multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

a refined conjecture of mazur tate type for heegner points in mt1 mazur and tate present a refined conjecture of birch and swinnerton dyer type for a modular elliptic curvee this conjecture relates congruences for certain integral homology cycles one c the modular symbols to the arithmetic ofe overq in this paper we formulate an analogous conjecture fore over a suitable imaginary quadratic fieldk in which the role of the modular symbols is played by heegner points a large part of this conjecture can be proved thanks to the ideas of kolyvagin on the euler system of heegner points in effect the main result of this paper can be viewed as a generalization of kolyvagin s result relating the structure of the selmer group ofe overk to the heegner points defined in the mordell weil groups ofe over ring class fields ofk an explicit application of our method to the galois module structure of heegner points is given in sect 2 2

an unusual stabilized finite element method for a generalized stokes problem an unusual stabilized finite element is presented and analyzed herein for a generalized stokes problem with a dominating zeroth order term the method consists in subtracting a mesh dependent term from the formulation without compromising consistency the design of this mesh dependent term as well as the stabilization parameter involved are suggested by bubble condensation stability is proven for any combination of velocity and pressure spaces under the hypotheses of continuity for the pressure space optimal order error estimates are derived for the velocity and the pressure using the standard norms for these unknowns numerical experiments confirming these theoretical results and comparisons with previous methods are presented

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

plus minus p adic l functions for hilbert modular forms abstract r pollack constructed in pollack 2003 13 plus minus p adic l functions for elliptic modular forms which are p adically bounded when the hecke eigenvalues at p are zero the most supersingular case the goal of this work is to generalize his construction to hilbert modular forms we find a suitable condition for hilbert modular forms corresponding to the vanishing of p th hecke eigenvalue in elliptic modular form case which guarantees the existence of plus minus p adic l functions which are p adically bounded as an application we construct cyclotomic plus minus p adic l functions for modular elliptic curves over a totally real field f under the assumption that a p e 0 for each prime p dividing p we formulate a cyclotomic plus minus iwasawa main conjecture for such elliptic curves

an anisotropic pressure stabilized finite element method for incompressible flow problems we consider a pressure stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a projection of the gradient of the discrete pressure onto the space of discrete functions equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

the main conjecture for cm elliptic curves at supersingular primes at a prime of ordinary reduction the iwasawa main conjecture for elliptic curves relates a selmer group to a p adic l function in the supersingular case the statement of the main conjecture is more complicated as neither the selmer group nor the p adic l function is well behaved recently kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is similar in structure to the ordinary case namely kobayashi s conjecture relates modified selmer groups which he defined with modified padic l functions defined by the first author in this paper we prove kobayashi s conjecture for elliptic curves with complex multiplication

iwasawa theory of elliptic curves at supersingular primes over ℤ p extensions of number fields in this paper we make a study of the iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary zp extension of a number field k in the case when p splits completely in k generalizing work of kobayashi 9 and perrin riou 17 we define restricted selmer groups and invariants we then derive asymptotic formulas describing the growth of the selmer group in terms of these invariants to be able to work with non cyclotomic zp extensions a new local result is proven that gives a complete description of the formal group of an elliptic curve at a supersingular prime along any ramified zp extension of qp

three dimensional finite element methods for the stokes problem the tetrahedral finite element approximation of the stokes problem is analyzed by means of polynomials piecewise of degree k 1 for the velocity and continuous piecewise of degree k for the pressure a stability result is given for every k ge 1

a relationship between stabilized finite element methods and the galerkin method with bubble functions abstract a relation between stabilized finite element methods and the galerkin method employing interpolations with bubble functions is established for the advective diffusive model and for the linearized compressible navier stokes equations the bubble functions are shown to help in stabilizing the advective operator without recourse to upwinding or any other numerical strategy in particular for the advective diffusive model the galerkin method employing piecewise linears with bubble functions is shown to be equivalent to the streamline upwind petrov galerkin supg method in the diffusive limit

