normalisation holomorphe d algebres de type cartan de champs de vecteurs holomorphes singuliers nous montrons un resultat de normalisation holomorphe d une famille commutative de champs de vecteurs holomorphes au voisinage de leur point singulier commun pour ce faire nous supposons qu une condition diophanti enne portant sur une algebre de lie commutative de champs lineaires associee est satisfaite d autre part nous imposons certaines conditions algebriques sur leur forme normale formelle les champs d une telle famille sauf un peuvent ne pas avoir de partie lineaire a l origine abstract we consider a commutative family of holomorphic vector fields in a neigh bourhood of a common singular point say 0 e cn let g be a commuta tive complex lie algebra of dimension 1 let a1 an e and let us set s g l ai g xi we assume that this lie morphism is diophantine in some sense let x1 be a holomorphic vector fields in a neighbourhood of 0 e cn we assume that its linear part s is regular relative to s that it belongs to s g and has the same formal centralizer as s let x2 xi be holomorphic vector fields vanishing at 0 and commuting with x1 then there exists a formal diffeomorphism of cn o such that the family of vector fields are in normal form in these formal coordinates this means that each element of the family commutes with s we assume that the normal forms of the xi s belong to 0s 0 s g os is the ring of formal first integral of s we also assume that the family of lowest order parts of the normal forms is free over 9s then we show that there exists a holomorphic diffeomorphism of cn 0 which transforms the family into a normal form the elements of the family

an optimization based approach for the design of pde solution algorithms we develop and analyze an optimization based approach for the robust and efficient solution of pde problems consisting of multiple physics operators with fundamentally different mathematical properties our approach relies on three essential steps decomposition of the original problem into subproblems for which robust solution algorithms are available integration of the subproblems into an equivalent pde constrained optimization problem and solution of the resulting optimization problem either directly as a fully coupled algebraic system or in the null space of the pde constraints this strategy gives rise to a general approach for synthesizing robust solvers for complex coupled problems from solvers for their simpler physics components

integrable almost symplectic hamiltonian systems we extend the notion of liouville integrability which is peculiar to hamiltonian systems on symplectic manifolds to hamiltonian systems on almost symplectic manifolds namely manifolds equipped with a nondegenerate but not closed 2 form the key ingredient is to require that the hamiltonian vector fields of the integrals of motion in involution or equivalently the generators of the invariant tori are symmetries of the almost symplectic form we show that under this hypothesis essentially all of the structure of the symplectic case from quasiperiodicity of motions to an analog of the action angle coordinates and of the isotropic coisotropic dual pair structure characteristic of the fibration by the invariant tori carries over to the almost symplectic case

birkhoff coordinates for the focusing nls equation in this paper we construct birkhoff coordinates for the focusing nonlinear schrodinger equation near the zero solution

optimization based remap and transport a divide and conquer strategy for feature preserving discretizations this paper examines the application of optimization and control ideas to the formulation of feature preserving numerical methods with particular emphasis on the conservative and bound preserving remap constrained interpolation and transport advection of a single scalar quantity we present a general optimization framework for the preservation of physical properties and specialize it to a generic optimization based remap obr of mass density the latter casts remap as a quadratic program whose optimal solution minimizes the distance to a suitable target quantity subject to a system of linear inequality constraints the approximation of an exact mass update operator defines the target quantity which provides the best possible accuracy of the new masses without regard to any physical constraints such as conservation of mass or local bounds the latter are enforced by the system of linear inequalities in so doing the generic obr formulation separates accuracy considerations from the enforcement of physical properties we proceed to show how the generic obr formulation yields the recently introduced flux variable flux target fvft 1 and mass variable mass target mvmt 2 formulations of remap and then follow with a formal examination of their relationship using an intermediate flux variable mass target fvmt formulation we show the equivalence of fvft and mvmt optimal solutions to underscore the scope and the versatility of the generic obr formulation we introduce the notion of adaptable targets i e target quantities that reflect local solution properties extend fvft and mvmt to remap on the sphere and use obr to formulate adaptable conservative and bound preserving optimization based transport algorithms for cartesian and latitude longitude coordinate systems a selection of representative numerical examples on two dimensional grids demonstrates the computational properties of our approach

