the stability in _ and _ of the projection onto finite element function spaces on etablit la stabilite de la l 2 projection sur quelques espaces a elements finis v h consideree comme une application dans l p et w p 1 1 p sous des hypotheses de regularite faibles

mizoguchi takahashi s fixed point theorem is a real generalization of nadler s abstract we give an example which says that mizoguchi takahashi s fixed point theorem for set valued mappings is a real generalization of nadler s we also give a very simple proof of mizoguchi takahashi s theorem

on really locking free mixed finite element methods for the transient incompressible stokes equations inf sup stable mixed methods for the steady incompressible stokes equations that relax the divergence constraint are often claimed to deliver locking free discretizations however this relaxation leads to a pressure dependent contribution in the velocity error which is proportional to the inverse of the viscosity thus giving rise to a different locking phenomenon however a recently proposed modification of the right hand side alone leads to a discretization that is really locking free i e its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity in this contribution we extend this approach to the transient incompressible stokes equations where besides the right hand side also the velocity time derivative requires an improved space discretization semidiscrete and fully discrete a priori velocity and pressure error estimates are derived which show remarkable robustness properties two numerical examples illustrate the superior accura

on the stability of the l 2 projection in h 1 ω we prove the stability in h1 of the l2 projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions we give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions in particular stability of the l2 projection in h1 holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically

a stable partitioned fsi algorithm for incompressible flow and deforming beams an added mass partitioned amp algorithm is described for solving fluid structure interaction fsi problems coupling incompressible flows with thin elastic structures undergoing finite deformations the new amp scheme is fully second order accurate and stable without sub time step iterations even for very light structures when added mass effects are strong the fluid governed by the incompressible navier stokes equations is solved in velocity pressure form using a fractional step method large deformations are treated with a mixed eulerian lagrangian approach on deforming composite grids the motion of the thin structure is governed by a generalized euler bernoulli beam model and these equations are solved in a lagrangian frame using two approaches one based on finite differences and the other on finite elements the key amp interface condition is a generalized robin mixed condition on the fluid pressure this condition which is derived at a continuous level has no adjustable parameters and is applied at the discrete level to couple the partitioned domain solvers special treatment of the amp condition is required to couple the finite element beam solver with the finite difference based fluid solver and two coupling approaches are described a normal mode stability analysis is performed for a linearized model problem involving a beam separating two fluid domains and it is shown that the amp scheme is stable independent of the ratio of the mass of the fluid to that of the structure a traditional partitioned tp scheme using a dirichlet neumann coupling for the same model problem is shown to be unconditionally unstable if the added mass of the fluid is too large a series of benchmark problems of increasing complexity are considered to illustrate the behavior of the amp algorithm and to compare the behavior with that of the tp scheme the results of all these benchmark problems verify the stability and accuracy of the amp scheme results for one benchmark problem modeling blood flow in a deforming artery are also compared with corresponding results available in the literature

stabilized reduced basis methods for parametrized steady stokes and navier stokes equations abstract it is well known in the reduced basis approximation of saddle point problems that the galerkin projection on the reduced space does not guarantee the inf sup approximation stability even if a stable high fidelity method was used to generate snapshots for problems in computational fluid dynamics the lack of inf sup stability is reflected by the inability to accurately approximate the pressure field in this context inf sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions the main goal of this work is to propose an alternative approach which relies on the residual based stabilization techniques customarily employed in the finite element literature such as brezzi pitkaranta franca hughes streamline upwind petrov galerkin galerkin least square in the spirit of offline online reduced basis computational splitting two such options are proposed namely offline only stabilization and offline online stabilization these approaches are then compared to and combined with the state of the art supremizer enrichment approach numerical results are discussed highlighting that the proposed methodology allows to obtain smaller reduced basis spaces i e neglecting supremizer enrichment for which a modified inf sup stability is still preserved at the reduced order level

an effective fluid structure interaction formulation for vascular dynamics by generalized robin conditions in this work we focus on the modeling and numerical simulation of the fluid structure interaction mechanism in vascular dynamics we first propose a simple membrane model to describe the deformation of the arterial wall which is derived from the koiter shell equations and is applicable to an arbitrary geometry secondly we consider a reformulation of the fluid structure problem in which the newly derived membrane model thanks to its simplicity is embedded into the fluid equations and will appear as a generalized robin boundary condition the original problem is then reduced to the solution of subsequent fluid equations defined on a moving domain and may be achieved with a fluid solver only we also derive a stability estimate for the resulting numerical scheme finally we propose new outflow absorbing boundary conditions which are easy to implement and allow us to reduce significantly the spurious pressure wave reflections that typically appear in artificially truncated computational domains we present several numerical results showing the effectiveness of the proposed approaches