convergence analyses of galerkin least squares methods for symmetric advective diffusive forms of the stokes and incompressible navier stokes equations abstract symmetric advective diffusive forms of the stokes and incompressible navier stokes equations are presented the galerkin least squares method for advective diffusive equations is used for both systems and is related to other stabilized methods previously studied the presentation reveals that the convergence analysis for advective diffusive equations as applied before to a linearized form of the compressible navier stokes equations carries over in a straightforward manner to the stokes problem and to a linearized form of the incompressible navier stokes equations

solutions of the divergence operator on john domains if rn is a bounded domain the existence of solutions u w01 p of divu f for f lp with vanishing mean value and 1 p is a basic result in the analysis of the stokes equations it is known that the result holds when is a lipschitz domain and that it is not valid for domains with external cusps in this paper we prove that the result holds for john domains our proof is constructive the solution u is given by an explicit integral operator acting on f to prove that u w01 p we make use of the calderon zygmund singular integral operator theory and the hardy littlewood maximal function for domains satisfying the separation property introduced in s buckley p koskela sobolev poincare implies john math res lett 2 5 1995 577 593 and 1 p n we also prove a converse result thus characterizing in this case the domains for which a continuous right inverse of the divergence exists in particular our result applies to simply connected planar domains because they satisfy the separation property

stabilized finite elements on anisotropic meshes a priori error estimates for the advection diffusion and the stokes problems stabilized finite elements on strongly anisotropic meshes are considered the design of the stability coefficients is addressed for both the advection diffusion and the stokes problems when using continuous piecewise linear finite elements on triangles using the polar decomposition of the jacobian of the affine mapping from the reference triangle to the current one k and from a priori error estimates a new definition of the stability coefficients is proposed our analysis shows that these coefficients do not depend on the element diameter hk but on a characteristic length associated with k via the polar decomposition a numerical assessment of the theoretical analysis is carried out

anisotropic mesh refinement in stabilized galerkin methods the numerical solution of a convection diffusion reaction model problem is considered in two and three dimensions a stabilized finite element method of galerkin least square type accomodates diffusion dominated as well as convection and or reaction dominated situations the resolution of boundary layers occuring in the singularly perturbed case is achieved using anisotropic mesh refinement in boundary layer regions in this paper the standard analysis of the stabilized galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement simplicial lagrangian elements of arbitrary order are used

error estimates for cq_1 isoparametric elements satisfying a weak angle condition we prove optimal order error estimates in the h1 norm for the cq 1 isoparametric interpolation on convex quadrilateral elements under rather weak hypotheses improving previously known results choose one diagonal and divide the element into two triangles we show that if the chosen diagonal is the longest one then the constant in the error estimate depends only on the maximum angle of the two triangles otherwise the constant depends on that maximum angle and on the ratio between the two diagonals in particular we obtain the optimal order error estimate under the maximum angle condition as in the case of triangular elements consequently the error estimate is uniformly valid for a rather general class of degenerate quadrilaterals

stabilized mixed methods for the stokes problem the solution of the stokes problem is approximated by three stabilized mixed methods one due to hughes balestra and franca and the other two being variants of this procedure in each case the bilinear form associated with the saddle point problem of the standard mixed formulation is modified to become coercive over the finite element space error estimates are derived for each procedure

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

stability of higher order hood taylor methods the stability of a higher order hood taylor method for the approximation of the stationary stokes equations using continuous piecewise polynomials of degree 3 to approximate velocities and continuo

lagrange and average interpolation over 3d anisotropic elements abstract an average interpolation is introduced for 3 rectangles and tetrahedra and optimal order error estimates in the h1 norm are proved the constant in the estimate depends weakly improving the results given in duran math comp 68 1999 187 199 on the uniformity of the mesh in each direction for tetrahedra the constant also depends on the maximum angle of the element on the other hand merging several known results acosta and duran siam j numer anal 37 1999 18 36 duran math comp 68 1999 187 199 krizek siam j numer anal 29 1992 513 520 al shenk math comp 63 1994 105 119 we prove optimal order error for the p 1 lagrange interpolation in w1 p p 2 with a constant depending on p as well as the maximum angle of the element again under the maximum angle condition optimal order error estimates are obtained in the h1 norm for higher degree interpolations

