construction of hyperelliptic function fields of high three rank we present several explicit constructions of hyperelliptic function fields whose jacobian or ideal class group has large 3 rank our focus is on finding examples for which the genus and the base field are as small as possible most of our methods are adapted from analogous techniques used for generating quadratic number fields whose ideal class groups have high 3 rank but one method applicable to finding large l ranks for odd primes l 3 is new and unique to function fields algorithms examples and numerical data are included

rates of convergence for discretizations of the stochastic incompressible navier stokes equations we show strong convergence with rates for an implicit time discretization a semi implicit time discretization and a related finite element based space time discretization of the incompressible navier stokes equations with multiplicative noise in two space dimensions we use higher moments of computed iterates to optimally bound the error on a subset omega kappa of the sample space omega where corresponding paths are bounded in a proper function space and mathbb p omega kappa to 1 holds for vanishing discretization parameters this implies convergence in probability with rates and motivates a practicable acception rejection criterion to overcome possible pathwise explosion behavior caused by the nonlinearity it turns out that it is the interaction of lagrange multipliers with the stochastic forcing in the scheme which limits the accuracy of general discretely lbb stable space discretizations and strategies to overcome this problem are proposed

a divergence free reconstruction of the nonconforming virtual element method for the stokes problem abstract in this paper we propose and investigate a divergence free reconstruction of the nonconforming virtual element for the stokes problem by constructing the computable raviart thomas like interpolation operator we guarantee the independence between the velocity error estimation u u h 1 h and the continuous pressure p as it happens for the divergence free flow solver moreover this modified scheme can also inherit the advantages of the classical nonconforming virtual element method such as very general meshes including non convex and degenerate elements a unified scheme for an arbitrary order approximation accuracy k etc then we provide the optimal l 2 error estimates for the velocity gradient and the pressure by taking advantage of the raviart thomas like interpolation operator and avoiding the use of a trace inequality finally three numerical experiments are presented to conform the theoretical analysis

topological entanglement in polymers the brereton shah problem of two polymer loops topologically linked is studied by field theoretical methods in the limit that one polymer is allowed to fill a macroscopic volume at finite density rho for l 1 it is shown that typically for winding numbers m2 lalpha rho alpha 2 epsilon nu universality is observed with the critical exponents of the unconstrained system despite the presence of a dangerous renormalisation group instability here l is the length of the smaller polymer loop whose mean square size r2 approximately l2 nu nu 1 2 for random flight statistics or nu approximately 3 5 for the swelled chain case

zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

a projection method for constructing a mass conservative velocity field abstract in the numerical modeling of fluid flow and transport problems the velocity field frequently needs to be projected from one finite dimensional space into another in certain applications especially those involving modeling of multi species transport the new projected velocity field should be accurate as well as locally mass conservative in this paper a velocity projection method has been developed that is both accurate and mass conservative element by element on the projected grid the velocity correction is expressed as gradient of a scalar pressure field and the resultant poisson equation is solved using a mixed hybrid finite element method and lowest order raviart thomas spaces the conservative projection method is applied to the system of shallow water equations and a theoretical error estimate is derived

a new divergence free interpolation operator with applications to the darcy stokes brinkman equations a new local interpolation operator preserving the divergence is constructed explicitly for the hsieh clough tocher divergence free element a divergence free finite element method is applied to the darcy stokes brinkman flow in a mixed region of both free and porous media the method is of optimal order as well for the darcy flow as for the stokes flow which is not the case for most other finite elements compared to the existing nonconforming elements the divergence free element method provides a continuous solution for the velocity which is also an orthogonal projection within a hilbert subspace of the true velocity numerical tests supporting the theory are presented

on really locking free mixed finite element methods for the transient incompressible stokes equations inf sup stable mixed methods for the steady incompressible stokes equations that relax the divergence constraint are often claimed to deliver locking free discretizations however this relaxation leads to a pressure dependent contribution in the velocity error which is proportional to the inverse of the viscosity thus giving rise to a different locking phenomenon however a recently proposed modification of the right hand side alone leads to a discretization that is really locking free i e its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity in this contribution we extend this approach to the transient incompressible stokes equations where besides the right hand side also the velocity time derivative requires an improved space discretization semidiscrete and fully discrete a priori velocity and pressure error estimates are derived which show remarkable robustness properties two numerical examples illustrate the superior accura

on a reduced sparsity stabilization of grad div type for incompressible flow problems abstract we introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems we show this new operator has a similar positive effect on velocity error as the well known and very successful grad div stabilization operator but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix that is while grad div produces fully coupled block matrices i e block full the matrices arising from the new operator are block upper triangular in two dimensions and in three dimensions the 2 1 and 3 1 blocks are empty moreover the diagonal blocks of the new operator s matrices are identical to those of grad div we provide error estimates and numerical examples for finite element simulations with the new operator which reveals the significant improvement in accuracy it can provide solutions found using the new operator are also compared to those using usual grad div stabilization and in all cases solutions are found to be very similar

new hybridized mixed methods for linear elasticity and optimal multilevel solvers in this paper we present a family of new mixed finite element methods for linear elasticity for both spatial dimensions n 2 3 which yields a conforming and strongly symmetric approximation for stress applying mathcal p k 1 mathcal p k as the local approximation for the stress and displacement the mixed methods achieve the optimal order of convergence for both the stress and displacement when k ge n for the lower order case n 2 le k n the stability and convergence still hold on some special grids the proposed mixed methods are efficiently implemented by hybridization which imposes the inter element normal continuity of the stress by a lagrange multiplier then we develop and analyze multilevel solvers for the schur complement of the hybridized system in the two dimensional case provided that no nearly singular vertex on the grids the proposed solvers are proved to be uniformly convergent with respect to both the grid size and poisson s ratio numerical experiments are provided to validate our theoretical results

