finite element solution of the stokes problem with dominating coriolis force abstract the objective of this paper is to present a finite element formulation to solve the stokes problem with coriolis force this force results in a skew symmetric term in the weak formulation of the problem that deteriorates the stability of the standard galerkin finite element method when the viscosity is small we show that the stability is worsened due to the presence of the pressure gradient to enforce the incompressibility of the flow the relevance of this effect depends on the relative importance of the viscous force and the coriolis force which is measured by the ekman number when it is small oscillations occur using the galerkin approach to overcome them we propose two different methods based on a consistent modification of the basic galerkin formulation both methods eliminate the oscillations keeping the accuracy of the formulation and enhancing its numerical stability

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

collision in a cross shaped domain a steady 2d navier stokes example demonstrating the importance of mass conservation in cfd abstract in the numerical simulation of the incompressible navier stokes equations different numerical instabilities can occur while instability in the discrete velocity due to dominant convection and instability in the discrete pressure due to a vanishing discrete ladyzhenskaya babuska brezzi lbb constant are well known instability in the discrete velocity due to a poor mass conservation at high reynolds numbers sometimes seems to be underestimated at least when using conforming galerkin mixed finite element methods like the taylor hood element the classical grad div stabilization for enhancing discrete mass conservation is often neglected in practical computations though simple academic flow problems showing the importance of mass conservation are well known these examples differ from practically relevant ones since specially designed force vectors are prescribed therefore we present a simple steady navier stokes problem in two space dimensions at reynolds number 1024 a colliding flow in a cross shaped domain where the instability of poor mass conservation is studied in detail and where no force vector is prescribed

stabilized finite element schemes with lbb stable elements for incompressible flows we study stabilized fe approximations of supg type to the incompressible navier stokes problem revisiting the analysis for the linearized model we show that for conforming lbb stable elements the design of the stabilization parameters for many practical flows differs from that commonly suggested in literature and initially designed for the case of equal order approximation then we analyze a reduced supg scheme often used in practice for lbb stable elements to provide the reduced scheme with appropriate stability estimates we introduce a modified lbb condition which is proved for a family of fe approximations the analysis is given for the linearized equations numerical experiments for some linear and nonlinear benchmark problems support the theoretical results

dyson s crank of a partition holds he was thus led to conjecture the existence of some other partition statistic which he called the crank this unknown statistic should provide a combinatorial interpretation of p lln 6 in the same way that 1 1 and 1 2 treat the primes 5 and 7 in 4 5 one of us was able to find a crank relative to vector partitions as follows for a partition 7r let 7r be the number of parts of ir and cr n be the sum of the parts of ir or the number ir is partitioning with the convention 0 for the empty partition 0 of 0 let

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

a right inverse for the divergence operator in spaces of piecewise polynomials in the first part of this paper we study in detail the properties of the divergence operator acting on continuous piecewise polynomials on some fixed triangulation more specifically we characterize the range and prove the existence of a maximal right inverse whose norm grows at most algebraically with the degree of the piecewise polynomials in the last part of this paper we apply these results to thep version of the finite element method for a nearly incompressible material with homogeneous dirichlet boundary conditions we show that thep version maintains optimal convergence rates in the limit as the poisson ratio approaches 1 2 this fact eliminates the need for any reduced integration such as customarily used in connection with the more standardh version of the finite element method

a gradient robust well balanced scheme for the compressible isothermal stokes problem abstract a novel notion for constructing a well balanced scheme a gradient robust scheme is introduced and a showcase application for the steady compressible isothermal stokes equations in a nearly hydrostatic situation is presented gradient robustness means that gradient fields in the momentum balance are well balanced by the discrete pressure gradient which is possible on arbitrary unstructured grids the scheme is asymptotic preserving in the sense that it reduces for low mach numbers to a recent inf sup stable and pressure robust discretization for the incompressible stokes equations the convergence of the coupled fem fvm scheme for the nonlinear isothermal stokes equations is proved by compactness arguments numerical examples illustrate the numerical analysis and show that the novel approach can lead to a dramatically increased accuracy in nearly hydrostatic low mach number flows numerical examples also suggest that a straightforward extension to barotropic situations with nonlinear equations of state is feasible

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 hybrid high order method for the incompressible navier stokes problem robust for large irrotational body forces abstract we develop a novel hybrid high order method for the incompressible navier stokes problem robust for large irrotational body forces the key ingredients of the method are discrete versions of the body force and convective contributions in the momentum equation formulated in terms of a globally divergence free velocity reconstruction two key properties are mimicked at the discrete level namely the invariance of the velocity with respect to irrotational body forces and the non dissipativity of the convective term a full convergence analysis is carried out showing optimal orders of convergence under a smallness condition involving only the solenoidal part of the body force the performance of the method is illustrated by a complete panel of numerical tests including comparisons that highlight the benefits with respect to more standard formulations

fuzzy sets in pattern recognition methodology and methods abstract an objective of the paper is to discuss a state of the art of methodology and algorithms of fuzzy sets in the field of pattern recognition in real world recognition and classification problems we are faced with fuzziness that is connected with diverse facets of cognitive activity of the human being an origin of sources of fuzziness is related to labels expressed in feature space as well as to labels of classes taken into account in classification procedures an evident difference between a way of information processing by means of probability and fuzzy set theory and a way of interpretation of results is explained in detail in sequel methods of pattern recognition are studied in two main streams namely supervised and unsupervised learning different approaches to designing of classification schemes as e g relation calculus decision making approach etc are put into account a method of feature selection with the aid of fuzzy measure and fuzzy integral is introduced in clustering techniques methods relying on minimization of objective function are analyzed in detail also a problem of cluster validity expressed in terms of clustering indices is addressed numerical examples are also provided

