numerical simulation of 3d viscoelastic flows with free surfaces a numerical model is presented for the simulation of viscoelastic flows with complex free surfaces in three space dimensions the mathematical formulation of the model is similar to that of the volume of fluid vof method but the numerical procedures are different a splitting method is used for the time discretization the prediction step consists in solving three advection problems one for the volume fraction of liquid which allows the new liquid domain to be obtained one for the velocity field one for the extra stress the correction step corresponds to solving an oldroyd b fluid flow problem without advection in the new liquid domain two different grids are used for the space discretization the three advection problems are solved on a fixed structured grid made out of small cubic cells using a forward characteristics method the oldroyd b problem without advection is solved using continuous piecewise linear stabilized finite elements on a fixed unstructured mesh of tetrahedrons efficient post processing algorithms enhance the quality of the numerical solution a hierarchical data structure reduces the memory requirements convergence of the numerical method is checked for the pure extensional flow and the filling of a tube numerical results are presented for the stretching of a filament fingering instabilities are obtained when the aspect ratio is large also results pertaining to jet buckling are reported

adaptive least squares finite element approximations to stokes equations this paper is concerned with adaptive least squares methods for stokes equations based on velocity pressure stress and velocity vorticity pressure formulations to capture the stokes flow region an adaptive algorithm based on mesh redistribution is developed for a nonlinear weighted least squares functional a redistribution approach is considered to generate the optimal grids model problems considered are the flow past a planar channel and a 4 to 1 contraction problems numerical results of model problems illustrating the efficiency of the proposed scheme are presented adaptive least squares methods are used to solve the stokes equations velocity pressure stress and velocity vorticity pressure formulations are used mesh redistribution for planar and 4 to 1 contraction flow problems are developed lower order basis functions in all variables are used in least squares methods hybrid meshes of regular and graded grids are used to reduce the grid effects

minimality and irreducibility of time invariant linear boundary value systems abstract minimality and irreducibility in the class of time invariant linear systems with well posed boundary conditions are introduced and characterized necessary and sufficient conditions for similarity of irreducible or minimal time invariant systems with well posed boundary conditions are given the problem of stable minimality is analysed some of the results are specified for stationary and displacement systems

on a fem method for a linearized version of the oldroyd s problem abstract we present a study of a finite element method fem for the approximation of a linearized version of the oldroyd s problem the method is based on a work in zamm 65 1985 449 concerning the continuous problem on a splitting of the momentum equation and on a modification of the streamline upwind petrov galerkin method in comput meth appl mech engrg 45 1984 285 it is also formally connected to the modified elastic viscous split stress method in j non newtonian fluid mech 60 1995 27 this method allows us to improve some numerical results expected for more standard methods in particular we obtain convergence results in the case of a null newtonian fraction of the viscosity

optimal space time adaptive wavelet methods for degenerate parabolic pdes we analyze parabolic pdes with certain type of weakly singular or degenerate time dependent coefficients and prove existence and uniqueness of weak solutions in an appropriate sense a localization of the pdes to a bounded spatial domain is justified for the numerical solution a space time wavelet discretization is employed an optimality result for the iterative solution of the arising systems can be obtained finally applications to fractional brownian motion models in option pricing are presented

mathematical and numerical analysis of a simplified time dependent viscoelastic flow a time dependent model corresponding to an oldroyd b viscoelastic fluid is considered the convective terms being disregarded global existence in time is proved in banach spaces provided the data are small enough using the implicit function theorem and a maximum regularity property for a three fields stokes problem a finite element discretization in space is then proposed existence of the numerical solution is proved for small data so as a priori error estimates using again an implicit function theorem

