optimal cycles in doubly weighted graphs and approximation of bivariate functions by univariate ones the problem of finding optimal cycles in a doubly weighted directed graph problem a is closely related to the problem of approximating bivariate functions by the sum of two univariate functions with respect to the supremum norm problem b the close relationship between problem a and problem b is detected by the characterization 7 4 of the distance dist f t of problem b in part 1 we construct an algorithm for problem a where the essential role is played by the minimal lengthsy j k defined by 2 3 if weight functiont 1 then the minimum of problem a is computed by equality 2 4 ift 1 then the minimum is obtained by a binary search procedure algorithm 3 in part 2 we construct our algorithms for solving problem b by following exactly the ideas of part 1 by algorithm 4 we compute the minimal pseudolengthsh k y m defined by 7 5 if weight functiont 1 then the infimum dist f t of problem b is obtained by equality 7 12 which is closely related to 2 4 ift 1 we compute the infimum dist f t by the binary search procedure algorithm 5 additionally algorithm 4 leads to a constructive proof of the existence of continuous optimal solutions of problem b see theorem 7 1e which is already known in caset 1 but unknown in caset 1 interesting applications to the steady state behaviour of industrial processes with interference sect 3 and the solution of integral equations problem c are included

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

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

velocity vorticity helicity formulation and a solver for the navier stokes equations for the three dimensional incompressible navier stokes equations we present a formulation featuring velocity vorticity and helical density as independent variables we find the helical density can be observed as a lagrange multiplier corresponding to the divergence free constraint on the vorticity variable similar to the pressure in the case of the incompressibility condition for velocity as one possible practical application of this new formulation we consider a time splitting numerical scheme based on an alternating procedure between vorticity helical density and velocity bernoulli pressure systems of equations results of numerical experiments include a comparison with some well known schemes based on pressure velocity formulation and illustrate the competitiveness on the new scheme as well as the soundness of the new formulation

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

iterative techniques for time dependent stokes problems abstract in this paper we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent stokes equations using a mixed finite element spatial discretization at each time step a two by two block system corresponding to a perturbed stokes problem must be solved although there are a number of techniques for iteratively solving this type of block system to be effective they require a good preconditioner for the resulting pressure operator schur complement in contrast to the time independent stokes equations where the pressure operator is well conditioned the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced and or the reynolds number is increased in this paper we shall describe and analyze preconditioners for the resulting pressure systems these preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as the time step and reynolds number parameter k

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 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

large deviation principle with generated pseudo measures contemporary large deviation theory uses various approaches one of possible recent approaches is based on the idempotent sup measure and related integrals based on operations sup and the usual product in this paper the large deviation convergence of a sequence of generated pseudo measures to sup decomposable measure is introduced the main result of the paper gives a necessary condition for introduced convergence

attractors via random perturbations we discuss conditions which ensure that weak limits of invariant measures of small random perturbations of dynamical systems have their supports on attractors

an adaptive pressure correction method without spurious velocities for diffuse interface models of incompressible flows in this article we propose to study two issues associated with the use of the incremental projection method for solving the incompressible navier stokes equation the first one is the combination of this time splitting algorithm with an adaptive local refinement method the second one is the reduction of spurious velocities due to the right hand side of the momentum balance we propose a new variant of the incremental projection method for solving the navier stokes equations with variable density and illustrate its behavior with the example of two phase flows simulations using a cahn hilliard navier stokes model

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

effective preconditioning of uzawa type schemes for a generalized stokes problem the schur complement of a model problem is considered as a preconditioner for the uzawa type schemes for the generalized stokes problem the stokes problem with the additional term alpha boldmath u in the motion equation the implementation of the preconditioned method requires for each iteration only one extra solution of the poisson equation with neumann boundary conditions for a wide class of 2d and 3d domains a theorem on its convergence is proved in particular it is established that the method converges with a rate that is bounded by some constant independent of alpha some finite difference and finite element methods are discussed numerical results for finite difference mac scheme are provided

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

dwell time and average dwell time methods based on the cycle ratio of the switching graph abstract a new method to find an upper bound on dwell time and average dwell time for switched linear systems is proposed the method is based on computing the maximum cycle ratio and the maximum cycle mean of the directed graph that governs switchings for planar switched systems an upper bound for dwell time and average dwell time can be estimated by considering only the cycles of length two

a conforming decomposition theorem a piecewise linear theorem of the alternative and scalings of matrices satisfying lower and upper bounds a scaling of a nonnegative matrixa is a matrixxay 1 wherex andy are nonsingular nonnegative diagonal matrices some condition may be imposed on the scaling for example whena is square x y or detx dety we characterize matrices for which there exists a scaling that satisfies predetermined upper and lower bound our principal tools are a piecewise linear theorem of the alternative and a theorem decomposing a solution of a system of equations as a sum of minimal support solutions which conform with the given solutions