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

new finite element methods in computational fluid dynamics by h div elements in this paper the authors present two formulations for the stokes problem which make use of the existing h rm div elements of the raviart thomas type originally developed for the second order elliptic problems in addition two new h rm div elements are constructed and analyzed particularly for the new formulations optimal order error estimates are established for the corresponding finite element solutions in various sobolev norms the finite element solutions feature a full satisfaction of the continuity equation when existing raviart thomas type elements are employed in the numerical scheme

superconvergence and a posteriori error estimation in triangular mixed finite elements in this paper we prove superconvergence results for the vector variable when lowest order triangular mixed finite elements of raviart thomas type 17 on uniform triangulations are used i e that the h mbox div omega distance between the approximate solution and a suitable projection of the real solution is of higher order than the h mbox div omega error we prove results for both dirichlet and neumann boundary conditions recently duran 9 proved similar results for rectangular mixed finite elements and superconvergence along the gauss lines for rectangular mixed finite elements was considered by douglas ewing lazarov and wang in 11 8 and 18 the triangular case however needs some extra effort using the superconvergence results a simple postprocessing of the approximate solution will give an asymptotically exact a posteriori error estimator for the l 2 omega error in the approximation of the vector variable

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

direct and inverse problem in comparison of fuzzy data abstract discussed is the direct and inverse problem of comparison of two linguistic quantities specified as fuzzy sets relations in the same universe of discourse the proposed way of comparison makes use of some logical properties of connectives and implication operators stemming from the foundations of fuzzy sets a list of plausible equality indices is provided the inverse problem is formulated a set of its solutions is provided direct impacts of the discussed approach on development of fuzzy modelling are clearly indicated also it is pointed out how fuzzy sets of higher type or interval valued sets can be generated as a result of analysis of performance of any fuzzy model

knowledge based networks in classification problems abstract the paper proposes a distributed computational structure called knowledge based network as a classification scheme in pattern recognition unlike the existing architectures and algorithms of pattern recognition the network allows for an explicit representation of domain classification knowledge while maintaining its learning capabilities the knowledge based network blends useful properties of knowledge based systems namely explicit knowledge representation with those advantageous for neural networks viz learning the network is composed of basic and and or neurons fuzzy clustering constitutes a preprocessing phase leading towards developing geometric constructs they contribute to a conceptual level around which numerical processing of the classifier is centred

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

values of the legendre chi and hurwitz zeta functions at rational arguments we show that the hurwitz zeta function a and the legendre chi function x z defined by formula math respectively form a discrete fourier transform pair many formulae involving the values of these functions at rational arguments most of them unknown are obtained as a corollary to this result among them is the further simplification of the summation formulae from our earlier work on closed form summation of some trigonometric series for rational arguments also these transform relations make it likely that other results can be easily recovered and unified in a more general context

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

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

the maximum angle condition for mixed and nonconforming elements application to the stokes equations for the lagrange interpolation it is known that optimal order error estimates hold for elements satisfying the maximum angle condition the objective of this paper is to obtain similar results for the raviart thomas interpolation arising in the analysis of mixed methods we prove that optimal order error estimates hold under the maximum angle condition for this interpolation both in two and three dimensions and moreover that this condition is indeed necessary to have these estimates error estimates for the mixed approximation of second order elliptic problems and for the nonconforming piecewise linear approximation of the stokes equations are derived from our results

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

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

asymptotic formulas for spt crank of partitions abstract in this paper using a variant of the hardy ramanujan circle method originally due to e wright we prove asymptotic formulas for spt crank of ordinary partitions

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

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

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

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

fuzzy neural networks with reference neurons as pattern classifiers a heterogeneous neural network consisting of logic neurons and realizing mappings in 0 1 hypercubes is presented the two kinds of neurons studied are utilized to perform matching functions equality or reference neurons and aggregation operations aggregation neurons all computations are driven by logic operations widely used in fuzzy set theory the network is heterogeneous in its nature and includes two types of neurons organized into a structure detecting individual regions of patterns using reference neurons and combining them to yield a final classification decision