moving mesh methods in multiple dimensions based on harmonic maps in practice there are three types of adaptive methods using the finite element approach namely the h method p method and r method in the h method the overall method contains two parts a solution algorithm and a mesh selection algorithm these two parts are independent of each other in the sense that the change of the pdes will affect the first part only however in some of the existing versions of the r method also known as the moving mesh method these two parts are strongly associated with each other and as a result any change of the pdes will result in the rewriting of the whole code in this work we will propose a moving mesh method which also contains two parts a solution algorithm and a mesh redistribution algorithm our efforts are to keep the advantages of the r method e g keep the number of nodes unchanged and of the h method e g the two parts in the code are independent a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in this work we will extend dvinsky s method to provide an efficient solver for the mesh redistribution algorithm the key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure each iteration step is to move the mesh closer to the harmonic map this procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration the numerical schemes are applied to a number of test problems in two dimensions it is observed that the mesh redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi dimensional computations

on the discrete evss method abstract this paper is a completion of a previous paper written by two of the authors m fortin r guenette r pierre numerical analysis of the modified evss method comput methods appl mech engrg 143 1997 79 95 concerning the devss method which is a discrete variant of the popular elastic viscous split stress evss method introduced by rajagopalan r a brown r c armstrong finite element methods for calculation of steady viscoelastic flow using constitutive equations with a newtonian viscosity j non newtonian fluid mech 36 1990 159 199 within the context of viscoelastic fluids the method uses the rate of deformation tensor as an additional variable in this paper it is shown that a simple modification of the theoretical framework introduced in the first paper m fortin r guenette r pierre numerical analysis of the modified evss method comput methods appl mech engrg 143 1997 79 95 allows the treatment of multi mode constitutive equations of viscoelastic flows as well as that of the velocity gradient introduced as the additional variable instead of the rate of deformation tensor moreover this general framework based on a generalized brezzi babuska theory allows different discretizations for the stress and the additional variable the paper presents a general criteria insuring the stability of the method three particular mixed finite elements satisfying this criteria are studied an error analysis is performed on stokes and viscoelastic flows

similarity and reduction for time varying linear systems with well posed boundary conditions the classical theory of similarity and reduction of causal and anti causal systems is generalized and extended to time varying linear systems with well posed boundary conditions two main problems are solved the first is to determine to what extent a system with boundary conditions can be simplified by similarity and reduction the second problem is to find the invariants of such a simplification special attention is given to the time invariant case

a comparative study of the discontinuous galerkin and continuous supg finite element methods for computation of viscoelastic flows the performance of the discontinuous galerkin dg method and the continuous streamline upwind petrov galerkin supg method in the computation of steady viscoelastic flows have been compared in this paper for the continuous supg method it is necessary to include the elastic viscous split stress evss formulation to satisfy the compatibility requirement on the velocity stress interpolation supg evss while for the discontinuous galerkin method both the pure mixed formulation dg mix and the evss formulation dg evss have been tested we first examine a benchmark smooth problem i e the steady flow of an upper convected maxwell ucm fluid around a sphere falling along the axis of a cylindrical tube by using high order finite element approximation dg mix is found not to be a stable discretization in this problem its limiting weissenberg number decreases with increasing the interpolation order on the other hand dg evss is an accurate and stable algorithm its solution converges as the approximation order increases the so called p convergence and it produces almost identical stress boundary layers to that of the supg evss algorithm at a weissenberg number as high as 2 0 to explore benefits of the discontinuous interpolation for the stress in the dg method we further examine two benchmark problems with stress singularities by using low order interpolations on sufficiently fine meshes they are the axisymmetric steady flows of the modified chilcott and rallison mcr fluid through a four to one abrupt contraction and in a tube with the stick to slip transition in the wall velocity boundary condition in the contraction problem the dg evss method is found to be a significant improvement over the supg evss method quality of the solutions is better for dg evss and a much higher weissenberg number limit is observed in the stick slip problem both dg evss and supg evss are very robust they confront no limit of the weissenberg number and predict asymptotic stress profiles as the number becomes sufficiently large in both problems the pure mixed method dg mix produces severe stress oscillations near the singularities and fails to converge at medium weissenberg numbers

stabilized finite element methods for the velocity pressure stress formulation of incompressible flows abstract formulated in terms of velocity pressure and the extra stress tensor the incompressible navier stokes equations are discretized by stabilized finite element methods the stabilized methods proposed are analyzed for a linear model and extended to the navier stokes equations the numerical tests performed confirm the good stability characteristics of the methods these methods are applicable to various combinations of interpolation functions including the simplest equal order linear and bilinear elements

