infinitely often autoreducible sets a set a is autoreducible if one can compute for all x the value a x by querying a only at places y neq x furthermore a is infinitely often autoreducible if for infinitely many x the value a x can be computed by querying a only at places y neq x for all other x the computation outputs a special symbol to signal that the reduction is undefined it is shown that for polynomial time turing and truth table autoreducibility there are a b c in the class exp of all exponential time computable sets such that a is not infinitely often turing autoreducible b is turing autoreducible but not infinitely often truth table autoreducible and c is truth table autoreducible with g n 1 queries but not infinitely often turing autoreducible with g n queries here n is the length of the input g is nondecreasing and there exists a polynomial p such that p n bounds both the computation time and the value of g at input of length n furthermore connections between notions of infinitely often autoreducibility and notions of approximability are investigated the hausdorff dimension of the class of sets which are not infinitely often autoreducible is shown to be 1

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

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

mixed stabilized finite element methods based on backward difference adams bashforth scheme for the time dependent variable density incompressible flows in this paper we consider a second order mixed stabilized finite element method based on pressure projection method for variable density incompressible flows the originality of the proposed approach is to use a stabilized method based on the difference between a consistent and an under integrated mass matrix of the pressure for variable density incompressible flows approximated by the lowest equal order finite element pairs a second order backward difference bdf for the temporal term and a second order adams bashforth ab for the explicit treatment of the nonlinear term lead to the presented second order bdf ab scheme the stability of the method was proved and the performance of the method is numerically illustrated finally comparison with some established methods a series of numerical experiments are given to show that this method has better stability and accuracy

unconditionally stable gauge uzawa finite element schemes for incompressible natural convection problems with variable density abstract we construct in this paper two gauge uzawa schemes one in conserved form and the other in convective form for solving natural convection problems with variable density and prove that the first order versions of both schemes are unconditionally stable we also show that a full discretized version of the conserved scheme with finite elements is also unconditionally stable these schemes lead to a sequence of decoupled elliptic equations to solve at each step hence they are very efficient and easy to implement we present several numerical tests to validate the analysis and demonstrate the effectiveness of these schemes for simulating natural convection problems with large density differences

a generalization of resource bounded measure with application to the bpp vs exp problem we introduce resource bounded betting games and propose a generalization of lutz s resource bounded measure in which the choice of the next string to bet on is fully adaptive lutz s martingales are equivalent to betting games constrained to bet on strings in lexicographic order we show that if strong pseudorandom number generators exist then betting games are equivalent to martingales for measure on e and exp however we construct betting games that succeed on certain classes whose lutz measures are important open problems the class of polynomial time turing complete languages in exp and its superclass of polynomial time turing autoreducible languages if an exp martingale succeeds on either of these classes or if betting games have the finite union property possessed by lutz s measure one obtains the nonrelativizable consequence mbox bpp neq mbox exp we also show that if mbox exp neq mbox ma then the polynomial time truth table autoreducible languages have lutz measure zero whereas if mbox exp mbox bpp they have measure one

a phase field model and its numerical approximation for two phase incompressible flows with different densities and viscosities modeling and numerical approximation of two phase incompressible flows with different densities and viscosities are considered a physically consistent phase field model that admits an energy law is proposed and several energy stable efficient and accurate time discretization schemes for the coupled nonlinear phase field model are constructed and analyzed ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for problems with large density and viscosity ratios

convergence of the discontinuous galerkin method for discontinuous solutions we consider linear first order scalar equations of the form rho t rm div rho v a rho f with appropriate initial and boundary conditions it is shown that approximate solutions computed using the discontinuous galerkin method will converge in ltwoltwo when the coefficients v and a and data f satisfy the minimal assumptions required to establish existence and uniqueness of solutions in particular v need not be lipschitz so characteristics of the equation may not be defined and the solutions being approximated my not have bounded variation

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

a posteriori error estimation and dynamic adaptivity for symmetric discontinuous galerkin approximations of reactive transport problems a symmetric discontinuous galerkin scheme is applied for solving reactive transport problems an explicit a posteriori error estimator in l 2 l 2 is derived for the semi discrete scheme applied to transport with general kinetic reactions explicit a posteriori error estimators in terms of linear functionals and negative norms are also obtained for transport problems with simple reactions an adaptive approach based on the a posteriori error estimators is proposed to modify the mesh dynamically numerical examples demonstrate the efficiency and effectivity of the estimators moreover it is shown in this study that the proposed adaptivity eliminates the need of slope limiters 2005 elsevier b v all rights reserved

