reduced basis error bound computation of parameter dependent navier stokes equations by the natural norm approach this work focuses on the a posteriori error estimation for the reduced basis method applied to partial differential equations with quadratic nonlinearity and affine parameter dependence we rely on natural norms local parameter dependent norms to provide a sharp and computable lower bound of the inf sup constant we prove a formulation of the brezzi rappaz raviart existence and uniqueness theorem in the presence of two distinct norms this allows us to relax the existence condition and to sharpen the field variable error bound we also provide a robust algorithm to compute the sobolev embedding constants involved in the error bound and in the inf sup lower bound computation we apply our method to a steady natural convection problem in a closed cavity with a grashof number varying from 10 to 10 7

a space time petrov galerkin certified reduced basis method application to the boussinesq equations we present a space time certified reduced basis method for long time integration of parametrized parabolic equations with quadratic nonlinearity which admit an affine decomposition in parameter but with no restriction on coercivity of the linearized operator we first consider a finite element discretization based on discontinuous galerkin time integration and introduce associated petrov galerkin space time trial and test space norms that yield optimal and asymptotically mesh independent stability constants we then employ an hp petrov galerkin or minimum residual space time reduced basis approximation we provide the brezzi rappaz raviart a posteriori error bounds which admit efficient offline online computational procedures for the three key ingredients the dual norm of the residual an inf sup lower bound and the sobolev embedding constant the latter are based respectively on a more round off resistant residual norm evaluation procedure a variant of the successive constraint method and

on the stability of the reduced basis method for stokes equations in parametrized domains we present an application of reduced basis method for stokes equations in domains with affine parametric dependence the essential components of the method are i the rapid convergence of global reduced basis approximations galerkin projection onto a space wn spanned by solutions of the governing partial differential equation at n selected points in parameter space ii the off line on line computational procedures decoupling the generation and projection stages of the approximation process the operation count for the on line stage in which given a new parameter value we calculate an output of interest depends only on n typically very small and the parametric complexity of the problem the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation design optimization and real time control particular attention is given i to the pressure treatment of incompressible stokes problem ii to find an equivalent inf sup condition that guarantees stability of reduced basis solutions by enriching the reduced basis velocity approximation space with the solutions of a supremizer problem iii to provide algebraic stability of the problem by reducing the condition number of reduced basis matrices using an orthonormalization procedure applied to basis functions iv to reduce computational costs in order to allow real time solution of parametrized problem

on the large time behavior of solutions of hamilton jacobi equations associated with nonlinear boundary conditions in this article we study the large time behavior of solutions of first order hamilton jacobi equations set in a bounded domain with nonlinear neumann boundary conditions including the case of dynamical boundary conditions we establish general convergence results for viscosity solutions of these cauchy neumann problems by using two fairly different methods the first one relies only on partial differential equations methods which provides results even when the hamiltonians are not convex and the second one is an optimal control dynamical system approach named the weak kam approach which requires the convexity of hamiltonians and gives formulas for asymptotic solutions based on aubry mather sets

natural norm a posteriori error estimators for reduced basis approximations we present a technique for the rapid and reliable prediction of linear functional outputs of coercive and noncoercive linear elliptic partial differential equations with affine parameter dependence the essential components are i rapidly convergent global reduced basis approximations galerkin projection onto a space wn spanned by solutions of the governing partial differential equation at n judiciously selected points in parameter space ii a posteriori error estimation relaxations of the error residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest and iii offiine online computational procedures methods which decouple the generation and projection stages of the approximation process the operation count for the online stage in which given a new parameter value we calculate the output of interest and associated error bound depends only on n typically very small and the parametric complexity of the problem in this paper we propose a new natural norm formulation for our reduced basis error estimation framework that a greatly simplifies and improves our inf sup lower bound construction offline and evaluation online a critical ingredient of our a posteriori error estimators and b much better controls significantly sharpens our output error bounds in particular through deflation for parameter values corresponding to nearly singular solution behavior we apply the method to two illustrative problems a coercive laplacian heat conduction problem which becomes singular as the heat transfer coefficient tends to zero and a non coercive helmholtz acoustics problem which becomes singular as we approach resonance in both cases we observe very economical and sharp construction of the requisite natural norm inf sup lower bound rapid convergence of the reduced basis approximation reasonable effectivities even for near singular behavior for our deflated output error estimators and significant several order of magnitude online computational savings relative to standard finite element procedures

