on error estimates of the penalty method for unsteady navier stokes equations the penalty method has been widely used for numerical computations of the unsteady navier stokes equations however the best error estimates available to the author s knowledge were not optimal an

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

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

on error estimates of some higher order projection and penalty projection methods for navier stokes equations this paper is a continuation of our previous work 10 on projection methods we study first existing higher order projection schemes in the semidiscretized form for the navier stokes equations one error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent neumann boundary condition satisfied by the pressure approximations we then propose a penalty projection scheme for which we obtain improved error estimates

on the topological cyclic homology of the integers this article provides a computation of the mod p homotopy groups of the fixed points of the topological hochschild homology of the ring of integers under the action of any finite subgroup of the cirle group whose order is a power of an odd prime p this leads to a computation of the topological cyclic homology groups of the ring of integers and determines also the 7 adic completion of the algebraic at theory of the p adic integers 1 introduction the topological hochschild homology j is a spec trum or infinite loop space associated functorially to each algebraic ring r or more generally to each strictly associative ring spectrum tl it was introduced originally by bokstedt in b the spectrum thh r comes equipped with a natural action of the circle group t the topological cyclic homology of r tc r is the homotopy inverse limit of a diagram of spectra with vertices the fixed point spectra thh r c of thh r under the action of the finite cyclic subgroups c of the circle the maps in this diagram are either inclusions of fixed points or certain frobenius maps particular to the topological hochschild homology construction see g the topological hochschild and cyclic homology of r are both related to the algebraic jt their relation can be depicted by a commutative diagram of spectra

projective resolutions of cohen macaulay algebras the problem of explicitly finding a free resolution minimal in some suitable sense of a module over a polynomial ring is solved in principle by the algorithm of hilbert h however this algorithm is of enormous computational difficulty if the module happens to be finite dimensional over the ground field and if the module structure is given by specifying the commuting linear transformations induced by the indeterminates then a little known result of scheja and storch resumed in sect 1 allows one to write down an explicit free resolution of the right length without computation the same idea can be used for many cohenmacaulay modules example 1 1 but although the scheja storch resolution is minimal in some cases it is not minimal in the main case of interest where the module is a factor ring of the polynomial ring and not the ground field itself

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

localization sequences for logarithmic topological hochschild homology we study the logarithmic topological hochschild homology of ring spectra with logarithmic structures and establish localization sequences for this theory our results apply for example to connective covers of periodic ring spectra like real and complex topological k theory mathematics subject classification 14f10 19d55 55p43

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

on the approximation of the unsteady navier stokes equations by finite element projection methods this paper provides an analysis of a fractional step projection method to compute incompressible viscous flows by means of finite element approximations the analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method such a theoretical distinction leads to a finite element projection method with a poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm error estimates in finite time are given an extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

a penalty projection algorithm for a monolithic fluid structure interaction solver in this paper we propose a new iterative penalty projection algorithm for a monolithic fluid structure interaction solver projection methods that split the computation of the velocity from the pressure are very popular in fluid dynamics since the boundary errors generated by the projection method are localized mainly near the boundary layers where the incorrect pressure boundary conditions are imposed however when solid regions are taken into account the pressure projected field cannot satisfy fully coupled boundary constraints imposed on external solid surfaces such as stress free conditions and due to the rigidity of the medium the boundary errors propagate deeply in the interior in order to reduce the projection errors we propose a one step penalty projection method in the fluid domain and an iterative penalty projection method in the solid region this technique decouples the pressure velocity degrees of freedom and as a consequence the computational cost in order to verify the accuracy and robustness of the proposed method we compare the results between this splitting velocity pressure algorithm and the coupled one these numerical results show stability and robustness of the proposed algorithm with a strong reduction of the computational effort a new penalty projection algorithm for a monolithic fsi solver is proposed projection method error is localized where pressure boundary conditions are imposed a one step penalty together with an iterative projection algorithm is used this technique reduces the problem degrees of freedom and the computational cost numerical results show stability and reduction of cpu effort for large problems

