on the classification of hyperovals abstract a hyperoval in the projective plane p 2 f q is a set of q 2 points no three of which are collinear hyperovals have been studied extensively since the 1950s with the ultimate goal of establishing a complete classification it is well known that hyperovals in p 2 f q are in one to one correspondence to polynomials with certain properties called o polynomials of f q we classify o polynomials of f q of degree less than 1 2 q 1 4 as a corollary we obtain a complete classification of exceptional o polynomials namely polynomials over f q that are o polynomials of infinitely many extensions of f q

on crank nicolson adams bashforth timestepping for approximate deconvolution models in two dimensions we consider a crank nicolson adams bashforth temporal discretization together with a finite element spatial discretization for efficiently computing solutions to approximate deconvolution models of incompressible flow in two dimensions we prove a restriction on the timestep that will guarantee stability and provide several numerical experiments that show the proposed method is very effective at finding accurate coarse mesh approximations for benchmark flow problems

on the existence of global attractors of the approximate deconvolution models of turbulence abstract in this report we consider the problem of the existence of global attractors of the weak solution of the approximate deconvolution models of turbulence the l 2 global attractors are known to exist for large eddy simulation models such as leray alpha leray deconvolution or ns omega models in the case of the approximate deconvolution model as the l 2 norm of the weak solution of the adm is not controlled by the l 2 norm of the initial data and the approximate deconvolution model is an approximation of the filtered nse we are led to the question of existence of h 1 global attractors we also prove better regularity of the weak solution of adm

numerical analysis of the leray reduced order model abstract standard roms generally yield spurious numerical oscillations in the simulation of convection dominated flows regularized roms use explicit rom spatial filtering to decrease these spurious numerical oscillations the leray rom is a recently introduced regularized rom that utilizes explicit rom spatial filtering of the convective term in the navier stokes equations this paper presents the numerical analysis of the finite element discretization of the leray rom error estimates for the rom differential filter which is the explicit rom spatial filter used in the leray rom are proved these rom filtering error estimates are then used to prove error estimates for the leray rom finally both the rom filtering error estimates and the leray rom error estimates are numerically investigated in the simulation of the two dimensional navier stokes equations with an analytic solution

a global branch of positive solutions above the continuous spectrum for problems with indefinite nonlinearities we prove the existence and bifurcation of a global branch of positive solutions for a nonlinear neumann eigenvalue problem on the half axis 0 the nonlinearity is assumed to have a superlinear growth multiplied by a weight function changing sign this leads to the existence of nontrivial solutions above the continuous spectrum of the linearised problem

on the accuracy of the finite element method plus time relaxation if u denotes a local spatial average of u then u u u is the associated fluctuation consider a time relaxation term added to the usual finite element method the simplest case for the model advection equation u t a u x t is u h t a u h v h u h v h f x t v h we analyze the error in this and more importantly higher order extensions and show that the added time relaxation term not only suppresses excess energy in marginally resolved scales but also increases the accuracy of the resulting finite element approximation

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

on the vreman filter based stabilization for the advection equation this report investigates a stabilization method for first order hyperbolic differential equations if the usual unstabilized finite element method is used to solve numerically first order hyperbolics on unstructured meshes using general elements it is expected a loss of accuracy by one power of the mesh size and also for non smooth solutions the computed approximation might contain spurious oscillations to stabilize the finite element method the authors consider adding to the first order hyperbolic system as a stabilization term the vreman filtering analog of the vms variational multiscale method vreman 2003 22 the resulting model is shown to have the expected highest attainable convergence rate o h k 0 5 when the filter radius is optimally chosen the theoretical rates are also checked numerically and comparison with another known scheme is provided

efficient unconditionally stable and optimally accurate fe algorithms for approximate deconvolution models this paper addresses an open question of how to devise numerical schemes for approximate deconvolution fluid flow models that are efficient unconditionally stable and optimally accurate we propose analyze and test a scheme for these models that has each of these properties for the case of homogeneous dirichlet velocity boundary conditions there are several important components of the derivation both at the continuous and discrete levels which allow for these properties to hold the proofs of stability and convergence are carried out through the use of a special choice of test function and some technical estimates numerical tests are provided that confirm the effectiveness of the scheme