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 spurious velocities in incompressible flow problems with interfaces we show that a non physical velocity may appear in the finite element calculation of incompressible two phase flows subjected to an external local force there are different sources responsible for this phenomenon approximation of the incompressibility constraint the interface and the local external force furthermore we demonstrate that this non physical velocity does not vanish with the expected order of convergence we evaluate different concepts and give recommendations for a proper handling of this phenomenon

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

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

segregated runge kutta time integration of convection stabilized mixed finite element schemes for wall unresolved les of incompressible flows in this work we develop a high performance numerical framework for the large eddy simulation les of incompressible flows the spatial discretization of the nonlinear system is carried out using mixed finite element fe schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint these additional terms introduced at the discrete level have been proved to act as implicit les models in order to perform meaningful wall unresolved simulations we consider a weak imposition of the boundary conditions using a nitsche s type scheme where the tangential component penalty term is designed to act as a wall law next segregated runge kutta srk schemes recently proposed by the authors for laminar flow problems are applied to the les simulation of turbulent flows by the introduction of a penalty term on the trace of the acceleration these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms srk schemes are excellent for large scale simulations since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level leading to two uncoupled systems the pressure system is a darcy type problem that can easily be preconditioned using a traditional block preconditioning scheme that only requires a poisson solver at the end only coercive systems have to be solved which can be effectively preconditioned by multilevel domain decomposition schemes which are both optimal and scalable the framework is applied to the taylor green and turbulent channel flow benchmarks in order to prove the accuracy of the convection stabilized mixed fes as les models and srk time integrators the scalability of the preconditioning techniques in space only has also been proven for one step of the srk scheme for the taylor green flow using uniform meshes moreover a turbulent flow around a naca profile is solved to show the applicability of the proposed algorithms for a realistic problem

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

uncertain dynamical systems defined by pseudomeasures this paper deals with uncertain dynamical systems in which predictions about the future state of a system are assessed by so called pseudomeasures two special cases are stochastic dynamical systems where the pseudomeasure is the conventional probability measure and fuzzy dynamical systems in which the pseudomeasure is a so called possibility measure new results about possibilistic systems and their relation to deterministic and to stochastic systems are derived by using idempotent pseudolinear algebra by expressing large deviation estimates for stochastic perturbations in terms of possibility measures we obtain a new interpretation of the freidlin wentzell quasipotentials for stochastic perturbations of dynamical systems as invariant possibility densities

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

quasipotentials for simple noisy maps with complicated dynamics the theory of nonequilibrium potentials or quasipotentials is a physically motivated approach to small random perturbations of dynamical systems leading to exponential estimates of invariant probabilities and mean first exit times in the present article we develop the mathematical foundation of this theory for discrete time systems following and extending the work of freidlin and wentzell and kifer we discuss strategies for calculating and estimating quasipotentials and show their application to one dimensionals unimodal maps the method proves to be especially suited for describing the noise scaling behavior of invariant probabilities e g for the map occurring as the limit of the feigenbaum period doubling sequence we show that the method allows statements about the scaling behavior in the case of localized noise too which does not originally lie within the scope of the quasipotential formalism

effects of vocalic formant transitions and vowel quality on the english s s boundary the effects of the vocalic portion of fricative vowel syllables on the perception of alveolar and palatal fricatives were examined the fricative noises were synthesized to represent a continuum from s to the vowels ranged from u to i through i and u the vocalic formant transitions were of two types those appropriate to s and those to all stimuli were presented in forced choice labeling tests the boundary between s and for english speaking listeners was found to vary as a function both of transitions and of vowel the effect of the transitions was clear and straightforward an ambiguous fricative noise was heard more often as s before s transitions and as before transitions the quality of the vowel clearly had an effect but the interpretation of the effect in terms of the perception of coarticulation was not clear the responses of listeners who were unfamiliar with languages which use u and or i distinctively were not significantly different from those of listeners who were familiar with such languages

acoustic characteristics of english fricatives this study constitutes a large scale comparative analysis of acoustic cues for classification of place of articulation in fricatives to date no single metric has been found to classify fricative place of articulation with a high degree of accuracy this study presents spectral amplitudinal and temporal measurements that involve both static properties spectral peak location spectral moments noise duration normalized amplitude and f2 onset frequency and dynamic properties relative amplitude and locus equations while all cues except locus equations consistently serve to distinguish sibilant from nonsibilant fricatives the present results indicate that spectral peak location spectral moments and both normalized and relative amplitude serve to distinguish all four places of fricative articulation these findings suggest that these static and dynamic acoustic properties can provide robust and unique information about all four places of articulation despite variation in speaker vowel context and voicing