boundary value descriptor systems well posedness reachability and observability in this paper we introduce the class of two point boundary value descriptor systems tpbvds discrete time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system state at the two endpoints of the system s interval of definition by introducing a standard form for regular pencils we obtain a new and simple generalized cayley hamilton theorem that simplifies our investigation of well posedness green s function solutions and reachability and observability for tpbvds there are two distinct notions of reachability and observability that one can define for tpbvds associated with processes that propagate inward from and outward toward the boundaries we investigate each of these in detail obtaining among other things far simpler forms for the reachability and observability results than found previously in the literature in addition we describe several methods for the efficient solution of tpbvds one involving recursions from each end of t

convergence analyses of galerkin least squares methods for symmetric advective diffusive forms of the stokes and incompressible navier stokes equations abstract symmetric advective diffusive forms of the stokes and incompressible navier stokes equations are presented the galerkin least squares method for advective diffusive equations is used for both systems and is related to other stabilized methods previously studied the presentation reveals that the convergence analysis for advective diffusive equations as applied before to a linearized form of the compressible navier stokes equations carries over in a straightforward manner to the stokes problem and to a linearized form of the incompressible navier stokes equations

stabilized finite elements on anisotropic meshes a priori error estimates for the advection diffusion and the stokes problems stabilized finite elements on strongly anisotropic meshes are considered the design of the stability coefficients is addressed for both the advection diffusion and the stokes problems when using continuous piecewise linear finite elements on triangles using the polar decomposition of the jacobian of the affine mapping from the reference triangle to the current one k and from a priori error estimates a new definition of the stability coefficients is proposed our analysis shows that these coefficients do not depend on the element diameter hk but on a characteristic length associated with k via the polar decomposition a numerical assessment of the theoretical analysis is carried out

finite element approximation of viscoelastic fluid flow existence of approximate solutions and error bounds we study a finite element approximation of viscoelastic fluid flow obeying an oldroyd b type constitutive law the approximate stress velocity and pressure are respectivelyp 1 discontinuous p 2 continuous p 1 continuous we use the method of lesaint raviart for the convection of the extra stress tensor we suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution we show by a fixed point method that the approximate problem has a solution and we give an error bound

optimal finite element interpolation on curved domains this paper is devoted to a general theory of approximation of functions in finite element spaces in particular the case is considered where the functions to be interpolated are on the one hand not very smooth and on the other are defined on curved domains thus a number of well known optimal interpolation results are generalized

on mixed finite element methods for first order elliptic systems a physically based duality theory for first order elliptic systems is shown to be of central importance in connection with the galerkin finite element solution of these systems using this theory in conjunction with a certain hypothesis concerning approximation spaces optimal error estimates for galerkin type approximations are demonstrated an example of a grid which satisfies the hypothesis is given and numerical examples which illustrate the theory are provided

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

lenses in skew brownian flow we consider a stochastic flow in which individual particles follow skew brownian motions with each one of these processes driven by the same brownian motion one does not have uniqueness for the solutions of the corresponding stochastic differential equation simultaneously for all real initial conditions due to this lack of the simultaneous strong uniqueness for the whole system of stochastic differential equations the flow contains lenses that is pairs of skew brownian motions which start at the same point bifurcate and then coalesce in a finite time the paper contains qualitative and quantitative distributional results on the geometry of the flow and lenses