stabilized approximation to degenerate transport equations via filtering abstract we analyze a stabilization technique for degenerate transport equations of particular interest are coupled parabolic hyperbolic problems when the diffusion coefficient is zero in part of the domain the unstabilized computed approximations of these problems are highly oscillatory and several techniques have been proposed and analyzed to mitigate the effects of the sub grid errors that contribute to the oscillatory behavior in this paper we modify a time relaxation algorithm proposed in 1 and further studied in 10 our modification introduces the relaxation operator as a post processing step the operator is not time dependent so the discrete relaxation system need only be factored once we provide convergence analysis for our algorithm along with numerical results for several model problems

multiple solutions of schrodinger equations with indefinite linear part and super or asymptotically linear terms abstract based on new information concerning strongly indefinite functionals without palais smale conditions we study existence and multiplicity of solutions of the schrodinger equation u v x u g x u for x r n u x 0 as x where v and g are periodic with respect to x and 0 lies in a gap of v supposing g is asymptotically linear as u and symmetric in u we obtain infinitely many geometrically distinct solutions we also consider the situation where g is super linear with mild assumptions different from those studied previously and establish the existence and multiplicity

on a class of semilinear schrodinger equations involving critical growth and discontinuous nonlinearities abstract in this paper we deal with a class of schrodinger equation in r n involving critical sobolev exponent and jump discontinuities the basic tool employed here is an approximation technique with periodic functions and variational arguments based on a linking theorem for locally lipschitz functionals

multiple solutions for a class of nonlinear schrodinger equations abstract in this paper we find new conditions to ensure the existence of infinitely many homoclinic type solutions for the schrodinger equation u v x u g x u for x r n assuming v x and g x u depend periodically on x we deal with the situations where g x u is as u asymptotically linear or superlinear with certain hypothesis different from ones used in previous related study our approach is variational and we use the cerami condition instead of the palais smale one for deformation arguments

post bifurcation and stability of a finitely strained hexagonal honeycomb subjected to equi biaxial in plane loading abstract the buckling and crushing mechanics of cellular honeycomb materials is an important engineering problem motivated by the pioneering experimental and numerical studies of papka and kyriakides 1994 1999a b we review the literature on finitely strained honeycombs subjected to in plane loading and identify two open questions i how does the mechanical response of the honeycomb depend on the applied loading device and ii what can the bloch wave representation of all bounded perturbations contribute to our understanding of the stability of post bifurcated equilibrium configurations to address these issues we model the honeycomb as a two dimensional infinite perfect periodic medium we use analytical group theory methods as opposed to the more common but less robust imperfection method to study the honeycomb s bifurcation behavior under three different far field loadings that produce initially the same equi biaxial contractive dilatation using an fem discretization of the honeycomb walls struts we solve the equilibrium equations to find the principal and bifurcated equilibrium paths for each of the three loading cases we evaluate the structure s stability using two criteria rank one convexity of the homogenized continuum long wavelength perturbations and bloch wave stability bounded perturbations of arbitrary wavelength we find that the post bifurcation behavior is extremely sensitive to the applied loading device in spite of a common principal solution we confirm that the flower mode is always unstable as previously reported however our first ever bloch wave stability analysis of the post bifurcated equilibrium paths shows that the flower mode is stable for all sufficiently short wavelength perturbations this new result provides a realistic explanation for why this mode has been observed in the finite size specimen experiments of papka and kyriakides 1999a

on an optimal finite element scheme for the advection equation in this report it is presented a numerical finite element scheme for the advection equation that attains the optimal l 2 convergence rate o h k 1 when order k finite elements are used improving the order o h k 0 5 of other previous regularization methods this result is also confirmed by the numerical test performed in the last section the scheme assumes unstructured grids periodic boundary conditions a constant advection field and a bit two units stronger regularity on the exact solution than in the classical suboptimal finite element theory