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

a finite element variational multiscale method for incompressible flow in this paper we present a numerical scheme prove stability existence of solutions uniqueness and convergence of the incompressible navier stokes equations it has the advantage of being defined from strictly algebraic considerations a significant feature of the present method is that the structure of the stabilization term based on the multiscale enrichment and derived from the navier stokes problem itself ample numerical experiments are carried out to confirm the theory and illustrate the effectiveness of the scheme on incompressible fluid

applications of shortest path algorithms to matrix scalings a symmetric scaling of a nonnegative square matrixa is a matrixxax 1 wherex is a nonsingular nonnegative diagonal matrix by associating a family of weighted directed graphs with the matrixa we are able to adapt the shortest path algorithms to compute an optimal scaling ofa where we call a symmetric scalinga ofa optimal if it minimizes the maximum of the ratio of non zero elements

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

pseudo additive measures and integrals abstract we suggest pseudo additive measures based on a pseudo addition and discuss integrals with respect to pseudo additive measures a pseudo additive measure is a special type of fuzzy measures to define an integral a multiplication corresponding to a pseudo addition is introduced the resulting integral is an extension of the lebesgue integral in this context radon nikodym like theorems are shown

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

analysis of some finite elements for the stokes problem we study some finite elements which are used in the approximation of the stokes problem so as to obtain error estimates of optimal order resume nous etudions deux elements finis utilises pour l approximation du probleme de stokes et obtenons des estimations d erreur d ordre optimal

the chebyshev solution of the linear matrix equationax yb c in this paper we investigate the properties of the chebyshev solutions of the linear matrix equationax yb c wherea b andc are given matrices of dimensionsm r s n andm n respectively wherer m ands n we separately consider two particular cases in the first case we assumem r 1 andn s 1 in the second caser s 1 andm n are arbitrary for these two cases under the assumption that the matricesa andb are full rank we formulate necessary and sufficient conditions characterizing the chebyshev solution ofax yb c and we give the formulas for the chebyshev error then we propose an algorithm which may be applied to compute the chebyshev solution ofax yb c for some particular cases some numerical examples are also given

local projection stabilization of equal order interpolation applied to the stokes problem the local projection stabilization allows us to circumvent the babuska brezzi condition and to use equal order interpolation for discretizing the stokes problem the projection is usually done in a two level approach by projecting the pressure gradient onto a discontinuous finite element space living on a patch of elements we propose a new local projection stabilization method based on possibly enriched finite element spaces and discontinuous projection spaces defined on the same mesh optimal order of convergence is shown for pairs of approximation and projection spaces satisfying a certain inf sup condition examples are enriched simplicial finite elements and standard quadrilateral hexahedral elements the new approach overcomes the problem of an increasing discretization stencil and thus is simple to implement in existing computer codes numerical tests confirm the theoretical convergence results which are robust with respect to the user chosen stabilization parameter

small random perturbations of dynamical systems and the definition of attractors the strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction in view of this we study a new definition of attractors based on ideas of conley we argue that the attractors observed in the presence of small random perturbations correspond to this new definition

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

analysis of the small viscosity and large reaction coefficient in the computation of the generalized stokes problem by a novel stabilized finite element method abstract in this paper we propose and analyze a novel stabilized finite element method fem for the system of generalized stokes equations arising from the time discretization of transient stokes problem the system involves a small viscosity which is proportional to the inverse of large reynolds number and a large reaction coefficient which is the inverse of small time step the proposed stabilized fem employs the c 0 piecewise linear elements for both velocity field and pressure on the same mesh and uses the residuals of the momentum equation and the divergence free equation to define the stabilization terms the stabilization parameters are fixed and element independent without a comparison of the viscosity the reaction coefficient and the mesh size using the finite element solution of an auxiliary boundary value problem as the interpolating function for velocity and the h 1 seminorm projection for pressure instead of the usual nodal interpolants we derive error estimates for the stabilized finite element approximations to velocity and pressure in the l 2 and h 1 norms and most importantly we explicitly establish the dependence of error bounds on the viscosity the reaction coefficient and the mesh size our analysis reveals that this stabilized fem is particularly suitable for the generalized stokes system with a small viscosity and a large reaction coefficient which has never been achieved before in the error analysis of other stabilization methods in the literature we then numerically confirm the effectiveness of the proposed stabilized fem comparisons made with other existing stabilization methods show that the newly proposed method can attain better accuracy and stability

