assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows in this work we study the performance of some variational multiscale models vms in the large eddy simulation les of turbulent flows we consider vms models obtained by different subgrid scale approximations which include either static or dynamic subscales linear or nonlinear multiscale splitting and different choices of the subscale space after a brief review of these models we discuss some implementation aspects particularly relevant to the simulation of turbulent flows namely the use of a skew symmetric form of the convective term and the computation of projections when orthogonal subscales are used we analyze the energy conservation and numerical dissipation of the alternative vms formulations which is numerically evaluated in the numerical study we have considered three well known problems the decay of homogeneous isotropic turbulence the taylor green vortex problem and the turbulent flow in a channel we compare the results obtained using different vms models paying special attention to the effect of using orthogonal subscale spaces the vms results are also compared against classical les scheme based on filtering and the dynamic smagorinsky closure altogether our results show the tremendous potential of vms for the numerical simulation of turbulence further we study the sensitivity of vms to the algorithmic constants and analyze the behavior in the small time step limit we have also carried out a computational cost comparison of the different formulations out of these experiments we can state that the numerical results obtained with the different vms formulations as far as they converge are quite similar however some choices are prone to instabilities and the results obtained in terms of computational cost are certainly different the dynamic orthogonal subscales model turns out to be best in terms of efficiency and robustness

on inf sup stabilized finite element methods for transient problems we consider the behavior of inf sup stabilization in the context of transient problems with multiple time scales our motivation for studying this setting is provided by reacting flows problems for which small time steps are necessary in the integration process we show that for algorithms defined through a process wherein spatial and temporal discretizations are separated the coupling of implicit time integration with spatial inf sup stabilization may lead to anomalous pressure behavior including the onset of spurious oscillations for very small time steps effectively this coupling introduces a stability criterion resulting in a dependence between the spatial grid size and the time step we illustrate our theoretical results by numerical examples that demonstrate the stability criterion 2004 elsevier b v all rights reserved

the dirichlet problem of the quantum drift diffusion model abstract we study the quantum drift diffusion model a fourth order parabolic system with dirichlet boundary conditions using a semi discretization approximate method with a compact argument and applying a new entropy estimate we prove the existence of global regular weak solutions

logarithmic conformation reformulation in viscoelastic flow problems approximated by a vms type stabilized finite element formulation abstract the log conformation reformulation originally proposed by fattal and kupferman 2004 allows computing incompressible viscoelastic problems with high weissenberg numbers which are impossible to solve with the typical three field formulation by following this approach in this work we develop a new stabilized finite element formulation based on the logarithmic reformulation using the variational multiscale vms method as stabilization technique together with a modified log conformation formulation our approach follows the term by term stabilization proposed by castillo and codina 2014 for the standard formulation which is more effective when there are stress singularities the formulation can be used when the relaxation parameter is set to zero and permits a direct steady numerical resolution the formulation is validated in the classical benchmark flow past a cylinder and in the well known planar contraction 4 1 achieving very accurate stable and mesh independent results for highly elastic fluids

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

stability of multiple pulses in discrete systems the stability of multiple pulse solutions to the discrete nonlinear schrodinger equation is considered a bound state of widely separated single pulses is rigorously shown to be unstable unless the phase shift delta phi between adjacent pulses satisfies delta phi pi this instability is accounted for by positive real eigenvalues in the linearized system the analysis leading to the instability result does not however determine the linear stability of those multiple pulses for which delta phi pi between adjacent pulses a direct variational approach for a two pulse predicts that it is linearly stable if delta phi pi and if the separation between the individual pulses satisfies a certain condition the variational approach can easily be generalized to study the stability of n pulses for any n or 3 the analysis is supplemented with a detailed numerical stability analysis

existence and positivity of solutions of a fourth order nonlinear pde describing interface fluctuations we study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system in that application x lies in r but here we study primarily the periodic case x e s we establish existence uniqueness and regularity of solutions locally in time for positive initial data in h s and prove the existence of several families of lyapunov functions for the evolution from the latter we establish a sharp connection between existence globally in time and positivity preservation if o t is a maximal half open interval of existence for a positive solution of the equation with t 1 16r2 then t 03 and lim m w r exists and is constant we discuss also some explicit solutions and propose a generalization to higher dimensions 0 1994 john wiley

classical solutions to the one dimensional stationary quantum drift diffusion model abstract the existence of classical solutions to the one dimensional stationary quantum drift diffusion model for semiconductor devices is investigated the proof is based on an exponential variable transformation and the leray schauder fixed point theorem furthermore the uniqueness of solutions to the isothermal model is proved if the current density and the planck constant are sufficiently small