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

error estimates and automatic time step control for nonlinear parabolic problems i in this note we extend results by one of the authors on time discretization error estimates and related automatic time step control for stiff ordinary differential equations to the case of a nonlinear parabolic problem the method for time discretization is the so called discontinuous galerkin method based on using piecewise polynomials of degree q geqq 0 we consider in this note the case q 0 corresponding to a variant of the backward euler method we prove a new almost optimal error estimate and present a related new algorithm for automatic time step control this algorithm is very simple but yet is efficient and gives control of the global error

nonlinear neumann boundary conditions for quasilinear degenerate elliptic equations and applications abstract we prove comparison results between viscosity sub and supersolutions of degenerate elliptic and parabolic equations associated to possibly nonlinear neumann boundary conditions these results are obtained under more general assumptions on the equation in particular the dependence in the gradient of the solution and they allow applications to quasilinear possibly singular elliptic or parabolic equations one of the main applications is the extension of the so called level set approach for equations set in bounded domains with nonlinear neumann boundary conditions in such a framework the level set approach provides a weak notion for the motion of hypersurfaces with curvature dependent velocities and a prescribed contact angle at the boundary

space time adaptive wavelet methods for parabolic evolution problems with respect to space time tensor product wavelet bases parabolic initial boundary value problems are equivalently formulated as bi infinite matrix problems adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate in case the spatial domain is of product type the use of spatial tensor product wavelet bases is proved to overcome the so called curse of dimensionality i e the reduction of the convergence rate with increasing spatial dimension

large time behavior of solutions to parabolic equations with neumann boundary conditions in this paper we are interested in the large time behavior as t of the viscosity solutions of parabolic equations with nonlinear neumann type boundary conditions in connection with ergodic boundary problems which have been recently studied by barles and the author in g barles f da lio on the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear neumann boundary conditions ann inst h poincare anal non lineaire 22 5 2005 521 541

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

multiplicity results for a superlinear dirichlet problem on considere le probleme de dirichlet v f v v t0 1 ou t r v x max v x o et b b 1 o est une boule ouverte unite dans r n n 2 ici 0 1 est une valeur propre positive correspondant a la premiere valeur propre 1 du probleme r v v n 1 r v v ou 0r1 v 0 v 1 0

large time asymptotics for one dimensional dirichlet problems for hamilton jacobi equations with noncoercive hamiltonians we study large time asymptotics for a class of noncoercive hamilton jacobi equations with dirichlet boundary condition in one space dimension we prove that the average growth rate of a solution is constant only in a subset of the whole domain and give the asymptotic profile in the subset we show that the large time behavior for noncoercive problems may depend on the space variable in general which is different from the usual results under the coercivity condition this work is an extension with more rigorous analysis of a recent paper by e yokoyama y giga and p rybka in which a growing crystal model is established and the asymptotic behavior described above is first discovered

fully nonlinear neumann type boundary conditions for first order hamilton jacobi equations we consider problems of the following type h x u u 0 in where is a smooth bounded open set in r n u is a scalar function from into r u denotes its frechet differential at x and h is a given continuous function on xr r n this equation is nothing but a general form of the classical first order hamilton jacobi equation in order to investigate existence and uniqueness questions the equation has to be complemented with a boundary condition on or on a part of we consider here for reasons described below a fully nonlinear boundary condition of the form f x u u 0 on that we describe as being of the neumann type

an improved error bound for reduced basis approximation of linear parabolic problems we consider a space time variational formulation for linear para bolic partial dierential equations we introduce an associated petrov galerkin truth nite element discretization with favorable discrete inf sup constant the inverse of which enters into error estimates is unity for the heat equa tion decreases only linearly in time for non coercive but asymptotically stable convection operators the latter in turn permits eective long time a posteriori error bounds for reduced basis approximations in sharp contrast to classical pessimistic exponentially growing energy estimates the paper contains a full analysis and various extensions for the formulation introduced briey in 13 as well as numerical results for a model reaction convection diusion equation

