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 general advancing front technique for filling space with arbitrary objects an advancing front space filling technique for arbitrary objects has been developed the input required consists of the specification of the desired mean point distance in space and an initial triangulation of the surface one object at a time is removed from the active front and if possible surrounded by admissible new objects this operation is repeated until no active objects are left two techniques to obtain maximum packing are discussed closest object placement during generation and move enlarge after generation different deposition or layering patterns can be achieved by selecting the order in which objects are eliminated from the active front timings show that for simple objects like spheres the scheme is considerably faster than volume mesh generators based on the advancing front technique making it possible to generate large 106 yet optimal clouds of points in a matter of minutes on a pc for more general objects the performance may degrade depending on the complexity of the penetration checks several examples are included that demonstrate the capabilities of the technique copyright 2004 john wiley sons ltd

hybrid subconvexity bounds for l left frac 1 2 hbox sym 2 f otimes g right fix an integer kappa geqslant 2 let p be prime and let k kappa be an even integer for f a holomorphic cusp form of weight k and full level and g a primitive holomorphic cusp form of weight 2 kappa and level p we prove hybrid subconvexity bounds for l left frac 1 2 text sym 2 f otimes g right in the k and p aspects when p frac 13 64 delta k p frac 3 8 delta for any 0 delta frac 11 128 these bounds are achieved through a first moment method with amplification when p frac 13 64 k leqslant p frac 4 13

pressure stability in fractional step finite element methods for incompressible flows abstract the objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows that use a pressure poisson equation for the classical first order projection method it is shown that there is a pressure control which depends on the time step size and therefore there is a lower bound for this time step for stability reasons the situation is much worse for a second order scheme in which part of the pressure gradient is kept in the momentum equation the pressure stability in this case is extremely weak to overcome these shortcomings a stabilized fractional step finite element method is also considered and its stability is analyzed some simple numerical examples are presented to support the theoretical results

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

a stabilized finite element method for generalized stationary incompressible flows abstract in this paper we describe a finite element formulation for the numerical solution of the stationary incompressible navier stokes equations including coriolis forces and the permeability of the medium the stabilized method is based on the algebraic version of the sub grid scale approach we first describe this technique for general systems of convection diffusion reaction equations and then we apply it to the linearized flow equations the important point is the design of the matrix of stabilization parameters that the method has this design is based on the identification of the stability problems of the galerkin method and a scaling of variables argument to determine which coefficients must be included in the stabilization matrix this together with the convergence analysis of the linearized problem leads to a simple expression for the stabilization parameters in the general situation considered in the paper the numerical analysis of the linearized problem also shows that the method has optimal convergence properties

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

implicit fully mesh less method for compressible viscous flow calculations a dual time implicit mesh less scheme is developed for solution of governing viscous flow equations the computational efficiency of the method is enhanced by adopting accelerating techniques such as local time stepping and residual smoothing the taylor series least square method is used for approximation of derivatives at each node which leads to a central difference spatial discretization the capabilities of the method are demonstrated by flow computations about standard cases at subsonic and transonic laminar flow conditions results are presented which indicate good agreements with the alternative numerical and experimental data the computational time is considerably reduced when using the proposed mesh less method compared with the explicit mesh less and finite volume counterparts using the same distribution of points

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

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

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

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

subconvexity for twisted l functions on gl 3 let q be a large prime and chi the quadratic character modulo q let phi be a self dual cuspidal hecke eigenform for rm sl 3 bbb z and f a hecke maa ss cusp form for gamma 0 q subseteq rm sl 2 bbb z we consider the twisted l functions l s phi times f times chi and l s phi times chi on rm gl 3 times rm gl 2 and rm gl 3 with conductors q 6 and q 3 respectively we prove the subconvexity bounds l 1 2 phi times f times chi ll phi f varepsilon q 5 4 varepsilon quad l 1 2 it phi times chi ll phi t varepsilon q 5 8 varepsilon for any varepsilon 0

a numerical method for incompressible viscous flow simulation we describe a numerical scheme for computing time dependent solutions of the incompressible navier stokes equations in the primitive variable formulation this scheme uses finite elements for the space discretization and operator splitting techniques for the time discretization the resulting discrete equations are solved using specialized nonlinear optimization algorithms that are computationally efficient and have modest storage requirements the basic numerical kernel is the preconditioned conjugate gradient method for symmetric positive definite sparse matrix systems which can be efficiently implemented on the architectures of vector and parallel processing supercomputers

coefficient of maass forms and the siegel zero with an appendix by dorian goldfeld jeffrey hoffstein and daniel lieman