solution of transient viscoelastic flow problems approximated by a term by term vms stabilized finite element formulation using time dependent subgrid scales abstract some finite element stabilized formulations for transient viscoelastic flow problems are presented in this paper these are based on the variational multiscale vms method following the approach introduced in castillo and codina et al 2019 for the navier stokes problem the main feature of the method being that the time derivative term in the subgrid scales is not neglected the main advantage of considering time dependent sub grid scales is that stable solutions for anisotropic space time discretizations are obtained however other benefits related with elastic problems are found along this study additionally a split term by term stabilization method is discussed and redesigned where only the momentum equation is approached using a term by term methodology and which turns out to be much more efficient than other residual based formulations the proposed methods are designed for the standard and logarithmic formulations in order to deal with high weissenberg number problems in addition to anisotropic space time discretizations ensuring stability in all cases the proposed formulations are validated in several benchmarks such as the flow over a cylinder problem and the lid driven cavity problem obtaining stable and accurate results a comparison between formulations and stabilization techniques is done to demonstrate the efficiency of time dependent sub grid scales and the term by term methodologies

variational multi scale stabilized formulations for the stationary three field incompressible viscoelastic flow problem abstract in this paper three field finite element stabilized formulations are proposed for the numerical solution of incompressible viscoelastic flows these methods allow one to use equal interpolation for the problem unknowns u p elastic deviatoric stress velocity pressure and to stabilize dominant convective terms starting from residual based stabilized formulations the proposed method introduces a term by term stabilization which is shown to have a superior behavior when there are stress singularities a general discontinuity capturing technique for the elastic stress component is also proposed which allows one to eliminate the local oscillations that can appear when the weissenberg number we is high and the fluid flow finds an abrupt change in the geometry the formulations are tested in the classical 4 1 planar contraction benchmark up to we 5 in the inertial case with reynolds number re 1 and up to we 6 5 in the quasi non inertial case with re 0 01 the standard oldroyd b constitutive model is used for the rheological behavior and linear and quadratic elements for the spatial approximation

long time behavior for a nonlinear fourth order parabolic equation we study the asymptotic behavior of solutions of the initial boundary value problem with periodic boundary conditions for a fourth order nonlinear degenerate diffusion equation with a logarithmic nonlinearity for strictly positive and suitably small initial data we show that a positive solution exponentially approaches its mean as time tends to infinity these results are derived by analyzing the equation verified by the logarithm of the solution

time dependent subscales in the stabilized finite element approximation of incompressible flow problems abstract in this paper we analyze a stabilized finite element approximation for the incompressible navier stokes equations based on the subgrid scale concept the essential point is that we explore the properties of the discrete formulation that results allowing the subgrid scales to depend on time this apparently natural idea avoids several inconsistencies of previous formulations and also opens the door to generalizations

stabilized stress velocity pressure finite element formulations of the navier stokes problem for fluids with non linear viscosity abstract the three field stress velocity pressure mixed formulation of the incompressible navier stokes problem can lead to two different types of numerical instabilities the first is associated with the incompressibility and loss of stability in the calculation of the stress field and the second with the dominant convection the first type of instabilities can be overcome by choosing an interpolation for the unknowns that satisfies the appropriate inf sup conditions whereas the dominant convection requires a stabilized formulation in any case this paper proposes two stabilized schemes of sub grid scale sgs type differing in the definition of the space of the sub grid scales and both allowing to use the same interpolation for the variables u p deviatoric stress velocity and pressure even in problems where the convection component is dominant and the velocity stress gradients are high another aspect considered in this work is the non linearity of the viscosity modeled with constitutive models of quasi newtonian type this paper includes a description of the proposed methods some of their implementation issues and a discussion about benefits and drawbacks of a three field formulation several numerical examples serve to justify our claims

first second and third order fractional step methods for the three field viscoelastic flow problem in this paper three different fractional step methods are designed for the three field viscoelastic flow problem whose variables are velocity pressure and elastic stress the starting point of our methods is the same as for classical pressure segregation algorithms used in the newtonian incompressible navier stokes problem these methods can be understood as an inexact lu block factorization of the original system matrix of the fully discrete problem and are designed at the pure algebraic level the final schemes allow one to solve the problem in a fully decoupled form where each equation for velocity pressure and elastic stress is solved separately the first order scheme is obtained from a straightforward segregation of pressure and elastic stress in the momentum equation whereas the key point for the second order scheme is a first order extrapolation of these variables the third order fractional step method relies on yosida s scheme referring to the spatial discretization either the galerkin method or a stabilized finite element formulation can be used we describe the fractional step methods first assuming the former and then we explain the modifications introduced by the stabilized formulation we employ and that has been proposed in a previous work this discretization in space shows very good stability permitting in particular the use of equal interpolation for all variables