a finite element formulation for the stokes problem allowing equal velocity pressure interpolation abstract in this paper we study a variational formulation of the stokes problem that accommodates the use of equal velocity pressure finite element interpolations the motivation of this method relies on the analysis of a class of fractional step methods for the navier stokes equations for which it is known that equal interpolations yield good numerical results the reason for this turns out to be the difference between two discrete laplacian operators computed in a different manner the formulation of the stokes problem considered here aims to reproduce this effect from the analysis of the finite element approximation of the problem we obtain stability and optimal error estimates using velocity pressure interpolations satisfying a compatibility condition much weaker than the inf sup condition of the standard formulation in particular this condition is fulfilled by the most common equal order interpolations

supercloseness on graded meshes for q1 finite element approximation of a reaction diffusion equation in this paper we analyze the standard piece wise bilinear finite element approximation of a model reaction diffusion problem we prove supercloseness results when appropriate graded meshes are used the meshes are those introduced in duran and lombardi 2005 8 but with a stronger restriction on the graduation parameter as a consequence we obtain almost optimal error estimates in the l 2 norm thus completing the error analysis given in duran and lombardi 2005 8

analysis of a pressure stabilized finite element approximation of the stationary navier stokes equations the purpose of this paper is to analyze a finite element approximation of the stationary navier stokes equations that allows the use of equal velocity pressure interpolation the idea is to introduce as unknown of the discrete problem the projection of the pressure gradient multiplied by suitable algorithmic parameters onto the space of continuous vector fields the difference between these two vectors pressure gradient and projection is introduced in the continuity equation the resulting formulation is shown to be stable and optimally convergent both in a norm associated to the problem and in the l 2 norm for both velocities and pressure this is proved first for the stokes problem and then it is extended to the nonlinear case all the analysis relies on an inf sup condition that is much weaker than for the standard galerkin approximation in spite of the fact that the present method is only a minor modification of this

an anisotropic gls stabilized finite element method for incompressible flow problems abstract we consider a stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a gls formulation equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios rather than using a single stabilization parameter within each element the jacobian matrix of the affine transformation from the reference element is employed for stabilization purposes stability and anisotropic error estimates in a suitable mesh dependent norm are proved for the linear stokes problem numerical results confirm the accuracy and robustness of the proposed method compared to similar previous stabilized schemes

stabilised vs stable mixed methods for incompressible flow abstract the accuracy of low order mixed finite element methods for the incompressible navier stokes equations is investigated in this work some numerical experiments suggest that the lowest order stabilised p 1 p 0 method linear velocity constant pressure is more efficient than the alternative a priori stable non conforming crouzeix raviart p 1 p 0 approach the relative accuracy of stabilised p 1 p 0 and p 1 p 1 linear velocity continuous linear pressure is assessed herein

on the equivalence of fractional order sobolev semi norms abstract we present various results on the equivalence and mapping properties under affine transformations of fractional order sobolev norms and semi norms of orders between zero and one main results are mutual estimates of the three semi norms of sobolev slobodeckij interpolation and quotient space types in particular we show that the former two are uniformly equivalent under affine mappings that ensure shape regularity of the domains under consideration

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

self improving properties of inequalities of poincare type on measure spaces and applications abstract we show that the self improving nature of poincare estimates persists for domains in rather general measure spaces we consider both weak type and strong type inequalities extending techniques of b franchi c perez and r wheeden as an application in spaces of homogeneous type we derive global poincare estimates for a class of domains with rough boundaries that we call john domains and we show that such domains have the requisite properties this class includes john or boman domains as well as s john domains further applications appear in a companion paper