an optimization based atomistic to continuum coupling method we present a new optimization based method for atomistic to continuum atc coupling the main idea is to cast the latter as a constrained optimization problem with virtual dirichlet controls on the interfaces between the atomistic and continuum subdomains the optimization objective is to minimize the error between the atomistic and continuum solutions on the overlap between the two subdomains while the atomistic and continuum force balance equations provide the constraints separation rather then blending of the atomistic and continuum problems and their subsequent use as constraints in the optimization problem distinguishes our approach from the existing atc formulations we present and analyze the method in the context of a one dimensional chain of atoms modeled using a linearized two body potential with next nearest neighbor interactions

composite finite elements for the approximation of pdes on domains with complicated micro structures usually the minimal dimension of a finite element space is closely related to the geometry of the physical object of interest this means that sometimes the resolution of small micro structures in the domain requires an inadequately fine finite element grid from the viewpoint of the desired accuracy this fact limits also the application of multi grid methods to practical situations because the condition that the coarsest grid should resolve the physical object often leads to a huge number of unknowns on the coarsest level we present here a strategy for coarsening finite element spaces independently of the shape of the object this technique can be used to resolve complicated domains with only few degrees of freedom and to apply multi grid methods efficiently to pdes on domains with complex boundary in this paper we will prove the approximation property of these generalized fe spaces

hp version composite discontinuous galerkin methods for elliptic problems on complicated domains in this paper we introduce the hp version discontinuous galerkin composite finite element method for the discretization of second order elliptic partial differential equations this class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features or microstructures while standard numerical methods can be devised for such problems the computational effort may be extremely high as the minimal number of elements needed to represent the underlying domain can be very large in contrast the minimal dimension of the underlying composite finite element space is independent of the number of geometric features the key idea in the construction of this latter class of methods is that the computational domain omega is no longer resolved by the mesh instead the finite element basis or shape functions are adapted to the geometric details present in omega in this paper we extend these ideas to the discontinuous galer

two scale composite finite element method for dirichlet problems on complicated domains in this paper we define a new class of finite elements for the discretization of problems with dirichlet boundary conditions in contrast to standard finite elements the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro structures for the proposed finite element method we prove the optimal order approximation up to logarithmic terms and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re entering corners of the domain boundary numerical experiments confirm the theoretical results and show the potential of our proposed method

an hp adaptive mixed discontinuous galerkin fem for nearly incompressible linear elasticity we develop the a posteriori error estimation of mixed hp version discontinuous galerkin finite element methods for nearly incompressible elasticity problems in two space dimensions computable upper and lower bounds on the error measured in terms of a natural mesh dependent energy norm are derived the bounds are explicit in the local mesh sizes and approximation orders and are independent of the locking parameter a series of numerical experiments are presented which demonstrate the performance of the proposed error estimator within an automatic hp adaptive refinement procedure

a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

a conceptual approach to the problem of action angle variables in this paper we develop a general conceptual approach to the problem of existence of action angle variables for dynamical systems which establishes and uses the fundamental conservation property of associated torus actions anything which is preserved by the system is also preserved by the associated torus actions this approach allows us to obtain among other things a the shortest and most easy to understand conceptual proof of the classical arnold liouville mineur theorem b basically all known results in the literature about the existence of action angle variables in various contexts can be recovered in a unifying way with simple proofs using our approach c new results on action angle variables in many different contexts including systems on contact manifolds systems on presymplectic and dirac manifolds action angle variables near singularities stochastic systems and so on even when there are no natural action variables our approach still leads to useful normal forms for dynamical systems which are not necessarily integrable

variational multiscale element free galerkin vmefg and local discontinuous galerkin ldg methods for solving two dimensional brusselator reaction diffusion system with and without cross diffusion abstract the finite element method fem is one of the basic methods for solving deterministic and stochastic partial differential equations this method is proposed in the 19 decade and after years several modifications for this well known technique such as mortal fem discontinuous galerkin fem extended fem least squares fem spectral fem mixed fem immersed fem adaptive fem etc have been proposed also for improving the accuracy and for considering complex domain shape functions of the moving least squares approximation have been replaced with the traditional shape function of the finite element technique the name of this improved method is element free galerkin efg method which is classified in the category of meshless methods the efg method has been applied for solving many problems in engineering in the current paper we select two numerical procedures that are extracted from fem the discontinuous galerkin approach is a useful technique for solving problems that their exact solutions have discontinuity it should be noted that many enriched ideas have been explained for improving accuracy of efg method in the current paper the local discontinuous galerkin and variational multiscale efg methods are applied for obtaining numerical solution of two dimensional brusselator system with and without cross diffusion the brusselator system is a theoretical model for a type of autocatalytic reaction up to best of authors knowledge the accuracy of the obtained numerical solutions using local discontinuous galerkin method is very related to the used time discrete scheme therefore fourth order exponential time differencing method has been employed for discretizing the time variable also to achieve a full discretization scheme the local discontinuous galerkin finite element method is used moreover several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes

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 note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

multigrid methods for pde optimization research on multigrid methods for optimization problems is reviewed optimization problems considered include shape design parameter optimization and optimal control problems governed by partial differential equations of elliptic parabolic and hyperbolic type

extended integrability and bi hamiltonian systems the current notion of integrability of hamiltonian systems was fixed by liouville in a famous 1855 paper it describes systems in a 2k dimensional phase space whose trajectories are dense on tori q or wind on toroidal cylinders m q m within liouville s construction the dimension q cannot exceed k and is the main invariant of the system in this paper we generalize liouville integrability so that trajectories can be dense on tori k of arbitrary dimensions q 1 2k 1 2k and an additional invariant v 2 q k v 2 q 2 can be recovered the main theorem classifies all k k 1 2 canonical forms of hamiltonian systems that are integrable in a newly defined broad sense an integrable physical problem having engineering origin is presented the notion of extended compatibility of two poisson structures is introduced the corresponding bi hamiltonian systems are shown to be integrable in the broad sense

hp adaptive composite discontinuous galerkin methods for elliptic eigenvalue problems on complicated domains in this paper we develop the a posteriori error estimation of hp adaptive discontinuous galerkin composite finite element methods dgfems for the discretization of second order elliptic eigenvalue problems dgfems allow for the approximation of problems posed on computational domains which may contain local geometric features the dimension of the composite finite element space is independent of the number of geometric features this is in contrast with standard finite element methods as the minimal number of elements needed to represent the underlying domain can be very large and so the dimension of the finite element space computable upper bounds on the error for both eigenvalues and eigenfunctions are derived numerical experiments highlighting the practical application of the proposed estimators within an automatic hp adaptive refinement procedure will be presented

convergence versus integrability in birkhoff normal form we show that any analytically integrable hamiltonian system near an equilibrium point admits a convergent birkhoff normalization the proof is based on a new geometric approach to the topic

