78 5 1999 473 503 and one of them the yosida4 method is proposed in this paper for the first time they rely on an approximate lu block factorization of the matrix obtained after the discretization in time and space of the navier stokes system yielding a splitting in the velocity and pressure computation in this paper we analyze the numerical performances of these schemes when the space discretization is carried out with a spectral element method with the aim of investigating the impact of the splitting on the global accuracy of the computation

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

a finite element penalty projection method for incompressible flows the penalty projection method for the solution of navier stokes equations may be viewed as a projection scheme where an augmentation term is added in the first stage namely the solution of the momentum balance equation to constrain the divergence of the predicted velocity field after a presentation of the scheme in the time semi discrete formulation then in fully discrete form for a finite element discretization we assess its behaviour against a set of benchmark tests including in particular prescribed velocity and open boundary conditions the results demonstrate that the augmentation always produces beneficial effects as soon as the augmentation parameter takes a significant value the projection method splitting error is reduced pressure boundary layers are suppressed and the loss of spatial convergence of the incremental projection scheme in case of open boundary conditions does not occur anymore for high values of the augmentation parameter the results of coupled solvers are recovered consequently in contrast with standard penalty methods there is no need for a dependence of the augmentation parameter with the time step and this latter can be kept to reasonable values to avoid to degrade too severely the conditioning of the linear operator associated to the velocity prediction step

a class of fully second order accurate projection methods for solving the incompressible navier stokes equations in this paper a continuous projection method is designed and analyzed the continuous projection method consists of a set of partial differential equations which can be regarded as an approximation of the navier stokes n s equations in each time interval of a given time discretization the local truncation error lte analysis is applied to the continuous projection methods which yields a sufficient condition for the continuous projection methods to be temporally second order accurate based on this sufficient condition a fully second order accurate discrete projection method is proposed a heuristic stability analysis is performed to this projection method showing that the present projection method can be stable the stability of the present scheme is further verified through numerical experiments the second order accuracy of the present projection method is confirmed by several numerical test cases

wild pfister forms over henselian fields k theory and conic division algebras the epicenter of this paper concerns pfister quadratic forms over a field f with a henselian discrete valuation all characteristics are considered but we focus on the most complicated case where the residue field has characteristic 2 but f does not we also prove results about round quadratic forms composition algebras generalizations of composition algebras we call conic algebras and central simple associative symbol algebras finally we give relationships between these objects and kato s filtration on the milnor k groups of f

on the error estimates for the rotational pressure correction projection methods in this paper we study the rotational form of the pressure correction method that was proposed by timmermans minev and van de vosse we show that the rotational form of the algorithm provides better accuracy in terms of the h1 norm of the velocity and of the l2 norm of the pressure than the standard form

efficient symmetric positive definite second order accurate monolithic solver for fluid solid interactions we introduce a robust and efficient method to simulate strongly coupled monolithic fluid rigid body interactions we take a fractional step approach where the intermediate state variables of the fluid and of the solid are solved independently before their interactions are enforced via a projection step the projection step produces a symmetric positive definite linear system that can be efficiently solved using the preconditioned conjugate gradient method in particular we show how one can use the standard preconditioner used in standard fluid simulations to precondition the linear system associated with the projection step of our fluid solid algorithm overall the computational time to solve the projection step of our fluid solid algorithm is similar to the time needed to solve the standard fluid only projection step the monolithic treatment results in a stable projection step i e the kinetic energy does not increase in the projection step numerical results indicate that the method is second order accurate in the l norm and demonstrate that its solutions agree quantitatively with experimental results

on a theorem of tignol for defectless extensions and its converse abstract let k v be a henselian valued field of arbitrary rank in 1990 tignol proved that if k v k v is a finite separable defectless extension of degree a prime number then the set a k k v tr k k v k 0 has a minimum element this paper extends tignol s result to all finite separable extensions moreover a characterization of finite separable defectless extensions is given by showing that k v k v is a defectless extension if and only if the set a k k has a minimum element our proof also leads to a new proof of the well known result that each finite extension of a formally adic field or more generally of a finitely ramified valued field is defectless

simulation of airflow past a 2d naca0015 airfoil using an isogeometric incompressible navier stokes solver with the spalart allmaras turbulence model abstract the work presented in this paper concerns the efforts of conducting a computational fluid dynamics cfd simulation of air flow past a fixed 2d naca0015 airfoil at high reynolds number r e 2 5 10 6 using an isogeometric finite element methodology with linear quadratic and cubic spline elements flow simulations at such high reynolds numbers require turbulence models or very high resolution the present work employs the spalart allmaras turbulence model combined with a navier stokes solver based on a chorin projection method the first development of its kind in an isogeometric finite element framework the obtained results from the simulations are compared with two sets of experimental results available in the literature