on conservation laws of navier stokes galerkin discretizations abstract we study conservation properties of galerkin methods for the incompressible navier stokes equations without the divergence constraint strongly enforced in typical discretizations such as the mixed finite element method the conservation of mass is enforced only weakly and this leads to discrete solutions which may not conserve energy momentum angular momentum helicity or vorticity even though the physics of the navier stokes equations dictate that they should we aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint and doing so leads us to a new formulation that conserves each of energy momentum angular momentum enstrophy in 2d helicity and vorticity for reference the usual convective formulation does not conserve most of these quantities several numerical experiments are performed which verify the theory and test the new formulation

on the determination of the grad div criterion abstract grad div stabilization adding a term grad div u to penalize violation of incompressibility has proven to be a useful tool in the simulation of incompressible flows such a term requires a choice of the coefficient and studies have begun appearing with various suggestions for its value we give an analysis herein that provides a restricted range of possible values for the coefficient in 3d turbulent flows away from walls if u l denote the large scale velocity and length respectively and is the signal to noise ratio of the body force estimates suggest that should be restricted to the range 2 24 l u 2 4 r e l u mesh independent case 2 24 l u 2 4 h l 4 3 l u mesh dependent case

on pressure approximation via projection methods for nonstationary incompressible navier stokes equations projection methods are an efficient tool to approximate strong solutions of the incompressible navier stokes equations as a major deficiency these methods often suffer from reduced accuracy for pressure iterates caused by nonphysical boundary data going along with suboptimal error estimates for pressure iterates we verify a rigorous bound for arising boundary layers in chorin s scheme under realistic regularity assumptions in a second step the new chorin penalty method is proposed where optimal rate of convergence for pressure iterates is shown

analysis of mixed finite elements methods for the stokes problem a unified approach we develop a method for the analysis of mixed finite element methods for the stokes problem in the velocity pressure formulation a technical macroelement condition which is sufficient for the classical babuska brezzi inequality to be valid is introduced using this condition we are able to verify the stability and optimal order of convergence of several known mixed finite element methods

numerical study of a regularization model for incompressible flow with deconvolution based adaptive nonlinear filtering abstract we study a trapezoidal in time finite element in space discretization of a new leray regularization model that locally chooses the filtering radius using a deconvolution based indicator function to identify regions where regularization is needed because this indicator function is mathematically based it allows us to establish a rigorous analysis of the resulting numerical algorithm we prove well posedness unconditional stability and convergence of the proposed algorithm and test the model on several benchmark problems

enforcing energy helicity and strong mass conservation in finite element computations for incompressible navier stokes simulations abstract we study a finite element scheme for the 3d navier stokes equations nse that globally conserves energy and helicity and through the use of scott vogelius elements enforces pointwise the solenoidal constraints for velocity and vorticity a complete numerical analysis is given including proofs for conservation laws unconditional stability and optimal convergence we also show the method can be efficiently computed by exploiting a connection between this method its associated penalty method and the method arising from using grad div stabilized taylor hood elements finally we give numerical examples which verify the theory and demonstrate the effectiveness of the scheme

an augmented lagrangian based approach to the oseen problem we describe an effective solver for the discrete oseen problem based on an augmented lagrangian formulation of the corresponding saddle point system the proposed method is a block triangular preconditioner used with a krylov subspace iteration like bicgstab the crucial ingredient is a novel multigrid approach for the 1 1 block which extends a technique introduced by schoberl for elasticity problems to nonsymmetric problems our analysis indicates that this approach results in fast convergence independent of the mesh size and largely insensitive to the viscosity we present experimental evidence for both isop2 p0 and isop2 p1 finite elements in support of our conclusions we also show results of a comparison with two state of the art preconditioners showing the competitiveness of our approach

a projection based stabilized finite element method for steady state natural convection problem abstract we formulate a projection based stabilization finite element technique for solving steady state natural convection problems in particular we consider heat transport through combined solid and fluid media this stabilization does not act on the large flow structures based on the projection stabilization idea finite element error analysis of the problem is investigated and optimal errors for the velocity temperature and pressure are established we also present some numerical tests which both verify the theoretical predictions and demonstrate the method s promise

grad div stablilization for stokes equations in this paper a stabilizing augmented lagrangian technique for the stokes equations is studied the method is consistent and hence does not change the continuous solution we show that this stabilization improves the well posedness of the continuous problem for small values of the viscosity coefficient we analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact uzawa method

a generic approach to searching for jacobians we consider the problem of finding cryptographically suitable jacobians by applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family we can search for jacobians containing a large subgroup of prime order for a suitable distribution of curves the complexity is subexponential in genus 2 and o n1 12 in genus 3 we give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size improving previous results our approach is particularly effective over low degree extension fields where in genus 2 we find jacobians over fp2 and trace zero varieties over fp3 with near prime orders up to 372 bits in size for p 261 1 the average time to find a group with 244 bit near prime order is under an hour on a pc

the effect of a sparse grad div stabilization on control of stationary navier stokes equations abstract this study presents a reduced sparsity grad div stabilization for the optimal control of an incompressible stationary flow problem the effect of the proposed stabilization is analyzed in terms of stability and velocity a priori error the main difference between this method and the usual grad div stabilization can be observed in numerical computations since the proposed method produces block upper triangular matrices an advantage can be seen in terms of implementation over the common grad div stabilization which produces fully coupled block matrices the stability and the convergence results are given for both the state and the adjoint state variables once the optimality conditions by using the lagrange approach are stated in numerical experiments solutions of the optimal control problem without any stabilization and the proposed method are presented in order to compare the effectiveness of the method the numerical convergence study is also presented