high order exactly divergence free hybrid discontinuous galerkin methods for unsteady incompressible flows abstract in this paper we present an efficient discretization method for the solution of the unsteady incompressible navier stokes equations based on a high order hybrid discontinuous galerkin formulation the crucial component for the efficiency of the discretization method is the distinction between stiff linear parts and less stiff non linear parts with respect to their temporal and spatial treatment exploiting the flexibility of operator splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub problems a corresponding hyperbolic transport problem and an unsteady stokes problem for the hyperbolic transport problem a spatial discretization with an upwind discontinuous galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation the treatment of the stokes part involves the solution of linear systems in this case a discretization with hybrid discontinuous galerkin methods is better suited we consider such a discretization for the stokes part with two important features h div conforming finite elements to guarantee exactly divergence free velocity solutions and a projection operator which reduces the number of globally coupled unknowns we present the method discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems

new combinatorial interpretations of ramanujan s partition congruences mod 5 7 and 11 soit p n le nombre de partitions non restreintes de n on donne de nouvelles interpretations combinatoires des congruences p 5n 4 p 7n 5 et p 11n 6 0 mod 5 7 et 11 respectivement

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

stokes complexes and the construction of stable finite elements with pointwise mass conservation two families of conforming finite elements for the two dimensional stokes problem are developed guided by two discrete smoothed de rham complexes which we coin stokes complexes we show that the finite element pairs are inf sup stable and also provide pointwise mass conservation on very general triangular meshes

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

error estimates for raviart thomas interpolation of any order on anisotropic tetrahedra we prove optimal order error estimates for the raviart thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition namely the so called three dimensional maximum angle condition and the regular vertex property for tetrahedra our techniques are different from those used in previous papers on the subject and the results obtained are more general in several aspects first intermediate regularity is allowed that is for the raviart thomas interpolation of degree k 0 we prove error estimates of order j 1 when the vector field being approximated has components in w j 1 p for triangles or tetrahedra where 0 j k and 1 p these results are new even in the two dimensional case indeed the estimate was known only in the case j k on the other hand in the three dimensional case results under the maximum angle condition were known only for k 0

numerical solution of the incompressible navier stokes equations with coriolis forces based on the discretization of the total time derivative in this paper we present a numerical formulation to solve the incompressible navier stokes equations written in a rotating frame of reference the method is based on a finite difference discretization in time and a finite element discretization in space when the viscosity is very small numerical oscillations may appear due both to the high reynolds number and to the presence of the coriolis forces to overcome these oscillations a special discretization in time is proposed the idea is to discretize the total time derivative in an inertial basis rather than only the partial time derivative in the rotating reference system after this is done a further high order approximation is introduced leading to a problem posed in the rotating frame of reference and in spatial coordinates in contrast with the straightforward discretization of the original equations some additional terms appear that enhance the stability of the numerical scheme in the absence of coriolis forces the method is a generalization of the characteristic galerkin technique for convection dominated flows

cranks and t cores new statistics on partitions calledcranks are defined which combinatorially prove ramanujan s congruences for the partition function modulo 5 7 11 and 25 explicit bijections are given for the equinumerous crank classes the cranks are closely related to thet core of a partition usingq series some explicit formulas are given for the number of partitions which aret cores some related questions for self conjugate and distinct partitions are discussed

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

sieved partition functions and q binomial coefficients the q binomial coefficient is a polynomial in q given an integer t and a residue class r modulo t a sieved q binomial coefficient is the sum of those terms whose exponents are congruent to r modu o t in this paper explicit polynomial identities in qt are given for sieved q binomial coefficients as a limiting case generating functions for the sieved partition function are found as multidimensional theta functions a striking corollary of this representation is the proof of ramanujan s congruences mod 5 7 and 11 by exhibiting symmetry groups of orders 5 7 and 11 of explicit quadratic forms we also verify the subbarao conjecture for t 3 t 5 and t 10

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

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

on dyson s crank conjecture and the uniform asymptotic behavior of certain inverse theta functions in this paper we prove a longstanding conjecture by freeman dyson concerning the limiting shape of the crank generating function we fit this function in a more general family of inverse theta functions which play a key role in physics

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

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material

on the asymptotic distribution of cranks and ranks of cubic partitions abstract cubic partitions are a special kind of bi partitions whose name is inspired by a connection between a cubic continued fraction and an arithmetic property of this bi partition there are two partition statistics namely rank and crank for cubic partitions which explain cubic partition congruences combinatorially we obtain asymptotics for the number of cubic partitions of rank resp crank m which reveal the distribution of the rank resp crank among cubic partitions as applications we derive asymptotic inequalities between cubic partition rank and crank functions and we remark upon the similarities and the differences between these and the corresponding inequalities for ordinary partitions

stable q k q k 1 mixed finite elements with discontinuous pressure mixed finite elements play a central role in many important cfd applications involving stokes solvers and alike a natural mixed finite element for the stokes equations is the q k q k 1 element on rectangular grids by which the velocity is approximated by continuous polynomials of separated degree k and the pressure is approximated by discontinuous polynomials of separated degree k 1 such an element is however not stable we propose in this paper three modified q k q k 1 elements with certain element wise divergence free property of velocity where the pressure space is slightly restricted to subspaces in c 1 q k 1 yet the optimal order of approximation is still retained the stability and approximation analysis for the new elements are presented comprehensive numerical experiments are also conducted to confirm the theoretical analysis and to reveal the super convergence for some of these new elements