projection method i convergence and numerical boundary layers this is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations the purpose of these papers is to provide a thorough understanding of the

the brauer group of a locally compact groupoid we define the brauer group br g of a locally compact groupoid g to be the set of morita equivalence classes of pairs a consisting of an elementary c bundle a over g 0 satisfying fell s condition and an action of g on a by isomorphisms when g is the transformation groupoid x h then br g is the equivariant brauer group br h x in addition to proving that br g is a group we prove three isomorphism results first we show that if g and h are equivalent groupoids then br g and br h are isomorphic this generalizes the result that if g and h are groups acting freely and properly on a space x say g on the left and h on the right then br g x h and br h g x are isomorphic secondly we show that the subgroup br0 g of br g consisting of classes a with a having trivial dixmier douady invariant is isomorphic to a quotient e g of the collection tw g of twists over g finally we prove that br g is isomorphic to the inductive limit ext g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of the groups e g x where x varies over all principal g spaces x and g x is the imprimitivity groupoid associated to x

analysis of velocity components decoupled projection method for the incompressible navier stokes equations we study the temporal accuracy and stability of the velocity components decoupled projection method vdpm for fully discrete incompressible navier stokes equations in particular we investigate the effect of three formulations of the nonlinear convection term which include the advective skew symmetric and divergence forms on the temporal accuracy and stability second order temporal accuracy of the vdpm for both velocity and pressure is verified by establishing global error estimates in terms of a discrete l 2 norm considering the energy evolution we demonstrate that the vdpm is stable when the time step is less than or equal to a constant stability diagrams which display the distributions of the maximum magnitude of the eigenvalues of the corresponding amplification matrices are obtained using von neumann analysis these diagrams indicate that the advective form is more stable than the other formulations of the nonlinear convection term numerical tests are performed in order to support the mathematical findings involving temporal accuracy and stability and the effects of the formulations of the nonlinear convection term are analyzed overall our results indicate that the vdpm along with an advective discrete convection operator is almost unconditionally stable second order accurate in time and computationally efficient because of the non iterative solution procedure in solving the decoupled momentum equations

t duality for torus bundles with h fluxes via noncommutative topology it is known that the t dual of a circle bundle with h flux given by a neveu schwarz 3 form is the t dual circle bundle with dual h flux however it is also known that torus bundles with h flux do not necessarily have a t dual which is a torus bundle a big puzzle has been to explain these mysterious missing t duals here we show that this problem is resolved using noncommutative topology it turns out that every principal t2 bundle with h flux does indeed have a t dual but in the missing cases which we characterize the t dual is non classical and is a bundle of noncommutative tori the duality comes with an isomorphism of twisted k theories just as in the classical case the isomorphism of twisted cohomology which one gets in the classical case is replaced by an isomorphism of twisted cyclic homology

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

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

an efficient spectral projection method for the navier stokes equations in cylindrical geometries an efficient and accurate numerical scheme is presented for the three dimensional navier stokes equations in primitive variables in a cylinder the scheme is based on a spectral galerkin approximation for the space variables and a second order projection scheme for time the new spectral projection scheme is implemented to simulate unsteady incompressible flows in a cylinder

a new class of massively parallel direction splitting for the incompressible navier stokes equations abstract we introduce in this paper a new direction splitting algorithm for solving the incompressible navier stokes equations the main originality of the method consists of using the operator i xx i yy i zz for approximating the pressure correction instead of the poisson operator as done in all the contemporary projection methods the complexity of the proposed algorithm is significantly lower than that of projection methods and it is shown the have the same stability properties as the poisson based pressure correction techniques either in standard or rotational form the first order in time version of the method is proved to have the same convergence properties as the classical first order projection techniques numerical tests reveal that the second order version of the method has the same convergence rate as its second order projection counterpart as well the method is suitable for parallel implementation and preliminary tests show excellent parallel performance on a distributed memory cluster of up to 1024 processors the method has been validated on the three dimensional lid driven cavity flow using grids composed of up to 2 109 points

convergence analysis of a class of massively parallel direction splitting algorithms for the navier stokes equations in simple domains we provide a convergence analysis for a new fractional timestepping technique for the incompressible navier stokes equations based on direction splitting this new technique is of linear complexity unconditionally stable and convergent and suitable for massive parallelization