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

rankin selberg l functions in the level aspect keywords moments rankin selberg convolution level aspect convexity breaking reference tan article 2002 003doi 10 1215 s0012 7094 02 11416 1 record created on 2008 11 14 modified on 2017 05 12

factorization methods for the numerical approximation of navier stokes equations we investigate a general approach for the numerical approximation of incompressible navier stokes equations based on splitting the original problem into successive subproblems which are cheaper to solve the splitting is obtained through an algebraic approximate factorization of the matrix arising from space and time discretization of the original equations several schemes based on approximate factorization are investigated for some of these methods a formal analogy with well known time advancing schemes such as the projection chorin temam s can be pointed out features and limits of this analogy that was earlier introduced in b perot j comp phys 108 1993 51 8 are addressed other new methods can also be formulated starting from this approach in particular we introduce the so called yosida method which can be investigated in the framework of quasi compressibility schemes numerical results illustrating the different performances of the different methods addressed are presented for a couple of test cases

error analysis of some galerkin least squares methods for the elasticity equations the recent technique of stabilizing mixed finite element methods by augmenting the galerkin formulation with least squares terms calculated separately on each element is considered the error analysis is performed in a unified manner yielding improved results for some methods introduced earlier in addition a new formulation is introduced and analyzed method is used within two different contexts the first is in connection with the plain strain or three dimensional problems near the incompressible limit for which the poisson ratio equals one half it is well known that for these problems a standard lower order displacement finite element method will break down by now it is also well known how a working method can be obtained a new unknown the pressure is introduced and discretized with a finite element space different from that used for the displacements the finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by brezzi and babuska 5 7 the application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition for a particular choice of displacement and pressure approximations it can be a difficult task to verify this babuska brezzi or inf sup condition during the last decade this problem has been intensively studied and the performances of many combinations are now known cf 18 and 23 26 for some recent extensions in addition there exists a general technique utilizing bubble functions by which a stable method can

hopf bifurcation of the unsteady regularized driven cavity flow abstract a numerical simulation of the unsteady incompressible flow in the unit cavity is performed by using a chebyshev tau approximation for the space variables the high accuracy of the spectral methods and the condensed distribution of the chebyshev collocation points near the boundary enable us to obtain reliable results for high reynolds numbers with a moderate number of modes it is found that the flow converges to a stationary state for reynolds numbers re up to 10 000 for reynolds numbers larger than a critical value 10 000

subconvex equidistribution of cusp forms reduction to eisenstein observables let traverse a sequence of cuspidal automorphic representations of gl2 with large prime level unramified central character and bounded infinity type for g gl1 pgl2 let h g denote the assertion that subconvexity holds for g twists of the adjoint l function of with polynomial dependence upon the conductor of the twist we show that h gl1 implies h pgl2 in geometric terms h pgl2 corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term h gl1 to the special case in which the relevant sequence of measures is tested against an eisenstein series

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

the circle method and bounds for l functions ii subconvexity for twists of gl 3 l functions let pi be a rm sl 3 bbb z hecke maass cusp form let chi chi 1 chi 2 be a dirichlet character with chi i primitive modulo m i suppose m 1 m 2 are primes such that sqrt m 2 m 4 delta m 1 m 2m 3 delta where m m 1m 2 and 0 delta 1 28 in this paper we will prove the following subconvex bound l left 1 over 2 pi otimes chi right ll pi varepsilon m 3 over 4 delta varepsilon

anisotropic error estimates for elliptic problems moving from the anisotropic interpolation error estimates derived in 12 we provide here both a priori and a posteriori estimates for a generic elliptic problem the a priori result is deduced by following the standard finite element theory for the a posteriori estimate the analysis extends to anisotropic meshes the theory presented in 3 5 numerical test cases validate the derived results

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

efficient and non reflecting far field boundary conditions for incompressible flow calculations abstract traditionally the artificial compressibility ac method of chorin is used for simulation of low speed and incompressible flows in this method the difficulty of continuity and momentum equations decoupling is removed by adding an artificial time derivative of pressure to the continuity equation for the first time a fully two dimensional upwind scheme was presented for ac equations by using characteristic structure of equations by the authors in this paper a new remote boundary calculation method is presented by using the idea of characteristics for these equations instead of simple boundary conditions which usually are employed for incompressible flows the flow quantities at the far field boundaries are evaluated by compatibility relations of characteristic equations this is implemented by assuming a row of ghost cells outside of the far field boundary and using the flux calculation method based on characteristics similar to the inside of computational domain the idea in conjunction with multidimensional characteristic based scheme was tested for incompressible flow around circular cylinder in comparison with conventional far field boundary condition and showed good improvements in the terms of accuracy and convergence speed