a projection fem for variable density incompressible flows abstract this work describes a new finite element projection method for the computation of incompressible viscous flows of nonuniform density one original idea of the proposed method consists in factorizing the density variable partly outside and partly inside the time evolution operator in the momentum equation to prevent spatial discretization errors in the mass conservation to affect the kinetic energy balance of the fluid it is shown that unconditional stability in the incremental version of the projection method is possible provided two projections are performed per time step in particular a second order accurate bdf projection method is presented and its numerical performance is illustrated by test computations and comparisons

decoupled energy stable schemes for phase field models of two phase incompressible flows in this paper we construct two classes based on stabilization and convex splitting of decoupled unconditionally energy stable schemes for cahn hilliard phase field models of two phase incompressible flows at each time step these schemes require solving only a sequence of elliptic equations including a pressure poisson equation furthermore all of these elliptic equations are linear for the schemes based on stabilization making them the first to the best of the authors knowledge totally decoupled linear unconditionally energy stable schemes for phase field models of two phase incompressible flows thus the schemes constructed in this paper are very efficient and easy to implement

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

gauge uzawa methods for incompressible flows with variable density two new gauge uzawa schemes are constructed for incompressible flows with variable density one is in the conserved form while the other is in the convective form it is shown that the first order versions of both schemes in their semi discretized form are unconditionally stable numerical experiments indicate that the first order resp second order versions of the two schemes lead to first order resp second order convergence rate for all variables and that these schemes are suitable for handling problems with large density ratios such as in the situation of air bubble rising in water

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

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

fully iterative ilu preconditioning of the unsteady navier stokes equations for gpgpu abstract in this work we investigate the numerical difficulties that arise in optimizing the efficiency of newtonian fluids simulations on a massively parallel computing hardware like a gpu in particular we will concentrate on the resulting algebraic problem we will present an approximate fully iterative ilu preconditioner and we will show that this solution is very efficient on a gpu if compared with an intrinsic massively parallel preconditioner like the diagonal preconditioner which indeed goes faster than more robust techniques like ilu despite their strong decrease in the number of iterations we refer to gmres as the iterative scheme used to solve the linear system in particular we will deal with the solution of incompressible flows with variable density and we will investigate the interplay between reynolds and atwood numbers we will show that the numerical simulation at medium high reynolds numbers produces linear systems whose matrices can be reasonably preconditioned with the diagonal preconditioner while at low reynolds numbers the higher viscosity of the fluid makes the diagonal preconditioner ineffective in the solution time requested from gmres and decreasing the reynolds number unable to let gmres converge at all in this situation we will show how an adequate iterative approach to the parallel solution of the triangular systems that result from the ilu preconditioning turns out to be robust and efficient we will show numerical results for variable density fluids discretized with the scheme described in calgaro et al 2008 in classical benchmarks and in particular in the well known rayleigh taylor instability

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

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 languages reducible to algorithmically randomlanguages in this paper languages bounded reducible to algorithmically random languages are studied these are the languages whose characteristic sequences are algorithmically random as defined by martin lof inform and control 9 1966 pp 602 619 here rand denotes the class of algorithmically random languages the reducibilities le cal r are very general but are defined so that if a in r b then there is a machine m with the properties that l m b a and every computation of m relative to any oracle halts book lutz and wagner math systems theory 27 1994 pp 201 209 studied sf almost r defined to be a mid for almost every b a le cal r b they showed that sf almost r r rand cap rec where rec denotes the class of recursive languages so that sf almost r is the recursive part of r rand in this paper this characterization is strengthened by showing that for every b in rand sf almost r r b cap rec a pair a b of languages is an independent pair of algorithmically random languages if a oplus b in rand in this paper it is shown that for every r and for every independent pair a b sf almost r r a cap r b

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

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

a new fractional time stepping method for variable density incompressible flows this paper describes a new method for solving variable density incompressible viscous flows we have dealt with the momentum equation and the divergence free constraint in a new manner by rewriting the original equations the originality of the proposed approach is that we have used different numerical methods to evaluate the evolution of the velocity and pressure compared with some established methods the proposed approach is parameter free more flexible and can avoid the difficulties caused by the original equations the stability analysis of the method is performed to show that our method is stable finally numerical experiments are given to show the accuracy efficiency and validness of this method for variable density incompressible flows

a priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems we analyze three discontinuous galerkin approximations for solving elliptic problems in two or three dimensions in each one the basic bilinear form is nonsymmetric the first one has a penalty term on edges the second has one constraint per edge and the third is totally unconstrained for each of them we prove hp error estimates in the h1 norm optimal with respect to h the mesh size and nearly optimal with respect to p the degree of polynomial approximation we establish these results for general elements in two and three dimensions for the unconstrained method we establish a new approximation result valid on simplicial elements l2 estimates are also derived for the three methods