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

error estimates for time discretizations of cahn hilliard and allen cahn phase field models for two phase incompressible flows we carry out rigorous error analysis for some energy stable time discretization schemes developed in shen and yang siam j sci comput 32 3 1159 1179 2010 for a cahn hilliard phase field model and in shen and yang chin ann math ser b 31 5 743 758 2010 for an allen cahn phase field model

error analysis of a fractional time stepping technique for incompressible flows with variable density in this paper we analyze the convergence properties of a new fractional time stepping technique for the solution of the variable density incompressible navier stokes equations the main feature of this method is that contrary to other existing algorithms the pressure is determined by just solving one poisson equation per time step first order error estimates are proved and stability of a formally second order variant of the method is established

an adaptive mesh redistribution method for the incompressible mixture flows using phase field model a phase field model which describes the motion of mixtures of two incompressible fluids is presented by liu and shen c liu j shen a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method phys d 179 2003 211 228 the model is based on an energetic variational formulation in this work we develop an efficient adaptive mesh method for solving a phase field model for the mixture flow of two incompressible fluids it is a coupled nonlinear system of navier stokes equations and allen cahn phase equation phase field equation through an extra stress term and the transport term the numerical strategy is based on the approach proposed by li et al r li t tang p w zhang moving mesh methods in multiple dimensions based on harmonic maps j comput phys 170 2001 562 588 to separate the mesh moving and pde evolution in the pde evolution part the phase field equation is numerically solved by a conservative scheme with a lagrange multiplier and the coupled incompressible navier stokes equations are solved by the incremental pressure correction projection scheme based on the semi staggered grid method in the mesh moving part the mesh points are iteratively redistributed by solving the euler lagrange equations with a parameter free monitor function in each iteration the pressure and the phase are updated on the resulting new grid by a conservative interpolation formula while the velocity is re mapped in a non conservative approach a simple method for preserving divergence free is obtained by projecting the velocity onto the divergence free space after generating the new mesh at the last iterative step numerical experiments are presented to demonstrate the effectiveness of the proposed method for solving the incompressible mixture flows

a fast algorithm for direct simulation of particulate flows using conforming grids this paper presents a development of the direction splitting algorithm for flows in complex geometries proposed in 2 to the case of flows containing rigid particles the main novelty of this method is that the grid is fit to the time dependent domain at each time step which allows for an optimal spatial approximation of the navier stokes equations in general this is a very hard task for multidimensional problems however it is significantly simplified if the momentum equations are discretized by a direction splitting scheme such a scheme requires only solving a set of one dimensional problems and the grid can be easily fit to the boundary in the direction of each of these problems here we use a mac discretization stencil but the same idea can be applied to other discretizations the equations of motion of each particle are discretized explicitly and the computed particle velocities are imposed as dirichlet boundary conditions for the navier stokes equations on the adapted grid the pressure is extended within the particles in a fictitious domain fashion but the divergence stencil is fit to the particle boundaries in this paper the direction splitting algorithm with boundary fitting is compared to pure fictitious domain methods

a dixmier douady theorem for fell algebras abstract we generalise the dixmier douady classification of continuous trace c algebras to fell algebras to do so we show that c diagonals in fell algebras are precisely abelian subalgebras with the extension property and use this to prove that every fell algebra is morita equivalent to one containing a diagonal subalgebra we then use the machinery of twisted groupoid c algebras and equivariant sheaf cohomology to define an analogue of the dixmier douady invariant for fell algebras and to prove our classification theorem

correspondences between valued division algebras and graded division algebras abstract if d is a tame central division algebra over a henselian valued field f then the valuation on d yields an associated graded ring gd which is a graded division ring and is also central and graded simple over gf after proving some properties of graded central simple algebras over a graded field including a cohomological characterization of its graded brauer group it is proved that the map d gd g yields an index preserving isomorphism from the tame part of the brauer group of f to the graded brauer group of gf this isomorphism is shown to be functorial with respect to field extensions and corestrictions and using this it is shown that there is a correspondence between f subalgebras of d with center tame over f and graded gf subalgebras of gd

on error estimates of projection methods for navier stokes equations first order schemes in this paper projection methods or fractional step methods are studied in the semi discretized form for the navier stokes equations in a two or three dimensional bounded domain error estimates for the velocity and the pressure of the classical projection scheme are established via the energy method a modified projection scheme which leads to improved error estimates is also proposed

error analysis of fully discrete velocity correction methods for incompressible flows a fully discrete version of the velocity correction method proposed by guermond and shen 2003 for the time dependent navier stokes equations is introduced and analyzed it is shown that when accounting for space discretization additional consistency terms which vanish when space is not discretized have to be added to establish stability and optimal convergence error estimates are derived for both the standard version and the rotational version of the method these error estimates are consistent with those by guermond and shen 2003 as far as time discretiztion is concerned and are optimal in space for finite elements satisfying the inf sup condition