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

weighted least squares approximationsto nonlinear hyperbolic equations some approximate methods for solving nonlinear hyperbolic equations are presented and analyzed the methods consist of discretizing with respect to time and solving the resulting hyperbolic equation for fixed time by least squares finite element methods stability of appropriate weighted least squares approximations is provided uniform and time varying optimal grids based on the mesh redistribution approach are considered numerical results for the inviscid burgers equation are also presented

stability stochastic stationarity and generalized lyapunov equations for two point boundary value descriptor systems the authors introduce the concept of internal stability for two point boundary value descriptor systems tpbvdss since tpbvdss are defined only over a finite interval the concept of stability is not easy to formulate the definition which is used consists of requiring that as the length of the interval of definition increases the effect of boundary conditions on states located close to the center of the interval should go to zero stochastic tpbvdss are studied and the property of stochastic stationarity is characterized in terms of a generalized lyapunov equation satisfied by the variance of the boundary vector a second generalized lyapunov equation satisfied by the state variance of a stochastically stationary tpbvds is also introduced and the existence and uniqueness of positive definite solutions to this equation are then used to characterize the property of internal stability

numerical solution of a singularly perturbed two point boundary value problem using equidistribution analysis of convergence abstract adaptive grid methods are becoming established as valuable computational techniques for the numerical solution of differential equations with near singular solutions adaptive methods are equally effective in approximating solutions of problems with boundary layers or interior layers see for example mulholland et al siam j sci comput 19 4 1998 1261 1289 much is now being done in developing error analyes for methods that are based on adaptivity in this paper we present a rigorous error analysis for the solution of a singularly perturbed two point boundary value problem on a grid that is constructed adaptively from a knowledge of the exact solution the discrete solutions are generated by an upwind finite difference scheme and the grid is formed by equidistributing a monitor function based on arc length an error analysis shows that the discrete solutions are uniformly convergent with respect to the perturbation parameter epsilon the epsilon uniform convergence is confirmed by numerical computations

grading functions and mesh redistribution we consider the problem of determining an appropriate grading of a mesh for piecewise polynomial interpolation and for approximate solution of two point boundary value problems by finite difference or finite element methods an analysis of optimality of the mesh and optimal grading functions in various norms and seminorms for the interpolation problem leads to the formulation of an adaptive mesh redistribution algorithm for boundary value problems in one dimension error estimates are given and numerical results presented to demonstrate the performance of the scheme and compare alternative redistribution criteria

efficient stress velocity least squares finite element formulations for the incompressible navier stokes equations abstract in this contribution three different mixed least squares finite element methods lsfems are investigated with respect to accuracy and efficiency with simultaneous consideration of regular and adaptive meshing strategies the deliberations are made for the incompressible navier stokes equations the generally known first order div grad system in terms of the total stress velocity and pressure svp formulation is the basis for two further div grad least squares formulations in terms of total stress and velocity sv whereby both formulations are derived from different fountainheads the extended sv formulation is an enrichment of the known stress velocity formulation proposed by cai et al 2004 the second sv formulation is based on the substitution of the pressure in the svp formulation thus it is based on a discontinuous pressure interpolation advantage of the sv formulations is a smaller system matrix size due to the reduction of the relevant degrees of freedom nevertheless the approximation quality has to last the demand of the established least squares formulations the drawback of a poor mass conservation is well investigated for the lsfem and known to overcome by using high order interpolations for all solution variables therefore the main attention of this contribution is focused on accuracy while aiming for high efficiency which is intrinsic for the fundamental idea in here in the light of efficient lsfem solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h type adaptive mesh refinement

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

on minimality and stable minimality of time varying linear systems with well posed boundary conditions for time varying linear systems with well posed boundary conditions the concept of stable minimality is introduced a complete characterization is given of stably minimal systems with analytic coefficients

adaptive wavelet algorithms for elliptic pde s on product domains with standard isotropic approximation by piecewise polynomials of fixed order in a domain d subset of r d the convergence rate in terms of the number n of degrees of freedom is inversely proportional to the space dimension d this so called curse of dimensionality can be circumvented by applying sparse tensor product approximation when certain high order mixed derivatives of the approximated function happen to be bounded in l 2 it was shown by nitsche 2006 that this regularity constraint can be dramatically reduced by considering best n term approximation from tensor product wavelet bases when the function is the solution of some well posed operator equation dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible applying piecewise smooth wavelets we verify this compressibility for general non separable elliptic pdes in tensor domains applications of the general theory developed include adaptive galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust i e independent of the scale parameters resp of the size of the anisotropy