on the approximation of the unsteady navier stokes equations by finite element projection methods this paper provides an analysis of a fractional step projection method to compute incompressible viscous flows by means of finite element approximations the analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method such a theoretical distinction leads to a finite element projection method with a poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm error estimates in finite time are given an extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method

hopf bifurcation and exchange of stability in diffusive media we consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses a continuous spectrum up to the imaginary axis for all values of the bifurcation parameter and that a pair of imaginary eigenvalues crosses the imaginary axis for a reaction diffusion convection system we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations prove the occurrence of a hopf bifurcation and the nonlinear stability of the bifurcating time periodic solutions again with respect to spatially localized perturbations

gap bifurcation for nonlinear perturbations of hill s equation keywords bifurcation nonlinear hill s equation testfunctions reference ana article 1990 002view record in web of science record created on 2008 12 10 modified on 2016 08 08

vortex families near a spectral edge in the gross pitaevskii equation with a two dimensional periodic potential we examine numerically vortex families near band edges of the bloch wave spectrum for the gross pitaevskii equation with two dimensional periodic potentials and for the discrete nonlinear schrodinger equation we show that besides vortex families that terminate at a small distance from the band edges via fold bifurcations there exist vortex families that are continued all the way to the band edges

numerical approximation of the voigt regularization for incompressible navier stokes and magnetohydrodynamic flows we study voigt regularizations for the navier stokes equations nses and magnetohydrodynamic mhd equations in the presence of physical boundary conditions in particular we develop the first finite element numerical algorithms for these systems prove stability and convergence of the algorithms and test them computationally on problems of practical interest it is found that unconditionally stable implementations of the voigt regularization can be made from a simple change to existing nse and mhd codes and moreover optimal convergence of the developed algorithms solutions to physical solutions can be obtained if lower order mixed finite elements are used finally we show that for several benchmark problems the voigt regularization on a coarse mesh produces good approximations to nse and mhd systems that is the voigt regularization provides accurate reduced order models for nse and mhd flows

bifurcation into gaps in the essential spectrum in this lecture bifurcation from any boundary point of the essential spectrum is studied for a class of nonlinear operators generalizing results concerning bifurcation from the lowest point of the continuous spectrum the study is motivated by a nonlinear hill s equation as a typical example for a problem with gaps in the purely continuous spectrum

periodic solutions for a class of non autonomous hamiltonian systems abstract we consider the existence of nontrivial periodic solutions for a superlinear hamiltonian system h j u a t u h t u 0 u r 2 n t r we prove an abstract result on the existence of a critical point for a real valued functional on a hilbert space via a new deformation theorem different from the works in the literature the new deformation theorem is constructed under the cerami type condition instead of palais smale type condition in addition the main assumption here is weaker than the usual ambrosetti rabinowitz type condition 0 h t u u h t u 2 u r 0 this result extends theorems given by li and willem j math anal appl 189 1995 6 32 and li and szulkin j differential equations 112 1994 226 238

shadowing for nonautonomous parabolic problems with applications to long time error bounds shadowing results for time discretizations of nonautonomous parabolic evolution equations z t a t z t f t are given the analysis is performed in an abstract banach space setting of uniformly sectorial operators a t under natural assumptions on the operators a t it is shown that the backward euler discretization of a smooth solution obtained with constant stepsize h is ch shadowed by a true solution of the system and vice versa every smooth trajectory is ch shadowed by a suitable backward euler discretization this validates numerical computations over long time intervals it is further explained how these results extend to higher order runge kutta methods

numerical analysis and testing of a stable and convergent finite element scheme for approximate deconvolution turbulence models abstract this report presents a stable and convergent finite element scheme for the approximate deconvolution turbulence models adm the adm is a popular turbulence model intensely studied lately but the computation of its numerical solution raises issues in terms of efficiency and accuracy this report addresses this question the proposed scheme presented herein is based on a new interpretation of the adm model recently introduced by the author following this interpretation the solution of the adm is viewed as the average of a perturbed navier stokes system the scheme uses the crank nicolson time discretization and the finite element spatial discretization and is proved to be stable and convergent provided a moderate choice of the time step is made numerical tests to verify the convergence rates and performance on a benchmark problem are also provided and they prove the correctness of this approach to numerically solve the adm