an efficient output error estimation for model order reduction of parametrized evolution equations in this work we present an a posteriori output error bound for model order reduction of parametrized evolution equations with the help of the dual system and a simple representation of the relationship between the field variable error and the residual of the primal system a sharp output error bound is derived such an error bound successfully avoids the accumulation of the residual over time which is a common drawback in the existing error estimations for time stepping schemes an estimation needs to be performed for practical computation of the error bound and as a result the output error bound reduces to an output error estimation the proposed error estimation is applied to four kinds of problems the first one is a linear convection diffusion equation which is used to compare the performance of the new error estimation and an existing primal dual error bound the second one is the unsteady viscous burgers equation an academic benchmark of nonlinear evolution equations in fluid dynamics often u

multiplicity results for ode s with nonlinearities crossing all but a finite number of eigenvalues on considere des equations differentielles du type suivant u x u x p u x te 1 x x 0 u 0 u 0 ou p 1 u max u 0 r et e 1 x sinx est la fonction propre correspondant a 1 ou 0 1 2 3 k denotent les valeurs propres de l operateur d 2 dx 2 a conditions aux limites nulles

a robust error estimator and a residual free error indicator for reduced basis methods abstract the reduced basis method rbm is a rigorous model reduction approach for solving parameterizedpartial differential equations it identifies a low dimensional subspace for approximation of the parametric solution manifold that is embedded in high dimensional space a reduced order model is subsequently constructed in this subspace rbm relies on residual based error indicators or a posteriori error bounds to guide construction of the reduced solution subspace to serve as a stopping criteria and to certify the resulting surrogate solutions unfortunately it is well known that the standard algorithm for residual norm computation suffers from premature stagnation at the level of the square root of machine precision in this paper we develop two alternatives to the standard offline phase of reduced basis algorithms first we design a robust strategy for computation of residual error indicators that allows rbm algorithms to enrich the solution subspace with accuracy beyond root machine precision secondly we propose a new error indicator based on the lebesgue function in interpolation theory this error indicator does not require computation of residual norms and instead only requires the ability to compute the rbm solution this residual free indicator is rigorous in that it bounds the error committed by the rbm approximation but up to an uncomputable multiplicative constant because of this the residual free indicator is effective in choosing snapshots during the offline rbm phase but cannot currently be used to certify error that the approximation commits however it circumvents the need for a posteriori analysis of numerical methods and therefore can be effective on problems where such a rigorous estimate is hard to derive

large time behavior of solutions of hamilton jacobi equations with periodic boundary data abstract we study the large time behavior of viscosity solutions of hamilton jacobi equations with periodic boundary data on bounded domains we establish a result on convergence of viscosity solutions to state constraint asymptotic solutions or periodic asymptotic solutions depending on the sign of critical value as time goes to infinity

neumann type boundary conditions for hamilton jacobi equations abstract in this paper we present a notion of viscosity solutions of hamilton jacobi equations for neumann type boundary conditions or more generally oblique derivative in particular we prove the existence uniqueness stability of such solutions and we show that the vanishing viscosity method yields such solutions next we check that value functions of control problems or differential games problems for reflected dynamical processes are solutions in that sense of the associated bellman or isaacs equations finally we consider the ergodic problems

certified reduced basis vms smagorinsky model for natural convection flow in a cavity with variable height abstract in this work we present a reduced basis vms smagorinsky boussinesq model applied to natural convection problems in a variable height cavity in which the buoyancy forces are involved we take into account in this problem both physical and geometrical parametrizations considering the rayleigh number as a parameter so as the height of the cavity we perform an empirical interpolation method to approximate the sub grid eddy viscosity term that lets us obtain an affine decomposition with respect to the parameters we construct an a posteriori error estimator based upon the brezzi rappaz raviart theory used in the greedy algorithm for the selection of the basis functions finally we present several numerical tests for different parameter configuration