multigrid solution and grid redistribution for convection diffusion we examine multigrid solution for convection diffusion problems in which convective effects are significant in particular we consider the case where the fine grid satisfies the associated cell or diagonal dominance condition but coarse grid levels violate this condition related numerical experiments are conducted we also introduce an alternative local elliptic projection technique and compare this with interpolating the error corrections from coarse to fine grids in addition we demonstrate the use of grid redistribution and optimization techniques in the multilevel context

minimal partial realization by descriptor systems abstract in this paper a generalization of kalman s partial realization theory is developed using partial realizations defined by descriptor systems the use of singular system realizations in contrast to regular linear systems enables us to circumvent certain technical difficulties inherent in the standard approach to partial realizations an existence and uniqueness result for minimal partial descriptor realizations is proven and a simple rank formula for the mcmillan degree is derived

a stabilized finite element method for the two field and three field stokes eigenvalue problems in this paper the stabilized finite element approximation of the stokes eigenvalue problems is considered for both the two field displacement pressure and the three field stress displacement pressure formulations the method presented is based on a subgrid scale concept and depends on the approximation of the unresolvable scales of the continuous solution in general subgrid scale techniques consist in the addition of a residual based term to the basic galerkin formulation the application of a standard residual based stabilization method to a linear eigenvalue problem leads to a quadratic eigenvalue problem in discrete form which is physically inconvenient as a distinguished feature of the present study we take the space of the unresolved subscales orthogonal to the finite element space which promises a remedy to the above mentioned complication in essence we put forward that only if the orthogonal projection is used the residual is simplified and the use of term by term stabilization is allowed thus we do not need to put the whole residual in the formulation and the linear eigenproblem form is recovered properly we prove that the method applied is convergent and present the error estimates for the eigenvalues and the eigenfunctions we report several numerical tests in order to illustrate that the theoretical results are validated

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

sweepline algorithm for unstructured grid generation on two dimensional non convex domains the sweepline algorithm has been modified to triangulate a set of nodes on a non convex domain the new method is based on a fragile sweepline which is split and rejoined in a manner which ensures that boundaries are preserved both the original and the modified algorithms are introduced using simple examples it is shown that a set of n nodes is triangulated in o n log n time the validity of the new method has been confirmed through testing on a variety of problems and two test cases are presented in this paper

decoupled algorithm for solving phan thien tanner viscoelastic fluid by finite element method in this paper we present a decoupled algorithm for the viscoelastic fluid flow obeying the phan thien tanner ptt differential model the method consists to solve alternatively a stokes like problem by weighted least squares wls finite element method and the constitutive equation by streamline upwind petrov galerkin supg method a priori error estimate for the wls supg finite element method is derived the existence and uniqueness of the approximate solution are obtained using fixed point techniques we show that the mapping of the decoupled algorithm is locally contracting the planar channel flow problem is used to illustrate our theoretical results we encounter no limit of the weissenberg number for reasonable values of e a dimensionless material parameter in our calculations

error analysis of some galerkin least squares methods for the elasticity equations the recent technique of stabilizing mixed finite element methods by augmenting the galerkin formulation with least squares terms calculated separately on each element is considered the error analysis is performed in a unified manner yielding improved results for some methods introduced earlier in addition a new formulation is introduced and analyzed method is used within two different contexts the first is in connection with the plain strain or three dimensional problems near the incompressible limit for which the poisson ratio equals one half it is well known that for these problems a standard lower order displacement finite element method will break down by now it is also well known how a working method can be obtained a new unknown the pressure is introduced and discretized with a finite element space different from that used for the displacements the finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by brezzi and babuska 5 7 the application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition for a particular choice of displacement and pressure approximations it can be a difficult task to verify this babuska brezzi or inf sup condition during the last decade this problem has been intensively studied and the performances of many combinations are now known cf 18 and 23 26 for some recent extensions in addition there exists a general technique utilizing bubble functions by which a stable method can