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

necessary and sufficient conditions for gap bifurcation keywords square root of a positive selfadjoint linear operator in a hilbert space csp 1 nonlinearity gap bifurcation of a nonlinear equation reference ana article 1992 004doi 10 1016 0362 546x 92 90230 cview record in web of science record created on 2008 12 10 modified on 2016 08 08

exceptional scattered polynomials abstract let f be an f q linear function over f q n if the f q subspace u x q t f x x f q n defines a maximum scattered linear set then we call f a scattered polynomial of index t as these polynomials appear to be very rare it is natural to look for some classification of them we say a function f is an exceptional scattered polynomial of index t if the subspace u associated with f defines a maximum scattered linear set in pg 1 q m n for infinitely many m our main results are the classifications of exceptional scattered monic polynomials of index 0 for q 5 and of index 1 the strategy applied here is to convert the original question into a special type of algebraic curves and then to use the intersection theory and the hasse weil theorem to derive contradictions

convergence of time averaged statistics of finite element approximations of the navier stokes equations when discussing numerical solutions of the navier stokes equations especially when turbulent flows are concerned there are at least two questions that can be raised what is meaningful to compute how to determine the fidelity of the computed solution with respect to the true solution this paper takes a step towards the answer of these questions for turbulent flows we consider long time averages of weak solutions of the navier stokes equations rather than strong solutions we present error estimates for the time averaged energy dissipation rate drag and lift most of them under the assumption of small reynolds generalized grashof number for shear flows we address the question of fidelity of the computed solution with respect to the true solution in view of kolmogorov s energy cascade theory

a finite element variational multiscale method for the navier stokes equations this paper presents a variational multiscale method vms for the incompressible navier stokes equations which is defined by a large scale space lh for the velocity deformation tensor and a turbulent viscosity nu t the connection of this method to the standard formulation of a vms is explained the conditions on lh under which the vms can be implemented easily and efficiently into an existing finite element code for solving the navier stokes equations are studied numerical tests with the smagorinsky large eddy simulation model for nu t are presented

analysis of mesh effects on turbulent flow statistics abstract turbulence models such as the smagorinsky model herein are used to represent the energy lost from resolved to under resolved scales due to the energy cascade i e non linearity analytic estimates of the energy dissipation rates of a few turbulence models have recently appeared but none yet study the energy dissipation restricted to resolved scales i e after spatial discretization with h micro scale we do so herein for the smagorinsky model upper bounds are derived on the computed time averaged energy dissipation rate e u h for an under resolved mesh h for turbulent shear flow for coarse mesh size o r e 1 h l it is proven e u h c s h 2 l 5 c s 4 h l 5 2 c s 4 h 3 2 u 3 l where u and l are global velocity and length scale and c s and are model parameters this upper bound being independent of the viscosity at high reynolds number is in accord with the equilibrium dissipation law kolmogrov s conventional turbulence theory this estimate suggests over dissipation for any of c s 0 and 0 consistent with numerical evidence on the effects of model viscosity without wall damping function moreover the analysis indicates that the turbulent boundary layer is a more important length scale for shear flow than the kolmogorov microscale

approximate deconvolution model in a bounded domain with vertical regularization abstract we study the global existence issue for a three dimensional approximate deconvolution model with a vertical filter we consider this model in a bounded cylindrical domain where we construct a unique global weak solution the proof is based on a refinement of the energy method given by berselli 2012 in 4