on the role of the helmholtz decomposition in mixed methods for incompressible flows and a new variational crime abstract according to the helmholtz decomposition the irrotational parts of the momentum balance equations of the incompressible navier stokes equations are balanced by the pressure gradient unfortunately nearly all mixed methods for incompressible flows violate this fundamental property resulting in the well known numerical instability of poor mass conservation the origin of this problem is the lack of l 2 orthogonality between discretely divergence free velocities and irrotational vector fields therefore a new variational crime for the nonconforming crouzeix raviart element is proposed where divergence free lowest order raviart thomas velocity reconstructions reestablish l 2 orthogonality this approach allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence free flow solvers in the stokes case optimal a priori error estimates for the velocity gradients and the pressure are derived moreover the discrete velocity is independent of the continuous pressure several detailed linear and nonlinear numerical examples illustrate the theoretical findings

on the accuracy of the rotation form in simulations of the navier stokes equations the rotation form of the navier stokes equations nonlinearity is commonly used in high reynolds number flow simulations it was pointed out by a few authors and not widely known apparently that it can also lead to a less accurate approximate solution than the usual email protected u form we give a different explanation of this effect related to i resolution of the bernoulli pressure and ii the scaling of the coupling between velocity and pressure error with respect to the reynolds number we show analytically that i the difference between the two nonlinearities is governed by the difference in the resolution of the bernoulli and kinematic pressures and ii a simple linear grad div stabilization ameliorates much of the bad scaling of the velocity error with respect to re the rotation form does have superior conservation properties to the alternatives and it appears to be amenable to more efficient preconditioners thus the rotational form with grad div stabilization is a promising method we also give experiments that show bad velocity approximation is tied to poor pressure resolution in either form

pressure robustness and discrete helmholtz projectors in mixed finite element methods for the incompressible navier stokes equations abstract recently it was understood how to repair a certain l 2 orthogonality of discretely divergence free vector fields and gradient fields such that the velocity error of inf sup stable discretizations for the incompressible stokes equations becomes pressure independent these new pressure robust stokes discretizations deliver a small velocity error whenever the continuous velocity field can be well approximated on a given grid on the contrary classical inf sup stable stokes discretizations can guarantee a small velocity error only when both the velocity and the pressure field can be approximated well simultaneously in this contribution pressure robustness is extended to the time dependent navier stokes equations in particular steady and time dependent potential flows are shown to build an entire class of benchmarks where pressure robust discretizations can outperform classical approaches significantly speedups will be explained by a new theoretical concept the discrete helmholtz projector of an inf sup stable discretization moreover different discrete nonlinear convection terms are discussed and skew symmetric pressure robust discretizations are proposed

a p4 bubble enriched p3 divergence free finite element on triangular grids abstract on triangular grids the continuous p k plus discontinuous p k 1 mixed finite element is stable for polynomial degree k 4 when k 3 the inf sup condition fails and the mixed finite element converges at an order that is two orders lower than the optimal order we enrich the continuous p 3 by adding some p 4 divergence free bubble functions to be exact one p 4 divergence free bubble function each component each edge we show that such an enriched p 3 p 2 mixed element is inf sup stable and converges at the optimal order numerical tests are presented comparing the new element with the p 4 p 3 element and the unstable p 3 p 2 element

computing discrete logarithms in real quadratic congruence function fields of large genus the discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems recently real quadratic congruence function fields were used to construct a public key distribution system the security of this public key system is based on the difficulty of a discrete logarithm problem in these fields in this paper we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus this algorithm is a generalization of similar algorithms for real quadratic number fields

a new family of stable mixed finite elements for the 3d stokes equations a natural mixed element approach for the stokes equations in the velocity pressure formulation would approximate the velocity by continuous piecewise polynomials and would approximate the pressure by discontinuous piecewise polynomials of one degree lower however many such elements are unstable in 2d and 3d this paper is devoted to proving that the mixed finite elements of this p k p k 1 type when k 3 satisfy the stability condition the babuska brezzi inequality on macro tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedra this type of mesh simplifies the implementation since it has no restrictions on the initial mesh the new element also suits the multigrid method

a unified framework of continuous and discontinuous galerkin methods for solving the incompressible navier stokes equation abstract in this paper we propose a unified numerical framework for the time dependent incompressible navier stokes equation which yields the h 1 h div conforming and discontinuous galerkin methods with the use of different viscous stress tensors and penalty terms for pressure robustness under minimum assumption on galerkin spaces the semi and fully discrete stability is proved when a family of implicit runge kutta methods are used for time discretization furthermore we present a unified discussion on the penalty term numerical experiments are presented to compare our schemes with classical schemes in the literature in both unsteady and steady situations it turns out that our scheme is competitive when applied to well known benchmark problems such as taylor green vortex kovasznay flow potential flow lid driven cavity flow and the flow around a cylinder

on velocity errors due to irrotational forces in the navier stokes momentum balance this contribution studies the influence of the pressure on the velocity error in finite element discretisations of the navier stokes equations four simple benchmark problems that are all close to real world applications convey that the pressure can be comparably large and is not to be underestimated in fact the velocity error can be arbitrarily large in such situations only pressure robust mixed finite element methods whose velocity error is pressure independent can avoid this influence indeed the presented examples show that the pressure dependent component in velocity error estimates for classical mixed finite element methods is sharp in consequence classical mixed finite element methods are not able to simulate some classes of real world flows even in cases where dominant convection and turbulence do not play a role