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

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

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 second order decoupled penalty projection method based on deferred correction for mhd in elsasser variable abstract we study the deferred correction method for the magnetohydrodynamics mhd system written in elsasser variables the proposed algorithm is based on the penalty projection with grad div stabilized taylor hood solutions of the elsasser formulation in this way second order accuracy of the method in time is obtained through the deferred correction method with excellent mass conservation properties for the proposed method stability is rigorously proven and numerical experiments are presented to verify the proposed scheme and theory

a novel vector potential formulation of 3d navier stokes equations with through flow boundaries by a local meshless method an alternative vector potential formulation is used to solve the navier stokes n s equations in 3d incompressible viscous flow problems with and without through flow boundaries difficulties of the vector potential formulation include the implementation of boundary conditions for through flow boundaries and the numerical treatment of fourth order partial differential equations the advantages on the other hand are the automatic satisfaction of the continuity equation and pressure is decoupled from the velocity the objective of this paper is to introduce the appropriate gauge and boundary conditions on the vector potential formulation by a localized meshless method to handle the divergence free property a coulomb gauge condition is enforced on the vector potential to ensure its existence and uniqueness mathematically we further improve the algorithm to through flow problems for the boundary conditions of vector potential by introducing the concept of stokes theorem based on this innovation there is no need to include an additional variable to tackle the through flow fields this process will greatly simplify the imposition of boundary conditions by the vector potential approach under certain conditions the coupled fourth order partial differential equations can be easily solved by using this meshless local differential quadrature ldq method due to the ldq capability to deal with the high order differential equations this algorithm is very attractive to solve this fourth order vector potential formulation for the n s equations as comparing to the conventional numerical schemes such as finite element or finite difference methods the proposed vector potential formulation is simpler and has improved accuracy and efficiency compared to other pressure free or pressure coupled algorithms this investigation can be regarded as the first complete study to obtain the n s solutions by vector potential formulation through a ldq method two classic 3d benchmark problems lid driven cavity and backward facing step flows are numerically solved to examine the feasibility of the improved algorithm results show the flexibility of the proposed vector potential formulation for the 3d navier stokes equations

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

implicit formulation of material point method for analysis of incompressible materials abstract material point method mpm is a particle based numerical technique with application to solid mechanics problems involving large deformations recently mpm has been successfully applied in many engineering applications including solid and solid fluid interaction problems due to its capability to model complex history dependent material behaviours however the conventional explicit mpm approach exhibits numerical limitations in incompressible or near incompressible material behaviour problems we study an implicit treatment of the pressure in mpm algorithm to simulate material incompressibility avoiding artificial pressure oscillations and significantly small time steps present in the explicit mpm approach applying chorin s projection method to the velocity pressure coupled equations an elliptic equation is solved for the pressure which is then used to solve the divergence free velocity field the results of dam break problem show no spurious pressure oscillations and large time steps compared to the traditional explicit mpm the ability of the present implicit mpm formulation to model incompressible materials will be an additional advantage in simulating materials with complex constitutive equations which exhibit a time dependent memory

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

diagram spaces and symmetric spectra abstract we present a general homotopical analysis of structured diagram spaces and discuss the relation to symmetric spectra the main motivating examples are the i spaces which are diagrams indexed by finite sets and injections and j spaces which are diagrams indexed by quillen s localization construction 1 on the category of finite sets and bijections we show that the category of i spaces provides a convenient model for the homotopy category of spaces in which every e space can be rectified to a strictly commutative monoid similarly the commutative monoids in the category of j spaces model graded e spaces using the theory of j spaces we introduce the graded units of a symmetric ring spectrum the graded units detect periodicity phenomena in stable homotopy and we show how this can be applied to the theory of topological logarithmic structures

on reducing the splitting error in yosida methods for the navier stokes equations with grad div stabilization abstract this paper analyzes the accuracy of the discretize then split yosida solver for incompressible flow problems when divergence free elements are used together with grad div stabilization with parameter the yosida method uses an inexact block lu factorization to create linear algebraic systems that are easier to solve but at the expense of accuracy we prove the difference between solutions of the exact and approximated linear algebraic systems is o 2 in the natural norms of the associated finite element problems and thus that full accuracy can be obtained by the yosida method if large is used 10 is sufficient in our numerical examples the proof is based on transforming the yosida inexact linear algebraic system into finite element problems and analyzing these problems with finite element techniques based on pointwise divergence free subspaces and their orthogonal complements