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 pseudorandom oracle characterization of bpp it is known from work of bennett and gill siam j comput 10 1981 pp 96 113 and of ambos spies in proc 1st structure in complexity theory conference 1986 pp 23 34 that the following conditions are equivalent i l in text bpp ii for almost all oracles a l in text p a it is shown here that the following conditions are also equivalent to i and ii iii the set of oracles a for which l in text p a has pspace measure 1 iv for every pspace random oracle a l in text p a it follows from this characterization and its proof that almost every a in text espace is leqslant t text p hard for text bpp a succinctly the main content of the proof is that pseudorandom generators exist relative to every pseudorandom oracle

error estimate of gauge uzawa methods for incompressible flows with variable density abstract in this paper we construct a positivity preserving gauge uzawa method for the semi discrete in time scheme of incompressible viscous flows with variable density and establish its stability and error estimates we also construct and implement a fully discrete scheme with finite elements in space and derive its positivity preserving and stability result

relative to a random oracle a bf p a ne bf np a ne text co bf np a with probability 1 let a be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to a with probability 1 each of the relativized classes textbf logspace a bf p a bf np a bf pp a and textbf pspace a is properly contained in the next also bf np a ne text co bf np a with probability 1 by contrast with probability 1 the class bf p a coincides with the class bf bpp a of languages recognized by probabilistic oracle machines with error probability uniformly bounded below tfrac 1 2 bf np a is shown with probability 1 to contain a bf p a immune set i e a set having no infinite subset in bf p a the relationship of bf p a immunity to p sparseness and bf np a completeness is briefly discussed bf p a immune sets in bf np a can be sparse or moderately dense but not co sparse relativization with respect to a random length preserving permutation pi instead of a rand

entropy viscosity method for nonlinear conservation laws a new class of high order numerical methods for approximating nonlinear conservation laws is described entropy viscosity method the novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand this new approach does not use any flux or slope limiters applies to equations or systems supplemented with one or more entropy inequalities and does not depend on the mesh type and polynomial approximation various benchmark problems are solved with finite elements spectral elements and fourier series to illustrate the capability of the proposed method

on the autoreducibility of random sequences a binary sequence a a 0 a 1 ldots is called infinitely often i o turing au to re duc ible if a is reducible to itself via an oracle turing machine that never queries its oracle at the current input outputs either a x or a don t know symbol on any given input x and outputs a x for infinitely many x if in addition the oracle turing machine terminates on all inputs and oracles a is called i o truth table autoreducible we obtain the somewhat counterintuitive result that every martin l of random sequence in fact even every rec random or p random sequence is i o truth table autoreducible furthermore we investigate the question of how dense the set of guessed bits can be when i o autoreducing a random sequence we show that rec random sequences are never i o truth table autoreducible such that the set of guessed bits has positive constant density in the limit and that a similar assertion holds for martin l of random sequences and i o turing autoreducibility on the other hand we show that for any rational valued computable function r that goes nonascendingly to zero any rec random sequence is i o truth table autoreducible such that on any prefix of length m at least a fraction of r m of the m bits in the prefix are guessed we include a self contained account of the hat problem a puzzle that has received some attention outside of theoretical computer science the hat problem asks for guessing bits of a finite sequence thus illustrating the notion of i o autoreducibility in a finite setting the solution to the hat problem is then used as a module in the proofs of the positive results on i o autoreducibility

on 1 factorizations of bipartite kneser graphs abstract it is a challenging open problem to construct an explicit 1 factorization of the bipartite kneser graph h v t which contains as vertices all t element and v t element subsets of v 1 v where v 2 t and an edge between any two vertices when one is a subset of the other in this paper for the special case where t 2 and v is an odd prime power we construct such 1 factorizations using perpendicular arrays we also revisit two known 1 factorizations of h 2 t 1 t the lexical factorization and modular factorization among other results we give interesting alternative definitions of these 1 factorizations and design an optimal algorithm for computing the lexical factorization

convergence of numerical approximations of the incompressible navier stokes equations with variable density and viscosity we consider numerical approximations of incompressible newtonian fluids having variable possibly discontinuous density and viscosity since solutions of the equations with variable density and viscosity may not be unique numerical schemes may not converge if the solution is unique then approximate solutions computed using the discontinuous galerkin method to approximate the convection of the density and stable finite element approximations of the momentum equation converge to the solution if the solution is not unique a subsequence of these approximate solutions will converge to a solution

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 conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

every sequence is reducible to a random one every infinite sequence is turing reducible to an infinite sequence which is random in the sense of martin lof