approximation of the incompressible navier stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes abstract this paper describes a finite element model to solve the incompressible navier stokes equations based on the stabilization with orthogonal subscales and a pressure segregation the former consists of adding a least square form of the component orthogonal to the finite element space of the convective and pressure gradient terms this allows to deal with convection dominated flows and to use equal velocity pressure interpolation the pressure segregation is inspired in fractional step schemes although the converged solution corresponds to that of a monolithic time integration likewise we put special emphasis on the use of anisotropic grids in particular we describe some possible choices for the calculation of the element length that appears in the stabilization parameters and check their behavior in two classical numerical examples the preconditioning strategy used to solve the resulting algebraic system for very anisotropic meshes is also briefly described

convergence to steady state solutions for stabilized finite element simulations of compressible flows abstract the main objective of the paper is to compare the numerical performance of several pseudo time integration algorithms used for driving finite element solutions of compressible flow equations towards the steady state the algorithms are implementations of four strategies explicit linear implicit multistage linear implicit and nonlinear all except the explicit scheme employ a particular implementation of gmres to solve systems of linear equations the nonlinear algorithm uses at each time step two iterations of the newton method with an approximate jacobian the finite element formulation is a version of the stabilized fem with h adaptivity the comparison is done for three flow problems in three flow regimes subsonic transonic and supersonic final conclusions concerning relative performance of the methods are drawn

on the main conjecture for cm fields the aim of this work is to prove a divisibility relation toward the main conjecture for cm fields our main tool is the study of congruences between an eisenstein series and cusp forms on the unitary group gu 2 1 we introduce an eisenstein ideal in the nearly ordinary universal hecke algebra for gu 2 1 and show that it divides the characteristic ideal of the cm iwasawa module

analysis of velocity flux first order system least squares principles for the navier stokes equations part i this paper develops a least squares approach to the solution of the incompressible navier stokes equations in primitive variables as with our earlier work on stokes equations we recast the navier stokes equations as a first order system by introducing a velocity flux variable and associated curl and trace equations we show that a least squares principle based on l2 norms applied to this system yields optimal discretization error estimates in the h1 norm in each variable including the velocity flux an analogous principle based on the use of an h 1 norm for the reduced system with no curl or trace constraints is shown to yield similar estimates but now in the l2 norm for velocity flux and pressure although the h 1 least squares principle does not allow practical implementation these results are critical to the analysis of a practical least squares method for the reduced system based on a discrete equivalent of the negative norm a practical method of this type is the subject of a companion paper finally we establish optimal multigrid convergence estimates for the algebraic system resulting from the l2 norm approach

coleman adapted rubin stark kolyvagin systems and supersingular iwasawa theory of cm abelian varieties the goal of this article is to study the iwasawa theory of an abelian variety a that has complex multiplication by a cm field f that contains the reflex field of a which has supersingular reduction at every prime above p to do so we make use of the signed coleman maps constructed in our companion article bl14 to introduce signed selmer groups as well as a signed p adic l function via a reciprocity conjecture we formulate for the conjectural rubin stark elements which is a natural extension of the reciprocity conjecture for elliptic units we then prove a signed main conjecture relating these two objects to achieve this we develop along the way a theory of coleman adapted rank g euler kolyvagin systems to be applied with rubin stark elements and deduce the main conjecture for the maximal zp power extension of f for the primes failing the ordinary hypothesis of katz

on l functions of elliptic curves and cyclotomic towers in this paper we shall strengthen a recent theorem of ralph greenberg 1 concerning l functions of elliptic curves and anticyclotomic towers let k be an imaginary quadratic field and m an abelian extension of k possibly of infinite degree we say that the extension m k is anticyclotomic if m is galois over q and if the nontrivial element of gal k q acts on gal m k by inversion now fix a finite set p of rational primes and consider the compositum in an algebraic closure of k of all anticyclotomic extensions of k which are unramified outside p we denote this compositum by l and refer to it as the maximal anticyclotomic extension of k unramified outside p the field l may also be described as the union of all ring class fields of k with conductor divisible only by primes in p class field theory shows that gal l k is isomorphic to the product of a finite group and the group

a term by term stabilization algorithm for finite element solution of incompressible flow problems this paper introduces a stabilization technique for finite element numerical solution of 2d and 3d incompressible flow problems it may be applied to stabilize the discretization of the pressure gradient and also of any individual operator term such as the convection curl or divergence operators with specific levels of numerical diffusion for each one of them its computational complexity is reduced with respect to usual residual based stabilization techniques we consider piecewise affine finite elements for which we obtain optimal error bounds for steady navier stokes and also for generalized stokes equations including convection we include some numerical experiment in well known 2d test cases that show its good performances

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

integral iwasawa theory of galois representations for non ordinary primes in this paper we study the iwasawa theory of a motive whose hodge tate weights are 0 or 1 thence in practice of a motive associated to an abelian variety at a non ordinary prime over the cyclotomic tower of a number field that is either totally real or cm in particular under certain technical assumptions we construct sprung type coleman maps on the local iwasawa cohomology groups and use them to define integral p adic l functions and one unconditionally and other conjecturally cotorsion selmer groups this allows us to reformulate perrin riou s main conjecture in terms of these objects in the same fashion as kobayashi s iwasawa theory for supersingular elliptic curves by the aid of the theory of coleman adapted kolyvagin systems we develop here we deduce parts of perrin riou s main conjecture from an explicit reciprocity conjecture

a d conversion with imperfect quantizers this paper analyzes mathematically the effect of quantizer threshold imperfection commonly encountered in the circuit implementation of analog to digital a d converters such as pulse code modulation pcm and sigma delta sigmadelta modulation sigmadelta modulation which is based on coarse quantization of oversampled redundant samples of a signal enjoys a type of self correction property for quantizer threshold errors bias that is not shared by pcm although classical sigmadelta modulation is inferior to pcm in the rate distortion sense this robustness feature is believed to be one of the reasons why sigmadelta modulation is preferred over pcm in a d converters with imperfect quantizers motivated by these facts other encoders are constructed in this paper that use redundancy to obtain a similar self correction property but that achieve higher order accuracy relative to bit rate compared to classical sigmadelta more precisely two different types of encoders are introduced that exhibit exponential accuracy in the bit rate in contrast to the polynomial type accuracy of classical sigmadelta while possessing the self correction property

on the p adic l function of a modular form at a supersingular prime in this paper we study the two p adic l functions attached to a modular form f sum a sb nq sp n at a supersingular prime p when a sb p 0 we are able to decompose both the sum and the difference of the two unbounded distributions attached to f into a bounded measure and a distribution that accounts for all of the growth moreover this distribution depends only upon the weight of f and the fact that a sb p vanishes from this description we explain how the p adic l function is controlled by two iwasawa functions and by two power series with growth which have a fixed infinite set of zeros theorem 5 1 asymptotic formulas for the p part of the analytic size of the tate shafarevich group of an elliptic curve in the cyclotomic direction are computed using this result these formulas compare favorably with results established by m kurihara in 11 and b perrin riou in 23 on the algebraic side moreover we interpret kurihara s conjectures on the galois structure of the tate shafarevich group in terms of these two iwasawa functions

iwasawa theory for elliptic curves at supersingular primes we give a new formulation in iwasawa theory for elliptic curves at good supersingular primes this formulation is similar to mazur s at good ordinary primes namely we define a new selmer group and show that it is of cotorsion then we formulate the iwasawa main conjecture as that the characteristic ideal is generated by pollack s p adic l function we show that this main conjecture is equivalent to kato s and perrin riou s main conjectures we also prove an inequality in the main conjecture by using kato s euler system in terms of the and the invariants of our selmer group we specify the numbers and in the asymptotic formula for the order of the tate shafarevich group by kurihara and perrin riou

improved poincare inequalities with weights abstract in this paper we prove that if r n is a bounded john domain the following weighted poincare type inequality holds inf a r f x a l q w 1 c f x d x l p w 2 where f is a locally lipschitz function on d x denotes the distance of x to the boundary of the weights w 1 w 2 satisfy certain cube conditions and 0 1 depends on p q and n this result generalizes previously known weighted inequalities which can also be obtained with our approach

on the passage from local to global in number theory would a reader be able to predict the branch of mathematics that is the subject of this article if its title had not included the phrase in number theory the distinction local versus global with various connotations has found a home in almost every part of mathematics local problems being often a stepping stone to the more difficult global problems to illustrate what the earthy geometric terms local and global signify in number theory let us consider any diophantine equation say

a consistent and stabilized continuous discontinuous galerkin method for fourth order incompressible flow problems this paper presents a new consistent and stabilized finite element formulation for fourth order incompressible flow problems the formulation is based on the c 0 interior penalty method the galerkin least square gls scheme which assures that the formulation is weakly coercive for spaces that fail to satisfy the inf sup condition and considers discontinuous pressure interpolations a stability analysis through a lemma establishes that the proposed formulation satisfies the inf sup condition thus confirming the robustness of the method this lemma indicates that at the element level there exists an optimal or quasi optimal gls stability parameter that depends on the polynomial degree used to interpolate the velocity and pressure fields the geometry of the finite element and the fluid viscosity term numerical experiments are carried out to illustrate the ability of the formulation to deal with arbitrary interpolations for velocity and pressure and to stabilize large pressure gradients

stabilized finite element method for the transient navier stokes equations based on a pressure gradient projection abstract in this paper we present a stabilized finite element formulation for the transient incompressible navier stokes equations the main idea is to introduce as a new unknown of the problem the projection of the pressure gradient onto the velocity space and to add to the incompresibility equation the difference between the laplacian of the pressure and the divergence of this new vector field this leads to a pressure stabilization effect that allows the use of equal interpolation for both velocities and pressures in the case of the transient equations we consider the possibility of treating the pressure gradient projection either implicitly or explicity in the first case the number of unknowns of the problem is substantially increased with respect to the standard galerkin formulation nevertheless iterative techniques may be used in order to uncouple the calculation of the pressure gradient projection from the rest of unknowns velocity and pressure when this vector field is treated explicitly the increment of computational cost of the stabilized formulation with respect to the galerkin method is very low we provide a stability estimate for the case of the simple backward euler time integration scheme for both the implicit and the explicit treatment of the pressure gradient projection

stabilization of incompressibility and convection through orthogonal sub scales in finite element methods two apparently different forms of dealing with the numerical instability due to the incompressibility constraint of the stokes problem are analyzed in this paper the first of them is the stabilization through the pressure gradient projection which consists of adding a certain least squares form of the difference between the pressure gradient and its l2 projection onto the discrete velocity space in the variational equations of the problem the second is a sub grid scale method whose stabilization effect is very similar to that of the galerkin least squares gls method for the stokes problem it is shown here that the first method can also be recast in the framework of sub grid scale methods with a particular choice for the space of sub scales this leads to a new stabilization procedure whose applicability to stabilize convection is also studied in this paper

large scale computation of incompressible viscous flow by least squares finite element method abstract the least square finite element method lsfem based on the velocity pressure vorticity formulation is applied to large scale three dimensional steady incompressible navier stokes problems this method can accommodate equal order interpolations and results in a symmetric positive definite algebraic system which can be solved effectively by simple iterative methods the first order velocity bernoulli pressure vorticity formulation for incompressible viscous flows is also tested the first order velocity pressure stress formulation is not elliptic in the ordinary sense so we do not recommend its use for newtonian flows for three dimensional flows a compatibility equation i e zero divergence of vorticity vector is included to make the first order system elliptic as a by product of proving the ellipticity of first order systems a rigorous mathematical technique has been developed to justify the number of permissible boundary conditions for the navier stokes equations the simple substitution or newton s method is employed to linearize the partial differential equations the lsfem is used to obtain discretized equations and the system of algebraic equations is solved using the jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices matrix free to achieve high efficiency to show the validity of this method for large scale computation we give numerical results for the 2d driven cavity problem at re 10000 with 408 400 bilinear elements the flow in a 3d cavity is calculated at re 100 400 and 1000 with 50 52 50 trilinear elements the taylor gortler like vortices are observed for re 1000