existence of multibreathers in chains of coupled one dimensional hamiltonian oscillators we prove the existence of monoparametric families of multibreathers in chains of hamiltonian oscillators of one degree of freedom with the total energy of the chain as a parameter at the same time we evaluate the neighborhood of the initial conditions for these solutions as well as their stability for the proof we use an idea originally proposed by poincare we apply our results to calculate families of multibreathers in a chain consisting of coupled morse oscillators

multistable solitons in the cubic quintic discrete nonlinear schrodinger equation we analyze the existence and stability of localized solutions in the one dimensional discrete nonlinear schrodinger dnls equation with a combination of competing self focusing cubic and defocusing quintic onsite nonlinearities we produce a stability diagram for different families of soliton solutions that suggests the co existence of infinitely many branches of stable localized solutions bifurcations that occur with an increase in the coupling constant are studied in a numerical form a variational approximation is developed for accurate prediction of the most fundamental and next order solitons together with their bifurcations salient properties of the model which distinguish it from the well known cubic dnls equation are the existence of two different types of symmetric solitons and stable asymmetric soliton solutions that are found in narrow regions of the parameter space the asymmetric solutions appear from and disappear back into the symmetric ones via loops of forward and backward pitchfork bifurcations

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

dynamic term by term stabilized finite element formulation using orthogonal subgrid scales for the incompressible navier stokes problem abstract in this paper we propose and analyze the stability and the dissipative structure of a new dynamic term by term stabilized finite element formulation for the navier stokes problem that can be viewed as a variational multiscale vms method under some assumptions the essential point of the formulation is the time dependent nature of the subscales and contrary to residual based formulations the introduction of two velocity subscale components they represent the components of the convective and the pressure gradient terms respectively of the momentum equation that cannot be captured by the finite element mesh a key idea of the proposed method is that the convective subscale is close to a solenoidal field and the pressure gradient subscale is close to a potential field the method ensures stability in anisotropic space time discretizations which is proved using numerical analysis for a linearized problem and demonstrated in classical numerical tests the work includes a detailed description of the proposed formulation and several numerical examples that serve to justify our claims

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

a positivity preserving numerical scheme for a nonlinear fourth order parabolic system noindent a positivity preserving numerical scheme for a fourth order nonlinear parabolic system arising in quantum semiconductor modeling is studied the system is numerically treated by introducing an additional nonlinear potential and a subsequent semidiscretization in time the resulting sequence of nonlinear second order elliptic systems admits at each time level strictly positive solutions which is proved by an exponential transformation of variables the stability of the scheme is shown and convergence is proved in one space dimension numerical results concerning the switching behavior of a resonant tunneling diode are presented

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

an algebraic approach to discrete breather construction we obtain useful analytic discrete breather solutions in the form of elliptic functions for a large class of physically relevant nonlinear lattices through the application of an algebraic approach the method we introduce is quite accurate and applies equally well to optic and acoustic chains we present explicit results for 4 and fermi pasta ulam lattices and discuss their implications to breather properties such as generation and mobility the method is useful also in cases where no explicit analytic solutions can be obtained in the context of the present approach discrete breathers are shown to be localized cnoidal modes of nonlinear lattices

long term stability estimates and existence of a global attractor in a finite element approximation of the navier stokes equations with numerical subgrid scale modeling variational multiscale methods lead to stable finite element approximations of the navier stokes equations dealing with both the indefinite nature of the system pressure stability and the velocity stability loss for high reynolds numbers these methods enrich the galerkin formulation with a subgrid component that is modeled in fact the effect of the subgrid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum thus they also act as effective large eddy simulation turbulence models and allow one to compute flows without the need to capture all the scales in the system in this article we consider a dynamic subgrid model that enforces the subgrid component to be orthogonal to the finite element space in the l 2 sense we analyze the long term behavior of the algorithm proving the existence of appropriate absorbing sets and a compact global attractor the improvements with respect to a finite element galerkin approximation are the long term estimates for the subgrid component which are translated to effective pressure and velocity stability thus the stabilization introduced by the subgrid model into the finite element problem does not deteriorate for infinite time intervals of computation