a new discrete evss method for the viscoelastic flows in this paper we propose a new discrete elastic viscous split stress evss method for the viscoelastic flows obeying an oldroyd b type constitutive equation supposing that the velocity and pressure are approximated by the taylor hood elements and applying the orthogonal projection technique we introduce two gauss integrations as a stabilizing term in the discrete evss method compared with the popular one the new method does not need to introduce any extra variable and reduces the degrees of freedom of the discrete system a lot but gets the same stabilized result additionally some numerical experiments are shown to verify its efficiency

numerical construction of optimal adaptive grids in two spatial dimensions abstract this article concerns a new procedure to generate a solution adaptive grid for convection dominated problems in two spatial dimensions based on finite element approximations the procedure extends a one dimensional equidistribution principle which minimizes the interpolation error in appropriate norms the idea in extending such a technique to two spatial dimensions is to select two directions which can reflect the physics of the problems and then the one dimensional equidistribution principle is applied to the chosen directions the final grids generated are connected through a sweep line based unstructured grid technique model problems considered are the two dimensional convection diffusion problems where boundary and interior layers occur numerical results of model problems illustrating the efficiency of the proposed scheme are presented comparisons of the solutions with an existing scheme will also be provided

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

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

error estimates for a stabilized finite element method for the oldroyd b model abstract in this paper we study a new approximation scheme of transient viscoelastic fluid flow obeying an oldroyd b type constitutive equation the new stabilized formulation bases on the choice of a modified euler method connected to the streamline upwinding petrov galerkin supg method m bensaada d esselaoui d sandri stabilization method for continuous approximation of transient convection problem numer methods partial differential equations 21 2004 170 189 in order to stabilize the tensorial transport term of the oldroyd derivative suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution a priori error estimates for the approximation in terms of the mesh parameter h and the time discretization parameter t are derived

realization of acausal weighting patterns with boundary value descriptor systems this paper examines the realization of acausal weighting patterns with two point boundary value descriptor systems tpbvdss attention is restricted to the subclass oftpbvdss that are stationary so that their input output weighting pattern is shift invariant and extendible i e their weighting pattern can be extended outwards indefinitely then given an infinite acausal shift invariant weighting pattern the realization problem consists of constructing a minimal tpbvds over a fixed interval whose extended weighting pattern matches the given pattern the realization method that is proposed relies on a new transform the s t transform which is better adapted to the analysis of descriptor dynamics than the standard z transform since it handles zero and infinite frequencies in a symmetric way this new transform is used to determine the dimension of a minimal realization and then to construct a minimal realization by obtaining state space representations for two homogeneous rational matrices in s and obtained from the causal and anticausal components of the weighting pattern

a robust adaptive method for a quasi linear one dimensional convection diffusion problem a quasi linear conservative convection diffusion two point boundary value problem is considered to solve it numerically an upwind finite difference scheme is applied the mesh used has a fixed number n 1 of nodes and is initially uniform but its nodes are moved adaptively using a simple algorithm of de boor based on equidistribution of the arc length of the current computed piecewise linear solution it is proved for the first time that a mesh exists that equidistributes the arc length along the polygonal solution curve and that the corresponding computed solution is first order accurate uniformly in varepsilon where varepsilon is the diffusion coefficient in the case when the boundary value problem is linear if n is sufficiently large independently of varepsilon it is shown that after o rm ln 1 varepsilon rm ln n iterations of the algorithm the piecewise linear interpolant of the computed solution achieves first order accuracy in the l infty 0 1 norm uniformly in varepsilon numerical experiments are presented that support our theoretical results

optimal adaptive grids of least squares finite element methods in two spatial dimensions this article concerns a procedure to generate optimal adaptive grids for convection dominated problems in two spatial dimensions based on least squares finite element approximations the procedure extends a one dimensional equidistribution principle which minimizes the interpolation error in some norms the idea is to select two directions which can reflect the physics of the problems and then apply the one dimensional equidistribution principle to the chosen directions model problems considered are the two dimensional convection diffusion problems where boundary and interior layers occur numerical results of model problems illustrating the efficiency of the proposed scheme are presented in addition to avoid skewed mesh in the optimal grids generated by the algorithm an unstructured local mesh smoothing will be considered in the least squares approximations comparisons with the gakerkin finite element method will also be provided