smooth ideals in hyperelliptic function fields recently several algorithms have been suggested for solving the discrete logarithm problem in the jacobians of high genus hyperelliptic curves over finite fields some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense we explicitly show how these density results can be derived all proofs are purely combinatorial and do not exploit analytic properties of generating functions

continuous mixed finite elements for the second order elliptic equation with a low order term abstract we propose a mixed finite element where the velocity in terms of darcy s law is approximated by the continuous p k lagrange elements and the pressure the prime variable is approximated by the discontinuous p k 1 elements for solving the second order elliptic equation with a low order term we show the quasi optimality for this mixed finite element method when a low order term is present the traditional inf sup condition is no longer required but the inclusion condition that the divergence of the discrete velocity space is a subspace of the discrete pressure space is required thus the taylor hood element and most other continuous pressure mixed elements do not converge numerical tests are provided on the new elements and most other popular mixed elements

a discontinuous skeletal method for the viscosity dependent stokes problem we devise and analyze arbitrary order nonconforming methods for the discretization of the viscosity dependent stokes equations on simplicial meshes we keep track explicitly of the viscosity and aim at pressure robust schemes that can deal with the practically relevant case of body forces with large curl free part in a way that the discrete velocity error is not spoiled by large pressures the method is inspired from the recent hybrid high order hho methods for linear elasticity after elimination of the auxiliary variables by static condensation the linear system to be solved involves only discrete face based velocities which are polynomials of degree k ge 0 and cell wise constant pressures our main result is a pressure independent energy error estimate on the velocity of order k 1 the main ingredient to achieve pressure independence is the use of a divergence preserving velocity reconstruction operator in the discretization of the body forces we also prove an l 2 pressure estimate of order k 1 and an l 2 velocity estimate of order k 2 the latter under elliptic regularity the local mass and momentum conservation properties of the discretization are also established finally two and three dimensional numerical results are presented to support the analysis

the continued fraction algorithm and regulator for quadratic function fields of characteristic 2 the continued fraction expansion and infrastructure for quadratic congruence function fields of odd characteristic have been well studied recently these ideas have even been used to produce cryptosystems much less is known concerning the continued fraction expansion and infrastructure for quadratic function fields of even characteristic we will explore these ideas and show that the situation is very similar to the odd characteristic case this exploration will result in a method for computing the regulator for quadratic function fields of characteristic 2

the analogue of grad div stabilization in dg methods for incompressible flows limiting behavior and extension to tensor product meshes abstract grad div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems for mitigating velocity errors that are sometimes called poor mass conservation such errors arise due to the relaxation of the divergence constraint in classical mixed methods and are excited whenever the spatial discretization has to deal with comparably large and complicated pressures in this contribution an analogue of grad div stabilization for discontinuous galerkin methods is studied here the key is the penalization of the jumps of the normal velocities over facets of the triangulation which controls the measure valued part of the distributional divergence of the discrete velocity solution our contribution is twofold first we characterize the limit for arbitrarily large penalization parameters which shows that the stabilized nonconforming discontinuous galerkin methods remain robust and accurate in this limit second we extend these ideas to the case of non simplicial meshes here broken grad div stabilization must be used in addition to the normal velocity jump penalization in order to get the desired pressure robustness effect the analysis is performed for the stokes equations and more complex flows and crouzeix raviart elements are considered in numerical examples that also show the relevance of the theory in practical settings

an extension of the crouzeix raviart space to general meshes with application to quasi incompressible linear elasticity and stokes flow in this work we introduce a discrete functional space on general polygonal or polyhedral meshes which mimics two important properties of the standard crouzeix raviart space namely the continuity of mean values at interfaces and the existence of an interpolator which preserves the mean value of the gradient inside each element the construction borrows ideas from both cell centered galerkin and hybrid finite volume methods the discrete function space is defined from cell and face unknowns by introducing a suitable piecewise affine reconstruction on a fictitious pyramidal subdivision of the original mesh two applications are considered in which the discrete space plays an important role namely i the design of a locking free primal as opposed to mixed method for quasi incompressible planar elasticity on general polygonal meshes ii the design of an inf sup stable method for the stokes equations on general polygonal or polyhedral meshes in this context we also propose a general modification applicable to any suitable discretization which guarantees that the velocity approximation is unaffected by the presence of large irrotational body forces provided a helmholtz decomposition of the right hand side is available the relation between the proposed methods and classical finite volume and finite element schemes on standard meshes is investigated finally similar ideas are exploited to mimic key properties of the lowest order raviart thomas space on general polygonal or polyhedral meshes

versatile mixed methods for the incompressible navier stokes equations abstract in the spirit of the principle of equipresence introduced by truesdell toupin the classical field theories 1960 we use the full version of the viscous stress tensor u u t 2 3 u i which was originally derived for compressible flows instead of the classical incompressible stress tensor u note that here is the dynamic viscosity coefficient and u is the velocity field in our approach the divergence free constraint for the viscous stress term is not enforced ahead of discretization instead our formulation allows the scheme itself to choose a consistent way to interpret the divergence free constraint i e the divergence free constraint is interpreted or enforced in a consistent fashion in both the mass conservation equation and the stress tensor term in the momentum equation furthermore our approach preserves the original symmetrical properties of the stress tensor e g its rotational invariance and it remains physically correct in the context of compressible flows as a result our approach facilitates versatility and code reuse in this paper we introduce our approach and establish some important mathematical properties for the resulting class of finite element schemes more precisely for general mixed methods which are not necessarily pointwise divergence free we establish the existence of a new norm induced by the full viscous bilinear form thereafter we prove the coercivity of the viscous bilinear form and the semi coercivity of a convective trilinear form in addition we demonstrate l2 stability of the discrete velocity fields for the general class of methods and by deduction the h div conforming methods finally we run some numerical experiments to illustrate the behavior of the versatile mixed methods and we make careful comparisons with a conventional h div conforming scheme