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

large deviations from the thermodynamic limit in globally coupled maps abstract systems of a large number n of globally coupled maps have become popular as a relatively simple prototype of high dimensional dynamics showing many interesting and typical phenomena like synchronisation cluster formation and multistability and having potential applications in systems like josephson junction arrays or in biophysical models while much progress has been made in the explanation of the macroscopic behaviour of such systems in the limit n there is still need for a sound theory about the asymptotic behaviour of finite n systems as n approaches infinity this article introduces a method by which it is possible to obtain asymptotic estimates for finite size effects that show up as long term deviations from the thermodynamic limit behaviour this method works for systems of globally coupled noisy maps and is based upon the concept of quasipotentials originally developed by freidlin wentzell and others for describing the influence of small random perturbations on the long term behaviour of dynamical systems the problems of explicitly computing quasipotentials in the present context and potential approximation schemes are discussed all the concepts described in this article are illustrated with a simple example

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

speaker normalization in the perception of mandarin chinese tones this study investigated speaker normalization in perception of mandarin tone 2 midrising and tone 3 low falling rising by examining listeners use of f0 range as a cue to speaker identity two speakers were selected such that tone 2 of the low pitched speaker and tone 3 of the high pitched speaker occurred at equivalent f0 heights production and perception experiments determined that turning point or inflection point of the tone and f0 the difference in f0 between onset and turning point distinguished the two tones three tone continua varying in either turning point f0 or both acoustic dimensions were then appended to a natural precursor phrase from each of the two speakers results showed identification shifts such that identical stimuli were identified as low tones for the high precursor condition but as high tones for the low precursor condition stimuli varying in turning point showed no significant shift suggesting that listeners normalize only when the precursor varies in the same dimension as the stimuli the magnitude of the shift was greater for stimuli varying only in f0 as compared to stimuli varying in both turning point and f0 indicating that normalization effects are reduced for stimuli more closely matching natural speech

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

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

approximation of coupled stokes darcy flow in an axisymmetric domain abstract in this article we investigate the numerical approximation of coupled stokes and darcy fluid flow equations in an axisymmetric domain the fluid flow is assumed to be axisymmetric rewriting the problem in cylindrical coordinates reduces the 3 d problem to a problem in 2 d this reduction to 2 d requires the numerical analysis to be studied in suitably weighted hilbert spaces in this setting we show that the proposed approximation scheme has a unique solution and derive corresponding a priori error estimate computations for an example with a know solution are presented which support the a priori error estimate computations are also given for a model of fluid flow in the eye

pseudo lp space and convergence in the framework of the pseudo analysis the classical l p space is generalized and there are proved important properties of introduced space three types of convergence of sequences of measurable functions are considered in this space the inequalities for integrals based on pseudo integral have been recently proposed as the holder minkowski and markov inequalities which are applied in observing relationship among introduced convergences

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

computing optimal scalings by parametric network algorithms asymmetric scaling of a square matrixa 0 is a matrix of the formxax 1 wherex is a nonnegative nonsingular diagonal matrix having the same dimension ofa anasymmetric scaling of a rectangular matrixb 0 is a matrix of the formxby 1 wherex andy are nonnegative nonsingular diagonal matrices having appropriate dimensions we consider two objectives in selecting a symmetric scaling of a given matrix the first is to select a scalinga of a given matrixa such that the maximal absolute value of the elements ofa is lesser or equal that of any other corresponding scaling ofa the second is to select a scalingb of a given matrixb such that the maximal absolute value of ratios of nonzero elements ofb is lesser or equal that of any other corresponding scaling ofb we also consider the problem of finding an optimal asymmetric scaling under the maximal ratio criterion the maximal element criterion is of course trivial in this case we show that these problems can be converted to parametric network problems which can be solved by corresponding algorithms