a stabilized pod model for turbulent flows over a range of reynolds numbers optimal parameter sampling and constrained projection abstract we present a reduced basis technique for long time integration of parametrized incompressible turbulent flows the new contributions are threefold first we propose a constrained galerkin formulation that corrects the standard galerkin statement by incorporating prior information about the long time attractor for explicit and semi implicit time discretizations our statement reads as a constrained quadratic programming problem where the objective function is the euclidean norm of the error in the reduced galerkin algebraic formulation while the constraints correspond to bounds for the maximum and minimum value of the coefficients of the n term expansion second we propose an a posteriori error indicator which corresponds to the dual norm of the residual associated with the time averaged momentum equation we demonstrate that the error indicator is highly correlated with the error in mean flow prediction and can be efficiently computed through an offline online strategy third we propose a greedy algorithm for the construction of an approximation space procedure valid over a range of parameters the greedy is informed by the a posteriori error indicator developed in this paper we illustrate our approach and we demonstrate its effectiveness by studying the dependence of a two dimensional turbulent lid driven cavity flow on the reynolds number

a space time variational approach to hydrodynamic stability theory we present a hydrodynamic stability theory for incompressible viscous fluid flows based on a space time variational formulation and associated generalized singular value decomposition of the linearized navier stokes equations we first introduce a linear framework applicable to a wide variety of stationary or time dependent base flows we consider arbitrary disturbances in both the initial condition and the dynamics measured in a data space time norm the theory provides a rigorous sharp realizable and eciently computed bound for the velocity perturbation measured in a solution spacetime norm we next present a generalization of the linear framework in which the disturbances and perturbation are now measured in respective selected space time seminorms the semi norm theory permits rigorous and sharp quantification of for example the growth of initial disturbances or functional outputs we then develop a brezzirappaz raviart nonlinear theory which provides for disturbances which satisfy a certain rather stringent amplitude condition rigorous finite amplitude bounds for the velocity and output perturbations finally we demonstrate the application of our linear and nonlinear hydrodynamic stability theory to unsteady moderate reynolds number flow in an eddy promoter channel

a microscopic time scale approximation to the behavior of the local slope on the faceted surface under a nonuniformity in supersaturation abstract the morphological stability of a growing faceted crystal is discussed we argue that the interplay between nonuniformity in supersaturation on a growing facet and anisotropy of surface kinetics derived from the lateral motion of steps leads to a faceted instability qualitatively speaking as long as the nonuniformity in supersaturation on the facet is not too large it can be compensated for by a variation of step density along the facet and the faceted crystal can grow in a stable manner the problem can be modeled as a hamilton jacobi equation for height of the crystal surface the notion of a maximal stable region of a growing facet is introduced for microscopic time scale approximation of the original hamilton jacobi equation it is shown that the maximal stable region keeps its shape determined by profile of the surface supersaturation with constant growth rate by studying large time behavior of solution of macroscopic time scale approximation a quantitative criterion for the facet stability is given

long time behavior of solutions of hamilton jacobi equations with convex and coercive hamiltonians we investigate the long time behavior of viscosity solutions of hamilton jacobi equations in mathbb r n with convex and coercive hamiltonians and give three general criteria for the convergence of solutions to asymptotic solutions as time goes to infinity we apply the criteria to obtain more specific sufficient conditions for the convergence to asymptotic solutions and then examine them with examples we take a dynamical approach based on tools from weak kam theory such as extremal curves aubry sets and representation formulas for solutions for these investigations

reduced order finite difference extrapolation model based on proper orthogonal decomposition for two dimensional shallow water equations including sediment concentration abstract in this study we employ a proper orthogonal decomposition pod method to establish a pod based reduced order finite difference fd extrapolating model with very few degrees of freedom for two dimensional shallow water equations that include the sediment concentration we provide estimates of the error between the accurate solution and classical fd solutions as well as those between the accurate solution and the pod based reduced order fd solutions moreover we present two numerical simulation experiments to demonstrate that the pod based reduced order fd extrapolating model can greatly reduce the computational load thus we validate both the feasibility and efficiency of the pod based reduced order fd extrapolating model

error bounds for pod expansions of parameterized transient temperatures abstract we focus on the convergence analysis of the pod expansion for the parameterized solution of transient heat equations the parameter of interest is the conductivity coefficient we prove that this expansion converges with exponential accuracy uniformly if the conductivity coefficient remains within a compact set of positive numbers this convergence result is independent of the regularity of the temperature with respect to the space and time variables we present some numerical experiments to show that a reduced number of modes allows to represent with high accuracy the family of solutions corresponding to parameters that lie in the compact set under study