short note new connections between finite element formulations of the navier stokes equations we show the velocity solutions to the convective skew symmetric and rotational galerkin finite element formulations of the navier stokes equations are identical if scott vogelius elements are used and thus all three formulations will be the same pointwise divergence free solution velocity a connection is then established between the formulations for grad div stabilized taylor hood elements under mild restrictions the formulations velocity solutions converge to each other and to the scott vogelius solution as the stabilization parameter tends to infinity thus the benefits of using scott vogelius elements can be obtained with the less expensive taylor hood elements and moreover the benefits of all the formulations can be retained if the rotational formulation is used numerical examples are provided that confirm the theory

a numerical study of the navier stokes αβ model abstract we present a numerical study of the ns model which is a recently proposed multiscale variation of the ns model that attempts to recapture scales lost through over regularization by separately modeling dissipation range scales we develop a similarity theory for the new model which shows that it is better equipped than the ns model to capture smaller scale behavior next we propose and study an unconditionally stable optimally accurate and efficient finite element implementation for the ns model rigorous proofs for stability and convergence are provided finally we present results from two numerical experiments that demonstrate the advantages of the ns model over the ns model

longer time accuracy for incompressible navier stokes simulations with the emac formulation abstract in this paper we consider the recently introduced emac formulation for the incompressible navier stokes ns equations which is the only known ns formulation that conserves energy momentum and angular momentum when the divergence constraint is only weakly enforced since its introduction the emac formulation has been successfully used for a wide variety of fluid dynamics problems we prove that discretizations using the emac formulation are potentially better than those built on the commonly used skew symmetric formulation by deriving a better longer time error estimate for emac while the classical results for schemes using the skew symmetric formulation have gronwall constants dependent on exp c r e t with r e the reynolds number it turns out that the emac error estimate is free from this explicit exponential dependence on the reynolds number additionally it is demonstrated how emac admits smaller lower bounds on its velocity error since incorrect treatment of linear momentum angular momentum and energy induces lower bounds for l 2 velocity error and emac treats these quantities more accurately results of numerical tests for channel flow past a cylinder and 2d kelvin helmholtz instability are also given both of which show that the advantages of emac over the skew symmetric formulation increase as the reynolds number gets larger and for longer simulation times

error analysis and iterative solvers for navier stokes projection methods with standard and sparse grad div stabilization abstract this paper shows that use of a recently introduced sparse grad div stabilization can increase the accuracy of projection methods for solving the navier stokes equations sparse grad div stabilization has recently been introduced as an alternative to standard grad div stabilization which has a sparser matrix representation for both sparse and standard grad div stabilized projection methods we prove error estimates and provide numerical experiments which reveal that both stabilizations can cause a significant decrease in the error we then compare iterative solvers for the linear systems of equations arising from the use of both of the stabilizations a theoretical analysis of a simplified model problem as well as numerical tests show that iterative solvers perform better for systems arising from sparse grad div compared to standard grad div stabilized systems

flux preserving enforcement of inhomogeneous dirichlet boundary conditions for strongly divergence free mixed finite element methods for flow problems abstract we investigate a flux preserving enforcement of inhomogeneous dirichlet boundary conditions for velocity u g for use with finite element methods for incompressible flow problems that strongly enforce mass conservation typical enforcement via nodal interpolation is not flux preserving in general and it can create divergence error even when divergence free elements are used we show with analysis and numerical tests that by slightly and locally changing nodal interpolation the enforcement recovers flux preservation is optimally accurate and delivers divergence free solutions when used with divergence free finite elements

explict bounds and heuristics on class numbers in hyperelliptic function fields in this paper we provide tight estimates for the divisor class number of hyperelliptic function fields we extend the existing methods to any hyperelliptic function field and improve the previous bounds by a factor proportional to g with the help of new results we thus obtain a faster method of computing regulators and class numbers furthermore we provide experimental data and heuristics on the distribution of the class number within the bounds on the class number these heuristics are based on recent results by katz and sarnak our numerical results and the heuristics imply that our approximation is in general far better than the bounds suggest

stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection abstract we consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation free forcing in the momentum equation we provide analysis that suggests for such problems obtaining accurate solutions necessitates either the use of pointwise divergence free finite elements such as scott vogelius or heavy grad div stabilization of weakly divergence free elements the theory is demonstrated in numerical experiments for a benchmark natural convection problem where large irrotational forcing occurs with high rayleigh numbers

the parallelized pollard kangaroo method in real quadratic function fields we show how to use the parallelized kangaroo method for computing invariants in real quadratic function fields specifically we show how to apply the kangaroo method to the infrastructure in these fields we also show how to speed up the computation by using heuristics on the distribution of the divisor class number and by using the relatively inexpensive baby steps in the real quadratic model of a hyperelliptic function field furthermore we provide examples for regulators and class numbers of hyperelliptic function fields of genus 3 that are larger than those ever reported before

stabilized finite element methods for the generalized oseen problem the numerical solution of the non stationary incompressible navierstokes model can be split into linearized auxiliary problems of oseen type we present in a unique way different stabilization techniques of finite element schemes on isotropic meshes first we describe the state of the art for the classical residual based supg pspg method then we discuss recent symmetric stabilization techniques which avoid some drawbacks of the classical method these methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation finally we give a critical comparison of these methods