on the penalty projection method for the navier stokes equations with the mac mesh we deal with the time dependent navier stokes equations nse with dirichlet boundary conditions on the whole domain or on a part of the domain and open boundary conditions on the other part it is shown numerically that combining the penalty projection method with spatial discretization by the marker and cell scheme mac yields reasonably good results for solving the above mentioned problem the scheme which has been introduced combines the backward difference formula of second order bdf2 namely gear s scheme for the temporal approximation the second order richardson extrapolation for the nonlinear term and the penalty projection to split the velocity and pressure unknowns similarly to the results obtained for other projection methods we estimate the errors for the velocity and pressure in adequate norms via the energy method

a gauge uzawa finite element method for the time dependent viscoelastic oldroyd flows abstract in this paper we present the gauge uzawa finite element method for the time dependent viscoelastic oldroyd flow this method is a fully discrete projection method whereas most projection methods have been studied without space discretization and the ensuing analysis may not extend to this case we also give the stability analysis and the optimal error analysis the analysis is based on an iterated time discrete system with which the error function is split into a temporal error and a spatial error in order to show the effect of the method we present some numerical results the numerical results confirm our analysis and show clearly the stability and optimal convergence of the gauge uzawa finite element method for the viscoelastic oldroyd flow

nonlinear magnetohydrodynamics in axisymmetric heterogeneous domains using a fourier finite element technique and an interior penalty method the maxwell equations in the mhd limit in heterogeneous axisymmetric domains composed of conducting and non conducting regions are solved by using a mixed fourier lagrange finite element technique finite elements are used in the meridian plane and fourier modes are used in the azimuthal direction parallelization is made with respect to the fourier modes continuity conditions across interfaces are enforced using an interior penalty technique the performance of the method is illustrated on kinematic and full dynamo configurations

a formula for tignol s constant abstract let k v be a henselian valued field and l w be a finite separable extension of k v in 2004 it was proved that the set a l k defined by a l k v tr l k w l 0 has a minimum element if and only if l k e f where e f are the ramification index and the residual degree of w v i e l w k v is defectless the constant min a l k was first introduced by tignol and is referred to as tignol s constant in 2005 k ota and khanduja gave a formula for min a l k when l w k v is an extension of local fields in this paper we give this formula when l w is any finite separable defectless extension of a henselian valued field of arbitrary rank and thereby generalize some well known results of dedekind regarding different of extensions of algebraic number fields and ramification of prime ideals

a coupled prediction scheme for solving the navier stokes and convection diffusion equations this paper presents a new algorithm for the numerical solution of the navier stokes equations coupled with the convection diffusion equation after establishing convergence of the semi discrete formulation at each time step we introduce a new iterative scheme based on a projection method called the coupled prediction scheme we show that even though the predicted temperature is advected by a velocity prediction which is not necessarily divergence free the theoretical time accuracy of the global scheme is conserved from a numerical point of view this new approach gives a faster and more efficient algorithm compared to the usual fixed point approaches

a finite element pressure correction scheme for the navier stokes equations with traction boundary condition abstract we consider the popular pressure correction scheme for the solution of the time dependent navier stokes equations with traction boundary condition a finite element based method to improve the performance of the classical approach is proposed the improvement is achieved by modifying the traction boundary condition for the provisional velocity u n 1 in each time step the corresponding term consists of a simple boundary functional involving the normal derivative of the pressure correction that can be evaluated in a natural and easy way in the context of finite elements computational results show a significant improvement of the solution in particular for the pressure in the case of smooth domains

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

optimal first order error estimates of a fully segregated scheme for the navier stokes equations abstract a first order linear fully discrete scheme is studied for the incompressible time dependent navier stokes equations in three dimensional domains this scheme is based on an incremental pressure projection method and decouples each component of the velocity and the pressure solving in each time step a linear convection diffusion problem for each component of the velocity and a poisson neumann problem for the pressure using an inf sup stable and continuous finite elements approach of order o h in space unconditional optimal error estimates of order o k h are deduced for velocity and pressure without imposing constraints on the mesh size h and the time step k finally some numerical results are performed to validate the theoretical analysis and also to compare the studied scheme with other current first order segregated schemes

error analysis of pressure correction schemes for the time dependent stokes equations with open boundary conditions the incompressible stokes equations with prescribed normal stress open boundary conditions on part of the boundary are considered it is shown that the standard pressure correction method is not suitable for approximating the stokes equations with open boundary conditions whereas the rotational pressure correction method yields reasonably good error estimates these results appear to be the first ever published for splitting schemes with open boundary conditions numerical results in agreement with the error estimates are presented