numerical procedure for solving a minimization eigenvalue problem a numerical procedure for solving the left eigenvalue problem lambda l k min j l j k j k and its right counterpart is described k is a known realn n matrix the first stage uses a modification of the karp and von golitschek procedure for finding the minimum cyclic average ofk along with a minimizing cycle the second stage employs this information to generatel by a very similar algorithm

the chebyshev solution of certain matrix equations the problem of approximating ann x m matrixc by matrices of the formx a b y wherea b x andy are of appropriate size is considered the measure of error is the supremum of the absolute values of the individual entries in the error matrix the problem is closely related to that of approximating a bivariate functionf by sums of functions of the formxh gy wherex andg are functions of the first variable alone andh andy are functions of the second variable an old algorithm for constructing best approximations is described and some of the properties of its convergence are discussed

possibility theory an integral theoretic approach this paper introduces possibility integrals as fuzzy integrals w r t possibility measures it is shown that this concept can be used to unify a lot of known results in possibility theory that up to now have been introduced on a more ad hoc basis the integral theoretic approach allows us amongst other things to introduce product and conditional possibility measures in very much the same way as product lebesgue measures and conditional probability measures are introduced furthermore to stress the fact that the concept of order relations is basic for possibility theory all the results are proved for a complete lattice as the evaluation set instead of the more familiar unit interval 0 1

analysis of iterative methods for saddle point problems a unified approach in this paper two classes of iterative methods for saddle point problems are considered inexact uzawa algorithms and a class of methods with symmetric preconditioners in both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems the obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature a few numerical tests are given which confirm the sharpness of the estimates

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

performance evaluation of max automata automata with multiplicities over the max semiring can be used to represent the behavior of timed discrete event systems this formalism which extends both conventional automata and max linear representations covers a class of systems with synchronization phenomena and variable schedules performance evaluation is considered in the worst mean and optimal cases a simple algebraic reduction is provided for the worst case the last two cases are solved for the subclass of deterministic series recognized by deterministic automata deterministic series frequently arise due to the finiteness properties of max linear projective semigroups the mean performance is given by the kolmogorov equation of a markov chain the optimal performance is given by a hamilton jacobi bellman equation

max balancing weighted directed graphs and matrix scaling a weighted directed graph g is a triple v a g where v a is a directed graph and g is an arbitrary real valued function defined on the arc set a let g be a strongly connected simple weighted directed graph we say that g is max balanced if for every nontrivial subset of the vertices w the maximum weight over arcs leaving w equals the maximum weight over arcs entering w we show that there exists a up to an additive constant unique potential pv for v v such that v a gp is max balanced where gap pu ga pv for a u v a we describe an o v 2 a algorithm for computing p using an algorithm for computing the maximum cycle mean of g finally we apply our principal result to the similarity scaling of nonnegative matrices

idempotent version of the frechet contingency array problem in this paper we study the idempotent version of the so called frechet correlation array problem the problem is studied using an algebraic approach the major result is that there exists a unique upper bound and several lower bounds the formula for the upper bound is given an algorithm is proposed to compute one lower bound another algorithm is provided to compute all lower bounds but the number of lower bounds may be a very large number note that all these results are only based on the distributive lattice property of the idempotent algebraic structure

locking to incommensurate structures a model with three competing lengths a classical pseudo spin model in one dimension is considered representing a variation on the frenkel kontorova model to include non convex interactions resulting in three competing length scales an exact algorithm is used numerically to determine the classical ground state as a function of a chemical potential approximate arguments suggest that only a first order transition should occur however when the lengths are not rationally related it is found that the mean lattice spacing appears to vary as a devil s staircase the plateaux of the staircase correspond to locking to incommensurate structures while there is no locking to the commensurate ones most of the locking values observed numerically belong to a series which can be analytically calculated on the basis of simple topological hypotheses which are also consistent with numerical observations

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

interface interactions in modulated phases and upsilon points certain features in frenkel kontorova and other models of phases with a one dimensional modulation can be analyzed by assuming parallel interfaces separating sets of lattice planes belonging to two different phases and treating the free energy to create interfaces as well as the interaction of two three or more interfaces as phenomenological parameters a strategy employed by fisher and szpilka for interacting defects can be extended to the case of interfaces allowing a systematic study of the phase diagram by ignoring all interface interactions and then successively taking into account pair triple and higher order terms the possible phase diagrams which can occur near the point where 0 include various sorts of endpoints analogous to critical endpoints an accumulation point of first order transitions and triple points and a self similar structure which we call an upsilon point which turns out to be an accumulation point of an infinite number of segments of first order transition lines each of which terminates in two upsilon points