a stabilized cut finite element method for the three field stokes problem we propose a nitsche based fictitious domain method for the three field stokes problem in which the boundary of the domain is allowed to cross through the elements of a fixed background mesh the dependent variables of velocity pressure and extra stress tensor are discretized on the background mesh using linear finite elements this equal order approximation is stabilized using a continuous interior penalty cip method on the unfitted domain boundary dirichlet boundary conditions are weakly enforced using nitsche s method we add cip like ghost penalties in the boundary region and prove that our scheme is inf sup stable and that it has optimal convergence properties independent of how the domain boundary intersects the mesh additionally we demonstrate that the condition number of the system matrix is bounded independently of the boundary location we corroborate our theoretical findings numerically

adaptive wavelet schemes for parabolic problems sparse matrices and numerical results a simultaneous space time variational formulation of a parabolic evolution problem is solved with an adaptive wavelet method this method is shown to converge with the best possible rate in linear complexity thanks to the use of tensor product bases there is no penalty in complexity due to the additional time dimension special wavelets are designed such that the bi infinite system matrix is sparse this sparsity largely simplifies the implementation and improves the quantitative properties of the adaptive wavelet method numerical results for an ode and the heat equation are presented

realization theory for multivariate stationary gaussian processes this paper collects in one place a comprehensive theory of stochastic realization for continuous time stationary gaussian vector processes which in various pieces has appeared in a number of our earlier papers it begins with an abstract state space theory based on the concept of splitting subspace these results are then carried over to the spectral domain and described in terms of hardy functions finally differential equations type stochastic realizations are constructed the theory is coordinate free and it accommodates infinite dimensional representations minimality and other systems theoretical concepts being defined by subspace inclusion rather than by dimension we have strived for conceptual completeness rather than generality and the same framework can be used for other types of stochastic realization problems

gls and evss methods for a three field stokes problem arising from viscoelastic flows in this paper order one finite elements together with galerkin least squares gls methods are used for solving a three held stokes problem arising from the numerical study of viscoelastic hows stability and convergence results are established even when the solvent viscosity is small compared to the viscosity due to the polymer chains an iterative algorithm decoupling velocity pressure and stress calculations is proposed the link with the modified elastic viscous split stress evss method studied in m fortin r guenette r pierre comput methods appl mech engrg 143 1997 79 95 r guenette m fortin j non newtonian fluid mech 60 1995 27 52 is presented numerical results are in agreement with theoretical predictions and with those presented in m fortin r guenette r pierre comput methods appl mech engrg 143 1997 79 95 c 2001 elsevier science b v all rights reserved

revisiting stabilized finite element methods for the advective diffusive equation we give a brief overview of stabilized finite element methods and illustrate the developments applied to the advection diffusion equation this article presents a concise perspective of the developments emanated from the works started in the 1980s through today 2005 elsevier b v all rights reserved pacs 47 11 j

variational mesh adaptation ii error estimates and monitor functions the key to the success of a variational mesh adaptation method is to define a proper monitor function which controls mesh adaptation in this paper we study the choice of the monitor function for the variational adaptive mesh method developed in the previous work j comput phys 174 2001 924 two types of monitor functions scalar matrix and non scalar matrix ones are defined based on asymptotic estimates of interpolation error obtained using the interpolation theory of finite element methods the choice of the adaptation intensity parameter is also discussed for each of these monitor functions asymptotic bounds on interpolation error are obtained for adaptive meshes that satisfy the regularity and equidistribution conditions two dimensional numerical results are given to verify the theoretical findings

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization

generalized inf sup conditions for chebyshev spectral approximation of the stokes problem a mixed problem and its approximation in an abstract framework are considered they are proved to be well posed if and only if several inf sup conditions are satisfied these results are applied to the stokes equations in a square formulated in chebyshev weighted sobolev spaces and their approximations two kinds of spectral discretizations are analyzed a galerkin method and a collocation method at the chebyshev nodes

a nonlinear weighted least squares finite element method for stokes equations the paper concerns a nonlinear weighted least squares finite element method for the solutions of the incompressible stokes equations based on the application of the least squares minimization principle to an equivalent first order velocity pressure stress system model problem considered is the flow in a planar channel the least squares functional involves the l 2 norms of the residuals of each equation multiplied by a nonlinear weighting function and mesh dependent weights using linear approximations for all variables by properly adjusting the importance of the mass conservation equation and a carefully chosen nonlinear weighting function the least squares solutions exhibit optimal l 2 norm error convergence in all unknowns numerical solutions of the flow pass through a 4 to 1 contraction channel will also be considered

multiresolution weighted norm equivalences and applications we establish multiresolution norm equivalences in weighted spaces l2w 0 1 with possibly singular weight functions w x 0 in 0 1 our analysis exploits the locality of the biorthogonal wavelet basis and its dual basis functions the discrete norms are sums of wavelet coefficients which are weighted with respect to the collocated weight function w x within each scale since norm equivalences for sobolev norms are by now well known our result can also be applied to weighted sobolev norms we apply our theory to the problem of preconditioning p version fem and wavelet discretizations of degenerate elliptic and parabolic problems from finance

numerical analysis of the modified evss method abstract in this paper we present a proof of the stability of a new mixed finite element method introduced recently by the authors 9 within the context of viscoelastic fluids the mixed formulation is related to the evss elastic viscous split stress method proposed by rajagopalan et al 14 and is based on the introduction of the rate of deformation tensor as an additional unknown the proof applies to the stokes flow and some linearized constitutive equations of slow viscoelastic flows existence and uniqueness of the continuous and the discrete problems are derived from a generalized brezzi babuska theory it is shown that no additional compatibility condition is required between the various variables except the usual one for the velocity and the pressure fields this result allows to choose low order finite element for the stress several numerical experiments on the 4 1 contraction stokes flow will be presented which will confirm the improved stability obtained with this new formulation

finite element approximation of the three field formulation of the stokes problem using arbitrary interpolations the stress displacement pressure formulation of the elasticity problem may suffer from two types of numerical instabilities related to the finite element interpolation of the unknowns the first is the classical pressure instability that occurs when the solid is incompressible whereas the second is the lack of stability in the stresses to overcome these instabilities there are two options the first is to use different interpolation for all the unknowns satisfying two inf sup conditions whereas there are several displacement pressure interpolations that render the pressure stable less possibilities are known for the stress interpolation the second option is to use a stabilized finite element formulation instead of the plain galerkin approach if this formulation is properly designed it is possible to use arbitrary interpolation for all the unknowns the purpose of this paper is precisely to present one of such formulations in particular it is based on the decomposition of the unknowns into their finite element component and a subscale which will be approximated and whose goal is to yield a stable formulation a singular feature of the method to be presented is that the subscales will be considered orthogonal to the finite element space we describe the design of the formulation and present the results of its numerical analysis

the adaptive lagrangian particle method for macroscopic and micro macro computations of time dependent viscoelastic flows we propose a new numerical technique referred to as the adaptive lagrangian particle method alpm for computing time dependent viscoelastic flows using either a differential constitutive equation macroscopic approach or a kinetic theory model micro macro approach in alpm the eulerian finite element solution of the conservation equations is decoupled from the lagrangian computation of the extra stress at a number of discrete particles convected by the how in the macroscopic approach the extra stress carried by the particles is obtained by integrating the constitutive equation along the particle trajectories in the micro macro approach the extra stress is computed by solving along the particle paths the stochastic differential equation associated with the kinetic theory model at each time step alpm automatically enforces that all elements of the mesh have a number of lagrangian particles ranging within a user specified interval results are given for the start up flow between highly eccentric rotating cylinders using the fene and fene p dumbbell models for dilute polymer solutions c 1999 elsevier science s a all rights reserved