an interface fitted subspace projection method for finite element simulations of particulate flows abstract a novel finite element method for the direct numerical simulation of particles in a newtonian carrier fluid is presented the proposed method is based on a fictitious or one domain formulation and a subspace projection method to account for the rigid body motion of the particles underlying equations are posed in an ale arbitrary lagrangian eulerian formulation allowing for moving computational meshes the mesh is adapted to the particles by a novel mesh smoothing approach guaranteeing both mesh optimality and a sharp representation of the particles boundaries we show that by using quadratic taylor hood finite elements for the discretization of the navier stokes equations and isoparametric elements to represent the geometry second order convergence with respect to the energy norm can be achieved as opposed to a sharp h 1 2 result which holds if no mesh adaptation is performed we present numerical examples in 2d confirming the theoretical convergence results furthermore we validate the approach by simulating the sedimentation process of a single particle and show the potential of the method by simulating a lid driven cavity flow with 100 particles

velocity correction projection methods for incompressible flows we introduce and study a new class of projection methods namely the velocity correction methods in standard form and in rotational form for solving the unsteady incompressible navier stokes equations we show that the rotational form provides improved error estimates in terms of the h1 norm for the velocity and of the l2 norm for the pressure we also show that the class of fractional step methods introduced in s a orsag m israeli and m deville j sci comput 1 1986 pp 75 111 and k e karniadakis m israeli and s a orsag j comput phys 97 1991 pp 414 443 can be interpreted as the rotational form of our velocity correction methods thus to the best of our knowledge our results provide the first rigorous proof of stability and convergence of the methods in those papers we also emphasize that contrary to those of the above groups our formulations are set in the standard l2 setting and consequently they can be easily implemented by means of any variational approximation techniques in particular the finite element methods

stability analysis of pressure correction schemes for the navier stokes equations with traction boundary conditions abstract we present a stability analysis for two different rotational pressure correction schemes with open and traction boundary conditions first we provide a stability analysis for a rotational version of the grad div stabilized scheme of bonito et al 2015 this scheme turns out to be unconditionally stable provided the stabilization parameter is suitably chosen we also establish a conditional stability result for the boundary correction scheme presented in bansch 2014 these results are shown by employing the equivalence between stabilized gauge uzawa methods and rotational pressure correction schemes with traction boundary conditions

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

shear rate projection schemes for non newtonian fluids abstract the operator splitting approach applied to the navier stokes equations gave rise to various numerical methods for the simulations of the dynamics of fluids the separate work of chorin and temam on this subject gave birth to the so called projection methods the most basic of those schemes the incremental and non incremental variant see guermond et al 2006 induce an artificial neumann boundary condition on the pressure the so called rotational incremental pressure correction scheme proposed by timmermans et al 1996 gives a consistent equation for the pressure in case of a newtonian fluids with homogeneous viscosity in this work we propose a family of projection methods for generalized newtonian fluids based on an extension of the rotational projection scheme called shear rate projections these methods produce consistent pressure when applied to generalized newtonian fluids accuracy of the methods will be illustrated using a manufactured solution numerical experiments for the flow past a cylinder with a carreau rheological model will also be presented

nodal based finite element methods with local projection stabilization for linearized incompressible magnetohydrodynamics abstract we consider the linearized resistive magnetohydrodynamics mhd model in incompressible media and its numerical solution conforming nodal based finite element approximations of the mhd model both for inf sup stable and equal order finite elements with respect to velocity and pressure are considered as opposed to a residual based stabilization method by badia et al 2013 we consider a local projection stabilization for the numerical solution a detailed stability and error analysis for the arising discrete problem is given some numerical experiments like hartmann s mhd problem and singular solutions are examined

a segregated implicit pressure projection method for incompressible flows in this paper a segregated implicit projection method is presented and investigated although high reynolds turbulent flow has been a motivating factor the present study is mainly restricted to the incompressible navier stokes equations the proposed method is analyzed for properties like stability accuracy and consistency aided by some test examples in addition two different flow cases are simulated to evaluate the ability and efficiency of the method compared with other results

projection method iii spatial discretization on the staggered grid in e liu siam j numer anal 1995 we studied convergence and the structure of the error for several projection methods when the spatial variable was kept continuous we call this the semi discrete case in this paper we address similar questions for the fully discrete case when the spatial variables are discretized using a staggered grid we prove that the numerical solution in velocity has full accuracy up to the boundary despite the fact that there are numerical boundary layers present in the semi discrete solutions