multigrid methods for a parameter dependent problem in primal variables in this paper we consider multigrid methods for the parameter dependent problem of nearly incompressible materials we construct and analyze multilevel projection algorithms which can be applied to the mixed as well as to the equivalent non conforming finite element scheme in primal variables for proper norms we prove that the smoothing property and the approximation property hold with constants that are independent of the small parameter thus we obtain robust and optimal convergence rates for the w cycle and the variable v cycle multigrid methods the numerical results pretty well conform the robustness and optimality of the multigrid methods proposed

augmented lagrangian preconditioner for large scale hydrodynamic stability analysis abstract hydrodynamic linear stability analysis of large scale three dimensional configurations is usually performed with a time stepping approach based on the adaptation of existing solvers for the unsteady incompressible navier stokes equations we propose instead to solve the nonlinear steady equations with the newton method and to determine the largest growth rate eigenmodes of the linearized equations using a shift and invert spectral transformation and a krylov schur algorithm the solution of the shifted linearized navier stokes problem which is the bottleneck of this approach is computed via an iterative krylov subspace solver preconditioned by the modified augmented lagrangian mal preconditioner benzi et al 2011 the well known efficiency of this preconditioned iterative strategy for solving the real linearized steady state equations is assessed here for the complex shifted linearized equations the effect of various numerical and physical parameters is investigated numerically on a two dimensional flow configuration confirming the reduced number of iterations over state of the art steady state and time stepping based preconditioners a parallel implementation of the steady navier stokes and eigenvalue solvers developed in the freefem language suitably interfaced with the petsc slepc libraries is described and made openly available to tackle three dimensional flow configurations its application on a small scale three dimensional problem shows the good performance of this iterative approach over a direct l u factorization strategy in regards of memory and computational time on a large scale three dimensional problem with 75 million unknowns a 80 parallel efficiency on 256 up to 2048 processes is obtained

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

natural vorticity boundary conditions on solid walls abstract we derive a new kind of boundary conditions for the vorticity equation with solid wall boundaries for fluid flow problems the formulation uses a dirichlet condition for the normal component of vorticity and neumann type conditions for the tangential components in a galerkin integral formulation the tangential condition is natural i e it is enforced by a right hand side functional and does not impose a boundary constraint on trial and test spaces the functional involves the pressure variable and we discuss several velocity vorticity formulations where the proposed condition is appropriate several numerical experiments are given that illustrate the validity of the new approach

analysis of a streamline diffusion finite element method for the stokes and navier stokes equations for the stokes equations with convection and the incompressible navier stokes equations the authors analyze a streamline diffusion finite element method that is capable of balancing both the convection and the pressure thus allowing the use of arbitrary pairs of velocity pressure spaces for the linear problem the authors obtain for all mesh peclet numbers optimal error estimates in natural norms including in particular the l 2 norm of the pressure the same holds for the nonlinear problem which close to a regular branch of solutions i e the linearized operator is an isomorphism of the norm of the inverse of which still depends on the reynolds number consequently the dependence of the error constants on the reynolds number is not completely resolved in this case

a subspace projection method for the implementation of interface conditions in a single drop flow problem we present a mesh moving method within a finite element context where the interface conditions of a two phase flow problem are conveniently included in suitable chosen subspaces of the general trial and testfunction spaces a weak formulation of the two phase flow problem including species transport is derived where the problem specific function spaces are replaced by appropriate projections operating on standard function spaces the transfer of the variational formulation to a finite element method is straightforward and results in one set of equations for both fluidic phases this subspace projection method is applicable to a wide range of interfacial conditions for multiphase flow problems the method is validated for single drop flow problems including species transfer furthermore an application to the simulation of stagnant caps is presented

phase diagram of a one dimensional model with nonconvex interactions using the method of effective potentials the ground states of a one dimensional system of the frenkel kontorova type but involving piecewise parabolic potentials including a nonconvex interatomic interaction have been studied numerically using the method of effective potentials part of the phase diagram is identical to one studied earlier for a convex interaction and part of it exhibits some new phases first order phase transitions multicritical points and an accumulation point of multicritical points all associated with the nonconvex interaction

a charge conservative finite element method for inductionless mhd equations part i convergence a charge conservative finite element method is proposed to solve the inductionless and incompressible magnetohydrodynamic mhd equations in three dimensions the method yields an exactly divergenc