existence and bifurcation of solutions for nonlinear perturbations of the periodic schrodinger equation abstract in this paper we apply previously obtained abstract bifurcation results to nonlinear perturbations of the periodic schrodinger equation we show that depending on the sign of the nonlinearity lower or upper end points of the continuous spectrum are bifurcation points the main part of the proof consists of the construction of suitable test functions which are required for the application of the abstract theorem instead of pursuing the notion of an eigenpaket of the differential operator which has successfully been used in the one dimensional case we now exploit a construction based on bloch waves of the linear schrodinger operator

well posedness of a reduced order approximate deconvolution turbulence model abstract we prove well posedness for a recently proposed turbulence model and derive an estimate for its modeling error the model is derived from the fourth order representation of the stolz adams approximate deconvolution model by making an additional deconvolution approximation to reduce the model to second order which also allows the model to be computed with efficient and unconditionally stable algorithms

existence of solutions for semilinear elliptic equations with indefinite linear part we study equations on rn lu w x u p 1 ulu f u where l is an indefinite schrodinger operator 0 w x 0 as and subcritical nonlinearities we prove existence of solutions u h1 rn via an inversion to a dual variational problem

equivalence and characterizations of linear rank metric codes based on invariants abstract we show that the sequence of dimensions of the linear spaces generated by a given rank metric code together with itself under several applications of a field automorphism is an invariant for the whole equivalence class of the code the same property is proven for the sequence of dimensions of the intersections of itself under several applications of a field automorphism these invariants give rise to easily computable criteria to check if two codes are inequivalent we derive some concrete values and bounds for these dimension sequences for some known families of rank metric codes namely gabidulin and generalized twisted gabidulin codes we then derive conditions on the length of the codes with respect to the field extension degree such that codes from different families cannot be equivalent furthermore we derive upper and lower bounds on the number of equivalence classes of gabidulin codes and twisted gabidulin codes improving a result of schmidt and zhou for a wider range of parameters in the end we use the aforementioned sequences to determine a characterization result for gabidulin codes

solvability of nonlinear equations in spectral gaps of the linearization keywords strongle indefinite nonlinear hill s equation reference ana article 1992 002doi 10 1016 0362 546x 92 90116 vview record in web of science record created on 2008 12 10 modified on 2016 08 08

numerical analysis of leray tikhonov deconvolution models of fluid motion this report develops and studies a new family of navier stokes equation regularizations leray tikhonov regularizations with time relaxation models this new family of turbulence models is based on a modification consistent with the large scales of tikhonov lavrentiev regularization with this approach we obtain an approximation of the unfiltered solution by one filtering step we introduce the modified tikhonov deconvolution operator and study its mathematical properties we also perform rigorous numerical analysis of a computationally attractive algorithm for this family of models and present numerical experiments using it

finite element error analysis of a zeroth order approximate deconvolution model based on a mixed formulation a suitable discretization for the zeroth order model in large eddy simulation of turbulent flows is sought this is a low order model but its importance lies in the insight that it provides for the analysis of the higher order models actually used in practice by the pioneers stolz and adams n a adams s stolz on the approximate deconvolution procedure for les phys fluids 2 1999 1699 1701 n a adams s stolz deconvolution methods for subgrid scale approximation in large eddy simulation in b j geurts ed modern simul strategies for turbulent flow edwards philadelphia 2001 pp 21 44 and others the higher order models have proven to be of high accuracy however stable discretizations of them have proven to be tricky and other stabilizations such as time relaxation and eddy viscosity are often added we propose a discretization based on a mixed variational formulation that gives the correct energy balance we show it to be unconditionally stable and prove convergence

low degree planar polynomials over finite fields of characteristic two abstract planar functions are mappings from a finite field f q to itself with an extremal differential property such functions give rise to finite projective planes and other combinatorial objects there is a subtle difference between the definitions of these functions depending on the parity of q and we consider the case that q is even we classify polynomials of degree at most q 1 4 that induce planar functions on f q by showing that such polynomials are precisely those in which the degree of every monomial is a power of two as a corollary we obtain a complete classification of exceptional planar polynomials namely polynomials over f q that induce planar functions on infinitely many extensions of f q the proof strategy is to study the number of f q rational points of an algebraic curve attached to a putative planar function our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear functions