a gridless boundary condition method for the solution of the euler equations on embedded cartesian meshes with multigrid the solution of the euler equations using a gridless boundary condition treatment on a patched embedded cartesian field mesh is described the gridless boundary treatment is implemented by means of a least squares fitting of the conserved flux variables using a cloud of nodes in the vicinity of the body the method allows for accurate treatment of the surface boundary conditions without the need for excessive refinement of the cartesian mesh additionally the method does not suffer from problems associated with thin body geometry or extremely fine cut cells near the body unlike some methods that consider a gridless or meshless treatment throughout the entire domain the use of a cartesian field mesh allows for effective implementation of multigrid acceleration and issues associated with global conservation are alleviated results are presented for transonic flow about single and dual naca 0012 airfoil configurations including a convergence study indicating the effectiveness of multigrid acceleration within the construct of the gridless boundary treatment where applicable comparisons to the flo52 body fitted euler code are presented to gauge the accuracy of the method

distribution of mass of holomorphic cusp forms we prove an upper bound for the l4 norm and for the l2 norm restricted to the vertical geodesic of a holomorphic hecke cusp form f of large weight k the method is based on watson s formula and estimating a mean value of certain l functions of degree 6 further applications to restriction problems of siegel modular forms and subconvexity bounds of degree 8 l functions are given

multicloud multigrid convergence with a meshless operator the primary objective of this work is to develop and test a new convergence acceleration technique we call multicloud multicloud is well founded in the mathematical basis of multigrid but relies on a meshless operator on coarse levels the meshless operator enables extremely simple and automatic coarsening procedures for arbitrary meshes using arbitrary fine level discretization schemes the performance of multicloud is compared with established multigrid techniques for structured and unstructured meshes for the euler equations on two dimensional test cases results indicate comparable convergence rates per unit work for multicloud and multigrid however because of its mesh and scheme transparency multicloud may be applied to a wide array of problems with no modification of fine level schemes as is often required with agglomeration techniques the implication is that multicloud can be implemented in a completely modular fashion allowing researchers to develop fine level algorithms independent of the convergence accelerator for complex three dimensional problems

finite volume solution of the two dimensional euler equations on a regular triangular mesh the two dimensional euler equations have been solved on a triangular grid by a multigrid scheme using the finite volume approach by careful construction of the dissipative terms the scheme is designed to be secondorder accurate in space provided the grid is smooth except in the vicinity of shocks where it behaves as firstorder accurate in its present form the accuracy and convergence rate of the triangle code are comparable to that of the quadrilateral mesh code of jameson

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points ii we prove a general subconvex bound in the level aspect for rankin selberg l functions associated with two primitive holomorphic or maass cusp forms over q we use this bound to establish the equidistribution of incomplete galois orbits of heegner points on shimura curves associated with indefinite quaternion algebras over q

a multidimensional characteristic based method for making incompressible flow calculations on unstructured grids this paper presents a new multidimensional characteristic based scheme mcb for numerical simulation of incompressible flows on unstructured grids in conjunction with the artificial compressibility method the original mcb scheme has been proposed by the authors already and this research is the extension of the aforementioned scheme to unstructured grids the significant difference between mcb and the conventional characteristic based scheme cb is the multidimensional nature of the scheme which allows information to propagate in any direction instead of only normal to the cell interface in addition local time stepping and the residual smoothing technique have been used for convergence acceleration the accuracy and utility of the proposed scheme have been studied by means of numerical tests for different reynolds numbers and the results obtained using the new scheme are in good agreement with the standard benchmark solution in the literature

stabilized finite element approximation of transient incompressible flows using orthogonal subscales abstract in this paper we present a stabilized finite element method to solve the transient navier stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales the latter are approximately accounted for so as to end up with a stable finite element problem which in particular allows to deal with convection dominated flows and the use of equal velocity pressure interpolations three main issues are addressed the first is a method to estimate the behavior of the stabilization parameters based on a fourier analysis of the problem for the subscales secondly the way to deal with transient problems discretized using a finite difference scheme is discussed finally the treatment of the nonlinear term is also analyzed a very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space in the transient case this simplifies considerably the numerical scheme

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

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points keywords rankin selberg l function subconvexity heegner points reference tan article 2004 002doi 10 4007 annals 2004 160 185 record created on 2008 11 14 modified on 2017 05 12

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

the behaviour of eigenstates of arithmetic hyperbolic manifolds in this paper we study some problems arising from the theory of quantum chaos in the context of arithmetic hyperbolic manifolds we show that there is no strong localization scarring onto totally geodesic submanifolds arithmetic examples are given which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom