an inf sup stable and residual free bubble element for the oseen equations we investigate the residual free bubble method for the linearized incompressible navier stokes equations starting with a nonconforming inf sup stable element pair for approximating the velocity and pressure we enrich the velocity space by discretely divergence free bubble functions to handle the influence of strong convection an important feature of the method is that the stabilization does not generate an additional coupling between the mass equation and the momentum equation as is the case for the streamline upwind petrov galerkin method applied to equal order interpolation furthermore the discrete solution is piecewise divergence free a property which is useful for the mass balance in transport equations coupled with the incompressible navier stokes equations

bubble stabilization of linear finite element methods for nonlinear evolutionary convection diffusion equations abstract a spatial stabilization via bubble functions of linear finite element methods for nonlinear evolutionary convection diffusion equations is discussed the method of lines with supg discretization in space leads to numerical schemes that are not only difficult to implement when considering nonlinear evolutionary equations but also do not produce satisfactory results the method we propose can be seen as an alternative to this kind of methods once the numerical approximation belonging to a linear finite element space enriched with bubble functions is obtained the bubble part is discarded the linear part is shown to give a stabilized approximation to the solution being approached the bubble functions are deduced using a linear steady convection diffusion model in such a way that the linear part of the approximation to the linear steady model after static condensation of bubbles reproduces the supg method in the convection dominated regime however for the nonlinear evolutionary equations we consider in the paper the method we propose is not equivalent to the supg method some numerical experiments are provided in the paper to show the efficiency of the procedure

bubble stabilization of finite element methods for the linearized incompressible navier stokes equations abstract in this paper we discuss the stabilization via bubble functions of a finite element method for stationary linearized incompressible navier stokes equations it is shown that residual free bubbles can reproduce supg

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

modelling friction near sharp edges using a eulerian reference frame application to aluminium extrusion when using a eulerian finite element approach to model the material deformation that occurs in e g forming processes the accurate capturing of friction is of crucial importance to the quality of the computational results for the algorithm that incorporates the frictional phenomena into the system of equations the direction of the contact surface normal in a node is an essential parameter however this normal is not uniquely defined in the nodes of a curved discretized surface therefore a substitute normal has to be reconstructed the commonly used averaging methods to determine the normal are either mesh or geometry dependent which renders poor results on coarse meshes therefore a new method is presented that reconstructs the direction of the normal from the flow field near the node comparing the flow fields on a coarse mesh with those obtained on a very fine mesh reveals that a more accurate solution field is obtained using the method introduced here

epigraphical and uniform convergence of convex functions we examine when a sequence of lsc convex functions on a banach space converges uniformly on bounded sets resp compact sets provided it converges attouch wets resp painlev e kuratowski we also obtain related results for pointwise convergence and uniform convergence on weakly compact sets some known results concerning the convergence of sequences of linear functionals are shown to also hold for lsc convex functions for example a sequence of lsc convex functions converges uniformly on bounded sets to a continuous affine function provided that the convergence is uniform on weakly compact sets and the space does not contain an isomorphic copy of ell 1

graded mesh approximation in weighted sobolev spaces and elliptic equations in 2d we study the approximation properties of some general finite element spaces constructed using improved graded meshes in our results either the approximating function or the function to be approximated or both are in a weighted sobolev space we consider also the l p version of these spaces the finite element spaces that we define are obtained from em conformally invariant families of finite elements no affine invariance is used stressing the use of elements that lead to higher regularity finite element spaces we prove that for a suitable grading of the meshes one obtains the usual optimal approximation results we provide a construction of these spaces that does not lead to long skinny triangles our results are then used to obtain l 2 error estimates and h m quasi optimal rates of convergence for the fem approximation of solutions of strongly elliptic interface boundary value problems

two robust nonconforming hbox h 2 elements for linear strain gradient elasticity we propose two finite elements to approximate a boundary value problem arising from strain gradient elasticity which is a high order perturbation of the linearized elastic system our elements are hbox h 2 nonconforming while hbox h 1 conforming we show both elements converge in the energy norm uniformly with respect to the perturbation parameter

convex optimization and the epi distance topology let 17 x denote the proper lower semicontinuous convex functions on a banach space x equipped with the completely metrizable topology t of uniform convergence of distance functions on bounded sets a function f in 17 x is called well posed provided it has a unique minimizer and each minimizing sequence converges to this minimizer we show that well posedness of f e r x is the minimal condition that guarantees strong convergence of approximate minima of t approximating functions to the minimum of f moreover we show that most functions in 17 x taw are well posed and that this fails if 17 x is topologized by the weaker topology of mosco convergence whenever x is infinite dimensional applications to metric projections are also given including a fundamental characterization of approximative compactness

mosco and slice convergence of level sets and graphs of linear functionals given that most results of interest for mosco convergence seem to require reflexivity see especially bb our results here suggest that although mosco convergence is well known and widely applied in the convex analysis community its essence is poorly understood in fact it is our view that mosco convergence in its historical formulation as well as the topology of mosco convergence introduced in bel is a misdefined concept

set convergences an attempt of classification we endow families of nonempty closed subsets of a metric space with uniformities defined by semimetrics such structure is completely deter mined by a class which is a family of closed sets and a type which is a semi metric two types are sufficient to define and classify almost all convergences known till now these two types offer the possibility of defining other set con vergences

some current cfd issues relevant to the incompressible navier stokes equations abstract the goals of this paper are to carefully define a particular class of well set incompressible navier stokes problems in the continuum partial differential equation pde setting and to discuss some relevant and sometimes poorly understood issues related to these well posed pde problems both in the continuum world and in its computer counterpart emphasized are three common formulations used to generate computer codes the primitive variable velocity pressure formulation and two derived variable formulations pressure poisson equation and vorticity transport equation

on a two level finite element method for the incompressible navier stokes equations we consider the galerkin finite element method for the incompressible navier stokes equations in two dimensions where the finite dimensional space s employed consist of piecewise polynomials enriched with residual free bubble rfb functions to find the bubble part of the solution a two level finite element method tlfem is described and its application to the navier stokes equation is displayed numerical solutions employing the tlfem are presented for three benchmark problems we compare the numerical solutions using the tlfem with the numerical solutions using a stabilized method

a mathcal c 0 interior penalty method for a fourth order elliptic singular perturbation problem in this paper we develop a mathcal c 0 interior penalty method for a fourth order singular perturbation elliptic problem in two dimensions on polygonal domains using some a posteriori error analysis techniques we are able to show that the method converges in the energy norm uniformly with respect to the perturbation parameter under minimal regularity assumptions in addition we analyze the convergence of the numerical solution to the unperturbed second order problem finally we perform some numerical experiments that back up the theoretical results

matched interface and boundary mib method for elliptic problems with sharp edged interfaces elliptic problems with sharp edged interfaces thin layered interfaces and interfaces that intersect with geometric boundary are notoriously challenging to existing numerical methods particularly when the solution is highly oscillatory this work generalizes the matched interface and boundary mib method previously designed for solving elliptic problems with curved interfaces to the aforementioned problems we classify these problems into five distinct topological relations involving the interfaces and the cartesian mesh lines flexible strategies are developed to systematically extends the computational domains near the interface so that the standard central finite difference scheme can be applied without the loss of accuracy fictitious values on the extended domains are determined by enforcing the physical jump conditions on the interface according to the local topology of the irregular point the concepts of primary and secondary fictitious values are introduced to deal with sharp edged interfaces for corner singularity or tip singularity an appropriate polynomial is multiplied to the solution to remove the singularity extensive numerical experiments confirm the designed second order convergence of the proposed method

mass lumping emanating from residual free bubbles a nodally exact scheme is derived for a model equation in 1d involving zeroth and second order terms the method is derived using residual free bubbles in conjunction with the galerkin approximation it is shown that this approach leads to the mass lumping scheme for sufficiently small mesh sizes

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

discontinuous galerkin isogeometric analysis for the biharmonic equation abstract we present and analyze an interior penalty discontinuous galerkin isogeometric analysis dg iga method for the biharmonic equation in computational domain in r d with d 2 3 the computational domain consists of several non overlapping sub domains or patches we construct b spline approximation spaces which are discontinuous across patch interfaces we present a priori error estimate in a discrete norm and numerical experiments to confirm the theory

optimal selection of the bubble function in the stabilization of the p1 p1 element for the stokes problem in this paper we propose a criterion for the selection of the local stabilizing function of the mini discretization of the stokes problem we then show how the application of this criterion to nonconforming approximation leads to a particularly simple solution

partial extensions of attouch s theorem with applications to proto derivatives of subgradient mappings attouch s theorem which gives on a reflexive banach space the equivalence between the mosco epi convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients has many important applications in convex optimization in particular generalized derivatives have been defined in terms of the epi convergence or graph convergence of certain difference quotient mappings and attouch s theorem has been used to relate these various generalized derivatives these relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems we prove in a hilbert space several partial extensions of attouch s theorem to functions more general than convex these functions are called primal lower nice furthermore we use our extensions to derive a relationship between the second order epi derivatives of primal lower nice functions and the proto derivative of their associated subgradient mappings

asymptotic behavior of solutions an application to stochastic nlp in this article we study the consistency of optimal and stationary kkt points of a stochastic non linear optimization problem involving expectation functionals when the underlying probability distribution associated with the random variable is weakly approximated by a sequence of random probability measures the optimization model includes constraints with expectation functionals those are not captured in direct application of the previous results on optimality conditions exist in the literature we first study the consistency of stationary points of a general nlp problem with convex and locally lipschitz data and then apply those results to the stochastic nlp problem and stochastic minimax problem moreover we derive an exponential bound for such approximations using a large deviation principle

spectral element method for three dimensional elliptic problems with smooth interfaces abstract in this paper we propose a least squares spectral element method for three dimensional elliptic interface problems the differentiability estimates and the main stability theorem using non conforming spectral element functions are proven the proposed method is free from any kind of first order reformulation a suitable preconditioner is constructed with help of the regularity estimate and proposed stability estimates which is used to control the condition number we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them we obtain the error estimates which show the exponential accuracy of the method numerical results are obtained for both straight and curved interfaces to show the efficiency of the proposed method

a fully discrete c0 interior penalty galerkin approximation of the extended fisher kolmogorov equation a fully discrete c 0 interior penalty finite element method is proposed and analyzed for the extended fisher kolmogorov efk equation u email protected d email protected u 3 u 0 with appropriate initial and boundary conditions where c is a positive constant we derive a regularity estimate for the solution u of the efk equation that is explicit in c and as a consequence we derive a priori error estimates that are robust in c

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

wijsman convergence of convex sets under renorming let x p be a metric space a natural way to define convergence for a sequence of nonempty closed subsets a of x to a nonempty closed subset a is to insist that we have pointwise convergence of the associated sequence of distance functionals for the sets a to the distance functional for the set a for each x e x c x a lim d x a convergence in this sense is just convergence with respect to the weak topology on the nonempty closed subsets cl x of x determined by the family of distance functionals d x e x e x the so called wijsman topology z l 8 determined by the metric p here of course each functional of the form d x 0 is a function of a closed set variable unfortunately different metrics may give rise to different wijsman topologies even if the metrics are uniformly equivalent see e g 4 section 41 it turns out 6 theorem 3 11 that if we vary our metrics over the class c x of metrics that give rise to the topology of x as well as varying x over x the resulting family

a multiscale finite element method for the incompressible navier stokes equations this paper presents a new multiscale finite element method for the incompressible navier stokes equations the proposed method arises from a decomposition of the velocity field into coarse resolved scales and fine unresolved scales modeling of the unresolved scales corrects the lack of stability of the standard galerkin formulation and yields a method that possesses superior properties like that of the streamline upwind petrov galerkin supg method and the galerkin least squares gls method the multiscale method allows arbitrary combinations of interpolation functions for the velocity and the pressure fields specifically the equal order interpolations that are easy to implement but violate the celebrated babuska brezzi condition a significant feature of the present method is that the structure of the stabilization tensor s appears naturally via the solution of the fine scale problem a family of 2 d elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed convergence studies for the method on uniform skewed as well as composite meshes are presented numerical simulations of the nonlinear steady and transient flow problems are shown that exhibit the good stability and accuracy properties of the method 2005 elsevier b v all rights reserved

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

the epi distance topology continuity and stability results with applications to convex optimization problems let i x denote the proper lower semicontinuous convex functions on a banach space x equipped with the completely metrizable topology of uniform convergence of distance functions on bounded sets we show subject to some standard constraint qualifications that the operations of addition and restriction are continuous these results are applied to convex well posed optimization problems as well as to convergence of approximated solutions in infinite dimensional convex programming to linear functionals exposing convex sets and to metric projections also for any given function in i x we obtain results regarding the convergence of its inf convolution with smoothing kernels

a c0 interior penalty method for the dirichlet control problem governed by biharmonic operator an energy space based dirichlet boundary control problem governed by biharmonic equation is investigated and subsequently a c0 interior penalty method is proposed and analyzed an abstract a priori error estimate is derived under the minimal regularity conditions the abstract error estimate guarantees optimal order of convergence whenever the solution is sufficiently regular further an optimal order l2 norm error estimate is derived numerical experiments illustrate the theoretical findings

three dimensional matched interface and boundary mib method for treating geometric singularities this paper reports the three dimensional 3d generalization of our previous 2d higher order matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and non smooth interfaces new mib algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier mib scheme for treating interfaces with sharp geometric singularities such as sharp edges sharp wedges and sharp tips the resulting 3d mib schemes are of second order accuracy for arbitrarily complex interfaces with sharp geometric singularities of fourth order accuracy for complex interfaces with moderate geometric singularities and of sixth order accuracy for curved smooth interfaces a systematical procedure is introduced to make the mib matrix optimally symmetric and banded by appropriately choosing auxiliary grid points consequently the new mib linear algebraic equations can be solved with fewer number of iterations the proposed mib method makes use of cartesian grids standard finite difference schemes lowest order interface jump conditions and fictitious values the interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation while a pair of fictitious values are determined along a mesh at a time an iterative procedure is proposed to determine all the required fictitious values for higher order schemes by repeatedly using the lowest order jump conditions a variety of mib techniques are developed to overcome geometric constraints the essential strategy of the mib method is to locally reduce a 2d or a 3d interface problem into 1d like ones the proposed mib method is extensively validated in terms of the order of accuracy the speed of convergence the number of iterations and cpu time numerical experiments are carried out to complex interfaces including the molecular surfaces of a protein a missile interface and van der waals surfaces of intersecting spheres

semicontinuous limits of nets of continuous functions in this paper we present a topology on the space of real valued functions defined on a functionally hausdorff space x that is finer than the topology of pointwise convergence and for which 1 the closure of the set of continuous functions mathcal c x is the set of upper semicontinuous functions on x and 2 the pointwise convergence of a net in mathcal c x to an upper semicontinuous limit automatically ensures convergence in this finer topology

quantitative stability of variational systems i the epigraphical distance this paper proposes a global measure for the distance between the elements of a variational system parametrized families of optimization problems

a finite element method for the buckling problem of simply supported kirchhoff plates the aim of this paper is to develop a finite element method to approximate the buckling problem of simply supported kirchhoff plates subjected to general plane stress tensor we introduce an auxiliary variable w u with u representing the displacement of the plate to write a variational formulation of the spectral problem we propose a conforming discretization of the problem where the unknowns are approximated by piecewise linear and continuous finite elements we show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues finally we present some numerical experiments supporting our theoretical results