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

random walks on multiconnected manifolds and conformal field theory we propose a simple geometrical method which enables us to link the topological properties of a random walk on the double punctured plane and the conformal field theory characterized by the central charge c 2 and the conformal dimension delta 1 8 we discuss briefly the connection between the topological invariants obtained from the conformal methods and the algebraic alexander invariants for the simplest non trivial braid b3

analysis of a stabilized finite element method for stokes equations of velocity boundary condition and of pressure boundary condition abstract in this paper a new pressure stabilized finite element method is analyzed for solving the stokes equations the key feature of the method is using the curl integral instead of the standard dirichlet integral and using the pressure stabilization instead of the inf sup condition thus the method is very flexibly applicable to two problems of stokes equations in terms of velocity and pressure with either pressure dirichlet boundary condition or velocity dirichlet boundary condition for both stokes problems the finite element scheme and the finite element space of velocity are the same and one where only the finite element spaces of pressure have small differences a general analysis of stability and error estimates is developed and applications to both stokes problems are further analyzed the method covers the low regularity solution of very weak non h 1 space solution of stokes problem of pressure dirichlet boundary condition as well as higher regularity solutions of stokes problems of either pressure dirichlet boundary condition or velocity dirichlet boundary condition numerical results are presented to illustrate the performance of the method and to confirm the theoretical results

optimal order nonnested multigrid methods for solving finite element equations i on quasi uniform meshes we prove that the multigrid method works with optimal computational order even when the multiple meshes are not nested when a coarse mesh is not a submesh of the finer one the coarse level correction usually does not have the a projection property and does amplify the iterative error in some components nevertheless the low frequency components of the error can still be caught by the coarse level correction since the amplified highfrequency errors will be damped out by the fine level smoothing efficiently the optimal work order of the standard multigrid method can still be maintained however unlike the case of nested meshes a nonnested multigrid method with one smoothing does not converge in general no matter whether it is a v cycle or a w cycle method it is shown numerically that the convergence rates of nonnested multigrid methods are not necessarily worse than those of nested ones since nonnested multigrid methods accept quite arbitrarily related meshes we may then combine the efficiencies of adaptive refinements and of multigrid algorithms

grad div stabilization and subgrid pressure models for the incompressible navier stokes equations in this paper the grad div stabilization for the incompressible navier stokes finite element approximations is considered from two different viewpoints i as a variational multiscale approach for the pressure subgrid modeling and ii as a stabilization procedure of least square type some new error estimates for the linearized problem with the grad div stabilization are proved with the help of norms induced by the pressure schur complement operator we discuss the stabilization parameter choice arising in the frameworks of least square and multiscale methods and consider assumptions which allow to relate both approaches

dual mixed finite element methods for the stationary boussinesq problem abstract we propose and analyze two mixed approaches for numerically solving the stationary boussinesq model describing heat driven flows for the fluid equations the velocity gradient and a bernoulli stress tensor are introduced as auxiliary unknowns for the heat equation we consider primal and mixed primal formulations the latter incorporating additionally the normal component of the temperature gradient on the dirichlet boundary both dual mixed formulations exhibit the same classical structure of the navier stokes equations we derive a priori estimates and the existence of continuous and discrete solutions for the formulations in addition we prove the uniqueness of solutions and optimal order error estimates provided the data is sufficiently small numerical experiments are given which back up the theoretical results and illustrate the robustness and accuracy of both methods for a classic benchmark problem

analysis of algebraic chorin temam splitting for incompressible nse and comparison to yosida methods abstract algebraic splitting methods are a common approach to solving the saddle point linear systems that arise at each time step of an incompressible flow simulation there are two main classes of these methods those of yosida type and those of algebraic chorin temam act type with the yosida methods being predominantly used in practice we show herein through new analysis and extensive numerical testing that act methods that include the viscous term stiffness matrix in the modified a block are unconditionally stable and can be superior to yosida methods in a range of problems particular situations where the act type solvers are advantageous include problems where numerical stability is a concern as well as problems where strong enforcement of the divergence constraint is important

a low order galerkin finite element method for the navier stokes equations of steady incompressible flow a stabilization issue and iterative methods a galerkin finite element method is considered to approximate the incompressible navier stokes equations together with iterative methods to solve a resulting system of algebraic equations this system couples velocity and pressure unknowns thus requiring a special technique for handling we consider the navier stokes equations in velocity kinematic pressure variables as well as in velocity bernoulli pressure variables the latter leads to the rotation form of nonlinear terms this form of the equations plays an important role in our studies a consistent stabilization method is considered from a new view point theory and numerical results in the paper address both the accuracy of the discrete solutions and the effectiveness of solvers and a mutual interplay between these issues when particular stabilization techniques are applied 2002 published by elsevier science b v

error analysis of a fully discrete finite element variational multiscale method for the natural convection problem abstract for the natural convection problem we propose a new projection based finite element variational multiscale method by defining the stabilization terms via two local gauss integrations at the element level based on the implicit backward euler and implicit crank nicolson schemes for temporal discretization and stabilized mixed finite element spatial discretization we establish two numerical schemes for the natural convection problem unconditional stabilities of the two numerical schemes are proved we derive error bounds of the fully discrete solution which are first and second order in time respectively the optimal error estimates in space could be achieved for velocity and temperature in the h 1 semi norm and for pressure in the l 2 norm with the proper choosing of stabilized parameters however the error estimates in space are suboptimal for velocity and temperature in the l 2 norm the derived theoretical results are supported by two numerical examples

generalized finite element systems for smooth differential forms and stokes problem we provide both a general framework for discretizing de rham sequences of differential forms of high regularity and some examples of finite element spaces that fit in the framework the general framework is an extension of the previously introduced notion of finite element systems and the examples include conforming mixed finite elements for stokes equation in dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes in dimension 3 we define one low order complex which may be branched into whitney forms at a chosen index stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension the finite element spaces all consist of composite polynomials the framework guarantees some nice properties of the spaces in particular the existence of commuting interpolators it also shows that some of the examples are minimal spaces

hybridized globally divergence free ldg methods part i the stokes problem we devise and analyze a new local discontinuous galerkin ldg method for the stokes equations of incompressible fluid flow this optimally convergent method is obtained by using an ldg method to discretize a vorticity velocity formulation of the stokes equations and by applying a new hybridization to the resulting discretization one of the main features of the hybridized method is that it provides a globally divergence free approximate velocity without having to construct globally divergence free finite dimensional spaces only elementwise divergence free basis functions are used another important feature is that it has significantly less degrees of freedom than all other ldg methods in the current literature in particular the approximation to the pressure is only defined on the faces of the elements on the other hand we show that as expected the condition number of the schur complement matrix for this approximate pressure is of order h 2 in the mesh size h finally we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates

mass corrections for the conservative coupling of flow and transport on collocated meshes buoyancy driven flow models demand a careful treatment of the mass balance equation to avoid spurious source and sink terms in the non linear coupling between flow and transport in the context of finite elements it is therefore commonly proposed to employ sufficiently rich pressure spaces containing piecewise constant shape functions to obtain local or even strong mass conservation in three dimensional computations this usually requires nonconforming approaches special meshes or higher order velocities which make these schemes prohibitively expensive for some applications and complicate the implementation into legacy code in this paper we therefore propose a lean and conservatively coupled scheme based on standard stabilized linear equal order finite elements for the stokes part and vertex centered finite volumes for the energy equation we show that in a weak mass balance it is possible to recover exact conservation properties by a local flux correction which can be computed efficiently on the control volume boundaries of the transport mesh we discuss implementation aspects and demonstrate the effectiveness of the flux correction by different two and three dimensional examples which are motivated by geophysical applications

a connection between scott vogelius and grad div stabilized taylor hood fe approximations of the navier stokes equations this article studies two methods for obtaining excellent mass conservation in finite element computations of the navier stokes equations using continuous velocity fields with a particular mesh construction the scott vogelius element pair has recently been shown to be inf sup stable and have optimal approximation properties while also providing pointwise mass conservation we present herein the first numerical tests of this element pair for the time dependent navier stokes equations we also prove that the limit of the grad div stabilized taylor hood solutions to the navier stokes problem converges to the scott vogelius solution as the stabilization parameter tends to infinity that is we provide theoretical justification that choosing the grad div parameter large does not destroy the solution numerical tests are provided which verify the theory and show how both scott vogelius and grad div stabilized taylor hood with large stabilization parameter elements can provide accurate results with excellent mass conservation for navier stokes approximations

real and imaginary quadratic representations of hyperelliptic function fields a hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field if at least one of the rational prime divisors is rational over the field of constants then it also can be represented as an imaginary quadratic extension of the rational function field the arithmetic in the divisor class group can be realized in the second case by cantor s algorithm we show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved furthermore we show how the two representations are connected and compare the computational complexity

a family of q_ k 1 k timesq_ k k 1 divergence free finite elements on rectangular grids it is shown that the conforming q k 1 k times q k k 1 q k mixed element is stable and provides optimal order of approximation for the stokes equations on rectangular grids for all k ge2 here q k 1 k stands for the space of continuous piecewise polynomials of degree k 1 or less in x direction and of degree k or less in y direction q k is the space of discontinuous polynomials of separated degree k or less with spurious modes filtered to be precise q k is the divergence of the discrete velocity space q k 1 k times q k k 1 polynomials of different degrees in x and y are used for different components of velocity so that the resulting finite element solution is also divergence free point wise this is the first divergence free element found on nontriangular grids an efficient iterative method is presented where the discrete pressure function is produced as a byproduct numerical tests are provided confirming the theory

on the convergence rate of grad div stabilized taylor hood to scott vogelius solutions for incompressible flow problems abstract it was recently proven in case et al 2010 2 that under mild restrictions grad div stabilized taylor hood solutions of navier stokes problems converge to the scott vogelius solution of that same problem however even though the analytical rate was only shown to be 1 2 where is the stabilization parameter the computational results suggest the rate may be improvable to 1 we prove herein the analytical rate is indeed 1 and extend the result to other incompressible flow problems including leray and mhd numerical results are given that verify the theory

refined a posteriori error estimation for classical and pressure robust stokes finite element methods recent works showed that pressure robust modifications of mixed finite element methods for the stokes equations outperform their standard versions in many cases this is achieved by divergence free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities in this paper we develop a posteriori error control which reflects this robustness the main difficulty lies in the volume contribution of the standard residual based approach that includes the l 2 norm of the right hand side however the velocity is only steered by the divergence free part of this source term an efficient error estimator must approximate this divergence free part in a proper manner otherwise it can be dominated by the pressure error to overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the navier stokes equations the novel error estimators only take the hbox curl of the right hand side into account and so lead to provably reliable efficient and pressure independent upper bounds in case of a pressure robust method in particular in pressure dominant situations this is also confirmed by some numerical examples with the novel pressure robust modifications of the taylor hood and mini finite element methods

explicit eigenvalue bounds of differential operators defined by symmetric positive semi definite bilinear forms abstract recently the eigenvalue problems formulated with symmetric positive definite bilinear forms have been well investigated with the aim of explicit bounds for the eigenvalues in this paper the existing theorems for bounding eigenvalues are further extended to deal with the case of eigenvalue problems defined by positive semi definite bilinear forms as an application the eigenvalue estimation theorems are applied to the error constant estimation for polynomial projections