multiscale analysis of heterogeneous domain decomposition methods for time dependent advection reaction diffusion problems abstract domain decomposition methods which use different models in different subdomains are called heterogeneous domain decomposition methods we are interested here in the case where there is an accurate but expensive model one should use in the entire domain but for computational savings we want to use a cheaper model in parts of the domain where expensive features of the accurate model can be neglected for the model problem of a time dependent advection reaction diffusion equation in one spatial dimension we study approximate solutions of three different heterogeneous domain decomposition methods with pure advection reaction approximation in parts of the domain using for the first time a multiscale analysis to compare the approximate solutions to the solution of the accurate expensive model in the entire domain we show that a recent heterogeneous domain decomposition method based on factorization of the underlying differential operator has better approximation properties than more classical variational or non variational heterogeneous domain decomposition methods we show with numerical experiments in two spatial dimensions that the performance of the algorithms we study is well predicted by our one dimensional multiscale analysis and that our theoretical results can serve as a guideline to compare the expected accuracy of heterogeneous domain decomposition methods already for moderate values of the viscosity

inexact preconditioned conjugate gradient method with inner outer iteration an important variation of preconditioned conjugate gradient algorithms is inexact preconditioner implemented with inner outer iterations g h golub and m l overton numerical analysis lecture notes in math 912 springer berlin new york 1982 where the preconditioner is solved by an inner iteration to a prescribed precision in this paper we formulate an inexact preconditioned conjugate gradient algorithm for a symmetric positive definite system and analyze its convergence property we establish a linear convergence result using a local relation of residual norms we also analyze the algorithm using a global equation and show that the algorithm may have the superlinear convergence property when the inner iteration is solved to high accuracy the analysis is in agreement with observed numerical behavior of the algorithm in particular it suggests a heuristic choice of the stopping threshold for the inner iteration numerical examples are given to show the effectiveness of this choice and to compare the convergence bound

boundary velocity control of incompressible flow with an application to viscous drag reduction an optimal boundary control problem for the navier stokes equations is presented the control is the velocity on the boundary which is constrained to lie in a closed convex subset of h 1 2 of the boundary a necessary condition for optimality is derived computations are done when the control set is actually finite dimensional resulting in an application to viscous drag reduction

optimization based additive decomposition of weakly coercive problems with applications we present an abstract mathematical framework for an optimization based additive decomposition of a large class of variational problems into a collection of concurrent subproblems the framework replaces a given monolithic problem by an equivalent constrained optimization formulation in which the subproblems define the optimization constraints and the objective is to minimize the mismatch between their solutions the significance of this reformulation stems from the fact that one can solve the resulting optimality system by an iterative process involving only solutions of the subproblems consequently assuming that stable numerical methods and efficient solvers are available for every subproblem our reformulation leads to robust and efficient numerical algorithms for a given monolithic problem by breaking it into subproblems that can be handled more easily an application of the framework to the oseen equations illustrates its potential

a new petrov galerkin smoothed aggregation preconditioner for nonsymmetric linear systems we propose a new variant of smoothed aggregation sa suitable for nonsymmetric linear systems the new algorithm is based on two key generalizations of sa restriction smoothing and local damping restriction smoothing refers to the smoothing of a tentative restriction operator via a damped jacobi like iteration restriction smoothing is analogous to prolongator smoothing in standard sa and in fact has the same form as the transpose of prolongator smoothing when the matrix is symmetric local damping refers to damping parameters used in the jacobi like iteration in standard sa a single damping parameter is computed via an eigenvalue computation here local damping parameters are computed by considering the minimization of an energy like quantity for each individual grid transfer basis function numerical results are given showing how this method performs on highly nonsymmetric systems

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

adaptive finite element methods for computing band gaps in photonic crystals in this paper we propose and analyse adaptive finite element methods for computing the band structure of 2d periodic photonic crystals the problem can be reduced to the computation of the discrete spectra of each member of a family of periodic hermitian eigenvalue problems on a unit cell parametrised by a two dimensional parameter the quasimomentum these eigenvalue problems involve non coercive elliptic operators with generally discontinuous coefficients and are solved by adaptive finite elements we propose an error estimator of residual type and show it is reliable and efficient for each eigenvalue problem in the family in particular we prove that if the error estimator converges to zero then the distance of the computed eigenfunction from the true eigenspace also converges to zero and the computed eigenvalue converges to a true eigenvalue with double the rate we also prove that if the distance of a computed sequence of approximate eigenfunctions from the true eigenspace approaches zero then so must the error estimator the results hold for eigenvalues of any multiplicity we illustrate the benefits of the resulting adaptive method in practice both for fully periodic structures and also for the computation of eigenvalues in the band gap of structures with defect using the supercell method

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

analysis of a two scale locally conservative subgrid upscaling for elliptic problems we present a two scale theoretical framework for approximating the solution of a second order elliptic problem the elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid but limits on computational power require that computations be performed on a coarse grid we consider the elliptic problem in mixed variational form over w times bf v subset l 2 times h rm div we base our scale expansion on local mass conservation over the coarse grid it is used to define a direct sum decomposition of w times bf v into coarse and subgrid subspaces w c times bf v c and delta w times delta bf v such that 1 nabla cdot bf v c w c and nabla cdot delta bf v delta w and 2 the space delta bf v is locally supported over the coarse mesh we then explicitly decompose the variational problem into coarse and subgrid scale problems the subgrid problem gives a well defined operator taking w c times bf v c to delta w times delta bf v which is localized in space and it is used to upscale that is to remove the subgrid from the coarse scale problem using standard mixed finite element spaces two scale mixed spaces are defined a mixed approximation is defined which can be viewed as a type of variational multiscale method or a residual free bubble technique a numerical green s function approach is used to make the approximation to the subgrid operator efficient to compute a mixed method pi operator is defined for the two scale approximation spaces and used to show optimal order error estimates

an optimization based domain decomposition method for the navier stokes equations an optimization based nonoverlapping domain decomposition method for the solution of the navier stokes equations is presented the crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains the constraints are the navier stokes equations in the subdomains with suitably chosen boundary conditions along the interfaces we show that solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined finite element approximations of the solutions of the optimality system are examined the domain decomposition method is also reformulated as a nonlinear least squares problem and the results of some numerical experiments are given

a gradient method approach to optimization based multidisciplinary simulations and nonoverlapping domain decomposition algorithms it has been shown recently that optimization based nonoverlapping domain decomposition algorithms are connected to many well known algorithms using a gradient type iterative strategy for the optimization problem we present further discussion on how to develop various algorithms that can integrate subdomain solvers into a solver for the problem in the whole domain in particular the algorithms we discuss can be used to develop efficient solvers of multidisciplinary problems which are constructed using existing subdomain solvers without the need for making changes in the latter

flexible conjugate gradients we analyze the conjugate gradient cg method with preconditioning slightly variable from one iteration to the next to maintain the optimal convergence properties we consider a variant proposed by axelsson that performs an explicit orthogonalization of the search directions vectors for this method which we refer to as flexible cg we develop a theoretical analysis that shows that the convergence rate is essentially independent of the variations in the preconditioner as long as the latter are kept sufficiently small we further discuss the real convergence rate on the basis of some heuristic arguments supported by numerical experiments depending on the eigenvalue distribution corresponding to the fixed reference preconditioner several situations have to be distinguished in some cases the convergence is as fast with truncated versions of the algorithm or even with the standard cg method whereas quite large variations are allowed without too much penalty in other cases the flexible variant effectively outperforms the standard method while the need for truncation limits the size of the variations that can be reasonably allowed

an hp adaptive strategy based on continuous sobolev embeddings a new hp adaptive approach for finite element methods is presented specifically the paper addresses the well known question of how to effectively decide between h and p refinement within an adaptive hp finite element procedure the main idea is based on testing the smoothness of the underlying pde solution by means of suitable continuous sobolev embeddings numerical experiments confirm the effectiveness of the proposed method

le probleme general des variables actions angles feuilletages symplectiquement complets monodromie des fibrations isotropes symplectiquement completes structure locale et theoreme des variables action angles classe de chern realisation d une variete de poisson au voisinage d une feuille compacte