point counting on picard curves in large characteristic we present an algorithm for computing the cardinality of the jacobian of a random picard curve over a finite field if the underlying field is a prime field fp the algorithm has complexity o p

pressure separation a technique for improving the velocity error in finite element discretisations of the navier stokes equations abstract this paper presents a technique to improve the velocity error in finite element solutions of the steady state navier stokes equations this technique is called pressure separation it relies upon subtracting the gradient of an appropriate approximation of the pressure on both sides of the navier stokes equations with this the finite element error estimate can be improved in the case of higher reynolds numbers for practical reasons the pressure separation can be applied above all for finite element discretisations of the navier stokes equations with piecewise constant pressure this paper presents a computational study of five ways to compute an appropriate approximation of the pressure these ways are assessed on two and three dimensional examples they are compared with respect to the error reduction in the discrete velocity and the computational overhead

non body fitted fluid structure interaction divergence conforming b splines fully implicit dynamics and variational formulation abstract immersed boundary ib methods deal with incompressible visco elastic solids interacting with incompressible viscous fluids a long standing issue of ib methods is the challenge of accurately imposing the incompressibility constraint at the discrete level we present the divergence conforming immersed boundary dcib method to tackle this issue the dcib method leads to completely negligible incompressibility errors at the eulerian level and various orders of magnitude of increased accuracy at the lagrangian level compared to other ib methods furthermore second order convergence of the incompressibility error at the lagrangian level is obtained as the discretization is refined in the dcib method the eulerian velocity pressure pair is discretized using divergence conforming b splines leading to inf sup stable and pointwise divergence free eulerian solutions the lagrangian displacement is discretized using non uniform rational b splines which enables to robustly handle large mesh distortions the data transfer needed between the eulerian and lagrangian descriptions is performed at the quadrature level using the same spline basis functions that define the computational meshes this conduces to a fully variational formulation sharp treatment of the fluid solid interface and a 0 5 increase in the convergence rate of the eulerian velocity and the lagrangian displacement measured in l 2 norm in comparison with using discretized dirac delta functions for the data transfer by combining the generalized method and a block iterative solution strategy the dcib method results in a fully implicit discretization which enables to take larger time steps various two and three dimensional problems are solved to show all the aforementioned properties of the dcib method along with mesh independence studies verification of the numerical method by comparison with the literature and measurement of convergence rates

new mixed elements for maxwell equations new inf sup stable mixed elements are proposed and analyzed for solving the maxwell equations in terms of electric field and lagrange multiplier nodal continuous lagrange elements of any order on simplexes in two and three dimensional spaces can be used for the electric field the multiplier is compatibly approximated always by the discontinuous piecewise constant elements a general theory of stability and error estimates is developed when applied to the eigenvalue problem we show that the proposed mixed elements provide spectral correct spurious free approximations essentially optimal error bounds only up to an arbitrarily small constant are obtained for eigenvalues and for both singular and smooth solutions numerical experiments are performed to illustrate the theoretical results

topological properties of a two dimensional polymer chain in the lattice of obstacles a new method of calculating the topological properties of a polymer chain in the lattice of obstacles on the plane is considered in detail it is shown that by applying the method of conformal transformations it is possible to reduce the problem of calculation of the partition function of the polymer chain in a given topological state with respect to the lattice of obstacles to a much more simple problem of the calculation of the green function of a free random walk without any topological restrictions on a riemann surface

computation of the class number and class group of a complex cubic field let h and g be respectively the class number and the class group of a complex cubic field of discriminant a a method is described which makes use of recent ideas of lenstra and schoof to develop fast algorithms for finding h and g under certain riemann hypotheses it is shown that these algorithms will compute h in o 1a 1 5 elementary operations and g in o 1a 1 4 elementary operations finally the results of running some computer programs to determine h and g for all pure cubic fields 9 vdi with 2 s d 30 000 are summarized

divergence free finite elements on tetrahedral grids for k 6 it was shown two decades ago that the p k p k 1 mixed element on triangular grids approximating the velocity by the continuous p k piecewise polynomials and the pressure by the discontinuous p k 1 piecewise polynomials is stable for all k 4 provided the grids are free of a nearly singular vertex the problem with the method in 3d was posted then and remains open the problem is solved partially in this work it is shown that the p k p k 1 element is stable and of optimal order in approximation on a family of uniform tetrahedral grids for all k 6 the analysis is to be generalized to non uniform grids when we can deal with the complicity of 3d geometry for the divergence free elements the finite element spaces for the pressure can be avoided in computation if a classic iterated penalty method is applied the finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity numerical tests are provided

an optimal order multigrid method for biharmonic c finite element equations in this paper we study a special multigrid method for solving large linear systems which arise from discretizing biharmonic problems by the hsieh clough tocher c1 macro finite elements or several otherc1 finite elements since the multiplec1 finite element spaces considered are not nested the nodal interpolation operator is used to transfer functions between consecutive levels in the multigrid method this method converges with the optimal computational order

isogeometric divergence conforming b splines for the unsteady navier stokes equations divergence conforming b splines are developed for application to the incompressible navier stokes equations on geometrically mapped domains these enable smooth pointwise divergence free solutions and thus satisfy mass conservation in the strongest possible sense semi discrete methods based on divergence conforming b splines are shown to conserve linear and angular momentum and satisfy balance laws for energy vorticity enstrophy and helicity these are geometric structure preserving quantities and numerical simulations that are sensitive to them are shown to be qualitatively correct and quantitatively accurate the methods developed are anticipated to open new doors to the practical calculation of complex flows and to studies of their physical behavior