continuous interior penalty finite element method for oseen s equations in this paper we present an extension of the continuous interior penalty method of douglas and dupont interior penalty procedures for elliptic and parabolic galerkin methods in computing methods in applied sciences lecture notes in phys 58 springer verlag berlin 1976 pp 207 216 to oseen s equations the method consists of a stabilized galerkin formulation using equal order interpolation for pressure and velocity to counter instabilities due to the pressure velocity coupling or due to a high local reynolds number we add a stabilization term giving l2 control of the jump of the gradient over element faces edges in two dimensions to the standard galerkin formulation boundary conditions are imposed in a weak sense using a consistent penalty formulation due to nitsche we prove energy type a priori error estimates independent of the local reynolds number and give some numerical examples recovering the theoretical results

stabilized finite element methods for the generalized oseen problem the numerical solution of the non stationary incompressible navierstokes model can be split into linearized auxiliary problems of oseen type we present in a unique way different stabilization techniques of finite element schemes on isotropic meshes first we describe the state of the art for the classical residual based supg pspg method then we discuss recent symmetric stabilization techniques which avoid some drawbacks of the classical method these methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation finally we give a critical comparison of these methods

analysis of the pspg method for the transient stokes problem abstract we propose a new analysis for the pspg method applied to the transient stokes problem stability and convergence are obtained under different conditions on the discretization parameters depending on the approximation used in space for the pressure we prove optimal stability and convergence only in the case of piecewise affine approximation under the standard condition on the time step finally we show that the stability problems of the pspg method can be circumvented using an appropriate discrete reconstruction of the laplace operator

a projection semi implicit scheme for the coupling of an elastic structure with an incompressible fluid we address the numerical simulation of fluid structure systems involving an incompressible viscous fluid this issue is particularly difficult to face when the fluid added mass acting on the structure is strong as it happens in hemodynamics for example indeed several works have shown that in such situations implicit coupling seems to be necessary in order to avoid numerical instabilities although significant improvements have been achieved during the last years solving implicit coupling often exhibits a prohibitive computational cost in this work we introduce a semi implicit coupling scheme which remains stable for a reasonable range of the discretization parameters the first idea consists in treating implicitly the added mass effect whereas the other contributions geometrical non linearities viscous and convective effects are treated explicitly the second idea relies on the fact that this kind of explicit implicit splitting can be naturally performed using a chorin temam projection scheme in the fluid we prove conditional stability of the scheme for a fully discrete formulation several numerical experiments point out the efficiency of the present scheme compared to several implicit approaches

existence of a solution to a fluid multi layered structure interaction problem abstract we study a nonlinear unsteady moving boundary fluid structure interaction fsi problem in which the structure is composed of two layers a thick layer and a thin layer which serves as a fluid structure interface with mass the fluid flow which is driven by the time dependent dynamic pressure data is modeled by the navier stokes equations for an incompressible viscous fluid defined on a 2d cylinder the elastodynamics of the cylinder wall is governed by the 1d linear wave equation modeling the thin structural layer and by the 2d equations of linear elasticity modeling the thick structural layer we prove existence of a weak solution to this nonlinear fsi problem as long as the cylinder radius is greater than zero the spaces of weak solutions presented in this manuscript reveal a striking new feature the presence of a thin fluid structure interface with mass regularizes solutions of the coupled problem

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

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100

semidiscrete finite element approximations of a linear fluid structure interaction problem semidiscrete finite element approximations of a linear fluid structure interaction problem are studied first results concerning a divergence free weak formulation of the interaction problem are reviewed next semidiscrete finite element approximations are defined and the existence of finite element solutions is proved with the help of an auxiliary discretely divergence free formulation a discrete inf sup condition is verified and the existence of a finite element pressure is established strong a priori estimates for the finite element solutions are also derived then by passing to the limit in the finite element approximations the existence of a strong solution is demonstrated and semidiscrete error estimates are obtained

convergence analysis of a projection semi implicit coupling scheme for fluid structure interaction problems in this paper we provide a convergence analysis of a projection semi implicit scheme for the simulation of fluid structure systems involving an incompressible viscous fluid the error analysis is performed on a fully discretized linear coupled problem a finite element approximation and a semi implicit time stepping strategy are respectively used for space and time discretization the fluid is described by the stokes equations the structure by the classical linear elastodynamics equations linearized elasticity plate or shell models and all changes of geometry are neglected we derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least sqrt delta t finally some numerical experiments that confirm the theoretical analysis are presented

on a multiscale approach to the transient stokes problem dynamic subscales and anisotropic space time discretization abstract in this article we analyze some residual based stabilization techniques for the transient stokes problem when considering anisotropic time space discretizations we define an anisotropic time space discretization as a family of time space partitions that does not satisfy the condition h 2 c t with c uniform with respect to h and t standard residual based stabilization techniques are motivated by a multiscale approach approximating the effect of the subscales onto the large scales one of the approximations is to consider the subscales quasi static neglecting their time derivative it is well known that these techniques are unstable for anisotropic time space discretizations we show that the use of dynamic subscales where the subscales time derivatives are not neglected solves the problem and prove optimal convergence and stability results that are valid for anisotropic time space discretizations also the improvements related to the use of orthogonal subscales are addressed

on a countable dimensional subalgebra of the equivalence algebra for equations vtt f x vx vxx g x vx the analysis of the equations vtt f x vx vxx g x vx initiated by ibragimov et al j math phys 32 2988 1991 is continued the preliminary group classification of these equations with respect to a countable dimensional subalgebra of their equivalence algebra is presented this subalgebra is an extension of the 10 dimensional subalgebra considered by ibragimov et al the result of the work is a table of nonequivalent equations possessing an additional symmetry

finite element approximation of nematic liquid crystal flows using a saddle point structure in this work we propose finite element schemes for the numerical approximation of nematic liquid crystal flows based on a saddle point formulation of the director vector sub problem it introduces a lagrange multiplier that allows to enforce the sphere condition in this setting we can consider the limit problem without penalty and the penalized problem using a ginzburg landau penalty function in a unified way further the resulting schemes have a stable behavior with respect to the value of the penalty parameter a key difference with respect to the existing schemes two different methods have been considered for the time integration first we have considered an implicit algorithm that is unconditionally stable and energy preserving the linearization of the problem at every time step value can be performed using a quasi newton method that allows to decouple fluid velocity and director vector computations for every tangent problem then we have designed a linear semi implicit algorithm i e it does not involve nonlinear iterations and proved that it is unconditionally stable verifying a discrete energy inequality finally some numerical simulations are provided

mizoguchi takahashi s type fixed point theorem recently eldred a a eldred j anuradha p veeramani on equivalence of generalized multi valued contactions and nadler s fixed point theorem j math anal appl 2007 doi 10 1016 j jmaa 2007 01 087 claimed that nadler s s b nadler jr multivalued contraction mappings pacific j math 30 1969 475 488 fixed point theorem is equivalent to mizoguchi takahashi s n mizoguchi w takahashi fixed point theorems for multivalued mappings on complete metric space j math anal appl 141 1989 177 188 fixed point theorem very recently suzuki t suzuki mizoguchi takahashi s fixed point theorem is a real generalization of nadler s j math anal appl 2007 doi 10 1016 j jmaa 2007 08 022 produced an example to disprove their claim and showed that mizoguchi takahashi s fixed point theorem is a real generalization of nadler s fixed point theorem we refine generalize mizoguchi takahashi s fixed point theorem our result improves a recent result by klim and wadowski d klim d wardowski fixed point theorems for set valued contractions in complete metric spaces j math anal appl 334 1 2007 132 139 and extends hicks and rhoades t l hicks b e rhoades a banach type fixed point theorem math japonica 24 1979 327 330 fixed point theorem to multivalued maps

a partitioned scheme for fluid composite structure interaction problems we present a benchmark problem and a loosely coupled partitioned scheme for fluid structure interaction with composite structures the benchmark problem consists of an incompressible viscous fluid interacting with a structure composed of two layers a thin elastic layer with mass which is in contact with the fluid and modeled by the koiter membrane shell equations and a thick elastic layer with mass modeled by the equations of linear elasticity an efficient modular partitioned operator splitting scheme is proposed to simulate solutions to the coupled nonlinear fsi problem without the need for sub iterations at every time step an energy estimate associated with unconditional stability is derived for the fully nonlinear fsi problem defined on moving domains two instructive numerical benchmark problems are presented to test the performance of numerical fsi schemes involving composite structures it is shown numerically that the proposed scheme is at least first order accurate both in time and space this work reveals a new physical property of fsi problems involving thin interfaces with mass the inertia of the thin fluid structure interface regularizes solutions to the full fsi problem

explicit robin neumann schemes for the coupling of incompressible fluids with thin walled structures abstract we introduce a class of explicit coupling schemes for the numerical solution of fluid structure interaction problems involving a viscous incompressible fluid and a general thin walled structure e g including damping and non linear behavior the fundamental ingredient in these methods is a parameter free explicit robin interface condition for the fluid which enables the fluid solid splitting through appropriate extrapolations of the solid velocity and fluid stress on the interface the resulting solution procedures are genuinely partitioned stability and error estimates are provided for all the variants depending on the extrapolations using energy arguments within a representative linear setting in particular we show that one of them simultaneously yields added mass free unconditional stability and optimal first order time accuracy a comprehensive numerical study involving different examples from the literature supports the theory

preliminary group classification of equations vtt f x vx vxx g x vx a classification is given of equations vtt f v vx vxx g x vx admitting an extension by one of the principal lie algebra of the equation under consideration the paper is one of few applications of a new algebraic approach to the problem of group classification the method of preliminary group classification the result of the work is a wide class of equations summarized in table ii

a convergence study of a new partitioned fluid structure interaction algorithm based on fictitious mass and damping we develop analyze and validate a new method for simulating fluid structure interactions fsis which is based on fictitious mass and fictitious damping in the structure equation we employ a partitioned method for the fluid and structure motions in conjunction with sub iteration and aitken relaxation in particular the use of such fictitious parameters requires sub iterations in order to reduce the induced error in addition to the local temporal truncation error to this end proper levels of tolerance for terminating the sub iteration procedure have been obtained in order to recover the formal order of temporal accuracy for the coupled fsi problem these fictitious terms have a significant effect leading to better convergence rate and hence substantially smaller number of sub iterations through analysis we identify the proper range of these parameters which we then verify by corresponding numerical tests we implement the method in the context of spectral element discretization which is more sensitive than low order methods to numerical instabilities arising in the explicit fsi coupling however the method we present here is simple and general and hence applicable to fsi based on any other discretization we demonstrate the effectiveness of the method in applications involving 2d vortex induced vibrations viv and in 3d flexible arteries with structural density close to blood density we also present 3d results for a patient specific aneurysmal flow under pulsatile flow conditions examining in particular the sensitivity of the results on different values of the fictitious parameters

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

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

a new partitioned staggered scheme for flexible multibody interactions with strong inertial effects abstract in generic flexible multibody interaction problems the differences in the mass damping and stiffness of the bodies can be very large of particular interest is a small change in the higher mass structure which triggers relatively large inertial effects on the lower mass structure in this work a new partitioned staggered time integration scheme is developed and its applicability is extended to the problems involving low mass ratios in coupled multibody problems the equation of motion of body with larger inertia is integrated by implicit newmark method whereas the smaller inertia is solved using a robust self starting explicit integration procedure the coupled formulation includes explicit correction terms that adjusts the amount of interfacial velocity which plays a key role in the stability and accuracy of the simulations a wide range of mass 10 6 10 1 damping and stiffness 10 3 10 3 ratios are chosen to assess the stability and accuracy characteristics of the proposed scheme the closed form expressions for the coupling parameter have been constructed for both matching and non matching time stepping through the godunov ryabenkii normal mode analysis the stability of the proposed method is observed as a function of the relative properties and temporal discretisation of the coupled system the time step of the heavier body significantly influences the stability more than the lighter body to maintain the staggering error minimal an optimal range of the coupling parameter is identified the error computed with uniform variation in the time step revealed that the present method is more accurate than the existing methods the partitioned method is an energy preserving integration method for the dynamic analysis of multibody systems as a potential application of this scheme flexible multibody problems in the field of offshore engineering viz wave energy converter and floater mooring systems are presented and validated with experimental measurements

stabilized finite element method for the transient navier stokes equations based on a pressure gradient projection abstract in this paper we present a stabilized finite element formulation for the transient incompressible navier stokes equations the main idea is to introduce as a new unknown of the problem the projection of the pressure gradient onto the velocity space and to add to the incompresibility equation the difference between the laplacian of the pressure and the divergence of this new vector field this leads to a pressure stabilization effect that allows the use of equal interpolation for both velocities and pressures in the case of the transient equations we consider the possibility of treating the pressure gradient projection either implicitly or explicity in the first case the number of unknowns of the problem is substantially increased with respect to the standard galerkin formulation nevertheless iterative techniques may be used in order to uncouple the calculation of the pressure gradient projection from the rest of unknowns velocity and pressure when this vector field is treated explicitly the increment of computational cost of the stabilized formulation with respect to the galerkin method is very low we provide a stability estimate for the case of the simple backward euler time integration scheme for both the implicit and the explicit treatment of the pressure gradient projection

generalized robin neumann explicit coupling schemes for incompressible fluid structure interaction stability analysis and numerics we introduce a new class of explicit coupling schemes for the numerical solution of fluid structure interaction problems involving a viscous incompressible fluid and an elastic structure these methods generalize the arguments reported in 13 10 to the case of the coupling with thick walled structures the basic idea lies in the derivation of an intrinsic interface robin consistency at the space semi discrete level using a lumped mass approximation in the structure the fluid solid splitting is then performed through appropriate extrapolations of the solid velocity and stress on the interface based on these methods a new parameter free robin neumann iterative procedure is also proposed for the partitioned solution of implicit coupling a priori energy estimates guaranteeing the added mass free stability of the schemes and the convergence of the iterative procedure are established the accuracy and robustness of the methods are illustrated in several numerical examples

stable loosely coupled type algorithm for fluid structure interaction in blood flow we introduce a novel loosely coupled type algorithm for fluid structure interaction between blood flow and thin vascular walls this algorithm successfully deals with the difficulties associated with the added mass effect which is known to be the cause of numerical instabilities in fluid structure interaction problems involving fluid and structure of comparable densities our algorithm is based on a time discretization via operator splitting which is applied in a novel way to separate the fluid sub problem from the structure elastodynamics sub problem in contrast with traditional loosely coupled schemes no iterations are necessary between the fluid and structure sub problems this is due to the fact that our novel splitting strategy uses the added mass effect to stabilize rather than to destabilize the numerical algorithm this stabilizing effect is obtained by employing the kinematic lateral boundary condition to establish a tight link between the velocities of the fluid and of the structure in each sub problem the stability of the scheme is discussed on a simplified benchmark problem and we use energy arguments to show that the proposed scheme is unconditionally stable due to the crucial role played by the kinematic lateral boundary condition the proposed algorithm is named the kinematically coupled scheme

existence of a weak solution to a nonlinear fluid structure interaction problem modeling the flow of an incompressible viscous fluid in a cylinder with deformable walls we study a nonlinear unsteady moving boundary fluid structure interaction fsi problem arising in modeling blood flow through elastic and viscoelastic arteries the fluid flow which is driven by the time dependent pressure data is governed by two dimensional incompressible navier stokes equations while the elastodynamics of the cylindrical wall is modeled by the one dimensional cylindrical koiter shell model two cases are considered the linearly viscoelastic and the linearly elastic koiter shell the fluid and structure are fully coupled two way coupling via the kinematic and dynamic lateral boundary conditions describing continuity of velocity the no slip condition and the balance of contact forces at the fluid structure interface we prove the existence of weak solutions to the two fsi problems the viscoelastic and the elastic case as long as the cylinder radius is greater than zero the proof is based on a novel semi discrete operator splitting numerical scheme known as the kinematically coupled scheme introduced in guidoboni et al j comput phys 228 18 6916 6937 2009 to numerically solve the underlying fsi problems the backbone of the kinematically coupled scheme is the well known marchuk yanenko scheme also known as the lie splitting scheme we effectively prove convergence of that numerical scheme to a solution of the corresponding fsi problem

local projection stabilization for the oseen problem and its interpretation as a variational multiscale method we propose to apply the recently introduced local projection stabilization to the numerical computation of the oseen equation at high reynolds number the discretization is done by nested finite element spaces using a priori error estimation techniques we prove the convergence of the method the a priori estimates are independent of the local peclet number and give a sufficient condition for the size of the stabilization parameters in order to ensure optimality of the approximation when the exact solution is smooth moreover we show how this method may be cast in the framework of variational multiscale methods we indicate what modeling assumptions must be made to use the method for large eddy simulations

treating inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses finite element approximations of the stokes and navier stokes equations with in homogeneous essential boundary conditions are considered boundary conditions are enforced weakly by introducing lagrange multipliers optimal error estimates including some for the stress vector on the boundary are derived under minimal regularity assumptions on the data particular attention is paid to the analysis of a practical choice of finite element spaces for which the lagrange multiplier calculation uncouples from that for the velocity and pressure the results are also applicable to general second order elliptic systems

parallel multilevel solution of nonlinear shell structures the analysis of large scale nonlinear shell problems asks for parallel simulation approaches one crucial part of effi cient and well scalable parallel fe simulations is the solver for the system of equations due to the inherent suitability for parallelization one is verymuch directed towards preconditioned iterative solvers however thin walled structures discretized byfinite elements lead to ill conditioned system matrices and therefore performance of iterative solvers is generallypoor this situation further deteriorates when the thickness change of the shell is taken into account a preconditioner for this challenging class of problems is presented combining two approaches in a parallel framework the first approach is a mechanicallymotivated improvement called scaled director conditioning sdc and is able to remove the extra ill conditioning that appears with three dimensional shell formulations as compared to formulations that neglect thickness change of the shell it is introduced at the element level and harmonizes well with the second approach utilizing a multilevel algorithm here a hierarchyof coarse grids is generated in a semi algebraic sense using an aggregation concept therebythe complicated and expensive explicit generation of course triangulations can be avoided the formulation of this combined preconditioning approach is given and the effects on the performance of iterative solvers is demonstrated via numerical examples

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

a loosely coupled scheme for the interaction between a fluid elastic structure and poroelastic material abstract we model the interaction between an incompressible viscous fluid thin elastic structure and a poroelastic material the poroelastic material is modeled using the biot s equations of dynamic poroelasticity the fluid elastic structure and the poroelastic material are fully coupled giving rise to a nonlinear moving boundary problem with novel energy estimates we present a modular loosely coupled scheme where the original problem is split into the fluid sub problem elastic structure sub problem and poroelasticity sub problem an energy estimate associated with the stability of the scheme is derived in the case where one of the coupling parameters is equal to zero we present numerical tests where we investigate the effects of the material properties of the poroelastic medium on the fluid flow our findings indicate that the flow patterns highly depend on the storativity of the poroelastic material and cannot be captured by considering fluid structure interaction only

numerical analysis of a linearised fluid structure interaction problem this paper describes the numerical analysis of a time dependent linearised fluid structure interaction problems involving a very viscous fluid and an elastic shell in small displacements for simplicity all changes of geometry are neglected a single variational formulation is proposed for the whole problem and generic discretisation strategies are introduced independently on the fluid and on the structure more precisely the space approximation of the fluid problem is realized by standard mixed finite elements the shell is approximated by dkt finite elements and time derivatives are approximated either by midpoint rules or by backward difference formula using fundamental energy estimates on the continuous problem written in a proper functional space on its discrete equivalent and on an associated error evolution equation we can prove that the proposed variational problem is well posed and that its approximation in space and time converges with optimal order to the continuous solution

an efficient preconditioner for monolithically coupled large displacement fluid structure interaction problems with pseudo solid mesh updates we present a block preconditioner for the efficient solution of the linear systems that arise when employing newton s method to solve monolithically coupled large displacement fluid structure interaction problems in which the update of the moving fluid mesh is performed by the equations of large displacement elasticity following a theoretical analysis of the preconditioner we propose an efficient implementation that yields a solver with near optimal computational cost in the sense that the time for the solution of the linear systems scales approximately linearly with the number of unknowns we evaluate the performance of the preconditioner in selected two and three dimensional test problems

analysis of finite element approximation of evolution problems in mixed form this paper deals with the finite element approximation of evolution problems in mixed form following d boffi f brezzi and l gastaldi math comp 69 2000 pp 121 140 we handle separately two types of problems a model for the first case is the heat equation in mixed form while the time dependent stokes problem fits within the second one for either case we give sufficient conditions for a good approximation in the natural functional spaces the results are not obvious in the first situation in this case the well known conditions for the well posedness and convergence of the corresponding steady problem are not sufficient for the good approximation of the time dependent problem this issue is demonstrated with a numerical counter example and justified analytically

well posedness regularity and convergence analysis of the finite element approximation of a generalized robin boundary value problem in this paper we propose the mathematical and finite element analysis of a second order partial differential equation endowed with a generalized robin boundary condition which involves the laplace beltrami operator by introducing a function space h 1 omega gamma of h 1 omega functions with h 1 gamma traces where gamma subseteq partial omega based on a variational method we prove that the solution of the generalized robin boundary value problem possesses a better regularity property on the boundary than in the case of the standard robin problem we numerically solve generalized robin problems by means of the finite element method with the aim of validating the theoretical rates of convergence of the error in the norms associated to the space h 1 omega gamma

fluid structure interaction with large structural displacements the calculation of viscous flows inside deformable structures faces major difficulties when enforcing the kinematic compatibility at the fluid structure interface and updating the geometry of the domain this paper proposes to overcome these problems by considering the fluid and the structure as a unique continuous medium to be studied in a fixed reference configuration the resulting problem is then split into a fluid and a structural part through an additive decomposition of the space of kinematically admissible test functions this approach treats the structure in a fully lagrangian way and uses an associated arbitrary lagrangian eulerian ale formulation for the fluid such a strategy has three advantages which are detailed in the present paper it can be discretised in time by implicit stable energy conserving time integration schemes and solved by simple iterative uncoupled algorithms the fluid and the structural problems can be approximated by independent finite element spaces the most recent geometrically exact nonlinear shell models can be used for the structures the full strategy has been implemented into an industrial cfd code results obtained in the simulation of industrial hydraulic shock absorbers are presented at the end of the paper

multi valued nonlinear contraction mappings abstract two concepts of nonlinear contractions for multi valued mappings in complete metric spaces are introduced and three fixed point theorems are proved presented theorems are generalizations of very recent fixed point theorems due to klim and wardowski d klim d wardowski fixed point theorems for set valued contractions in complete metric spaces j math anal appl 334 2007 132 139 and feng and liu y feng s liu fixed point theorems for multi valued contractive mappings and multi valued caristi type mappings j math anal appl 317 2006 103 112 as well as a fixed point theorem of mizoguchi and takahashi n mizoguchi w takahash fixed point theorems for multi valued mappings on complete metric spaces j math anal appl 141 1989 177 188 and several others an example is given to show that presented results are genuine generalizations our results are also some contribution to an open problem raised by reich s reich fixed points of contractive functions boll unione mat ital 5 1972 26 42 s reich some fixed point problems atti acad naz lincei 57 1974 194 198 s reich some problems and results in fixed point theory contemp math 21 1983 179 187

on finite element approximations of problems having inhomogeneous essential boundary conditions abstract the analysis and implementation of finite element methods for problems with inhomogeneous essential boundary conditions are considered the results are given for linear second order elliptic partial differential equations and for the nonlinear stationary navier stokes equations for certain easily implemented boundary treatments optimal error estimates and numerical examples are provided for problems posed on polyhedral domains

explicit strategies for incompressible fluid structure interaction problems nitsche type mortaring versus robin robin coupling we discuss explicit coupling schemes for fluid structure interaction problems where the added mass effect is important in this paper we show the close relation between coupling schemes using nitsche s method and a robin robin type coupling in the latter case the method may be implemented either using boundary integrals of the stresses or the more conventional discrete lifting operators recalling the explicit method proposed in comput methods appl mech engrg 198 5 8 766 784 2009 we make the observation that this scheme is stable under a hyperbolic type cfl condition but that optimal accuracy imposes a parabolic type cfl conditions due to the splitting error two strategies to enhance the accuracy of the coupling scheme under the hyperbolic cfl condition are suggested one using extrapolation and defect correction and one using a penalty free non symmetric nitsche method finally we illustrate the performance of the proposed schemes on some numerical examples in two and three space dimensions

a nitsche cut finite element method for the oseen problem with general navier boundary conditions abstract in this work a nitsche based imposition of generalized navier conditions on cut meshes for the oseen problem is presented other methods from literature dealing with the generalized navier condition impose this condition by means of substituting the tangential robin condition in a classical galerkin way these methods work fine for a large slip length coefficient but lead to conditioning and stability issues when it approaches zero we introduce a novel method for the weak imposition of the generalized navier condition which remains well posed and stable for arbitrary choice of slip length including zero the method proposed here builds on the formulation done by juntunen and stenberg 2009 they impose a robin condition for the poisson problem by means of nitsche s method for an arbitrary combination of the dirichlet and neumann parts of the condition the analysis conducted for the proposed method is done in a similar fashion as in massing et al 2018 but is done here for a more general type of boundary condition the analysis proves stability for all flow regimes and all choices of slip lengths also an l 2 optimal estimate for the velocity error is shown which was not conducted in the previously mentioned work a numerical example is carried out for varying slip lengths to verify the robustness and stability of the method with respect to the choice of slip length even though proofs and formulations are presented for the more general case of an unfitted grid method they can easily be reduced to the simpler case of a boundary fitted grid with the removal of the ghost penalty stabilization terms

fluid structure partitioned procedures based on robin transmission conditions in this article we design new partitioned procedures for fluid structure interaction problems based on robin type transmission conditions the choice of the coefficient in the robin conditions is justified via simplified models the strategy is effective whenever an incompressible fluid interacts with a relatively thin membrane as in hemodynamics applications we analyze theoretically the new iterative procedures on a model problem which represents a simplified blood vessel system in particular the robin neumann scheme exhibits enhanced convergence properties with respect to the existing partitioned procedures the theoretical results are checked using numerical experimentation

local projection stabilized method on unsteady navier stokes equations with high reynolds number using equal order interpolation in this paper we propose and analyze a stabilized method for unsteady navier stokes equations with high reynolds number using local projection stabilized method to control spurious oscillations in the velocities due to dominant convection or in the pressure due to the velocity pressure coupling using equal order conforming elements in space and crank nicolson difference in time we derive a fully discrete formulation we prove stability and convergence of the approximate solution the error estimates hold irrespective of the reynolds number provided the exact solution is smooth this result is comparable with the streamline diffusion and continuous interior penalty methods

on some time marching schemes for the stabilized finite element approximation of the mixed wave equation abstract in this paper we analyze time marching schemes for the wave equation in mixed form the problem is discretized in space using stabilized finite elements on the one hand stability and convergence analyses of the fully discrete numerical schemes are presented using different time integration schemes and appropriate functional settings on the other hand we use fourier techniques also known as von neumann analysis in order to analyze stability dispersion and dissipation numerical convergence tests are presented for various time integration schemes polynomial interpolations for the spatial discretization stabilization methods and variational forms to analyze the behavior of the different schemes considered a 1d wave propagation problem is solved

nitsche xfem for the coupling of an incompressible fluid with immersed thin walled structures abstract in this paper we introduce a nitsche xfem method for fluid structure interaction problems involving a thin walled elastic structure lagrangian formalism immersed in an incompressible viscous fluid eulerian formalism the fluid domain is discretized with an unstructured mesh not fitted to the solid mid surface mesh weak and strong discontinuities across the interface are allowed for the velocity and pressure respectively the fluid solid coupling is enforced consistently using a variant of nitsche s method with cut elements robustness with respect to arbitrary interface intersections is guaranteed through suitable stabilization several coupling schemes with different degrees of fluid solid time splitting implicit semi implicit and explicit are investigated a series of numerical test in 2d involving static and moving interfaces illustrates the performance of the different methods proposed

truly monolithic algebraic multigrid for fluid structure interaction the coupling of flexible structures to incompressible fluids draws a lot attention during the last decade many different solution schemes have been proposed in this contribution we concentrate on strong coupling fluid structure interaction by means of monolithic solution schemes therein a newton krylov method is applied to the monolithic set of nonlinear equations such schemes require good preconditioning to be efficient we propose two preconditioners that apply algebraic multigrid techniques to the entire fluid structure interaction system of equations the first is based on a standard block gauss seidel approach where approximate inverses of the individual field blocks are based on a algebraic multigrid hierarchy tailored for the type of the underlying physical problem the second is based on a monolithic coarsening scheme for the coupled system that makes use of prolongation and restriction projections constructed for the individual fields the resulting nonsymmetric monolithic algebraic multigrid method therefore involves coupling of the fields on coarse approximations to the problem yielding significantly enhanced performance

comparisons between reduced order models and full 3d models for fluid structure interaction problems in haemodynamics when modelling the cardiovascular system the effect of the vessel wall on the blood flow has great relevance arterial vessels are complex living tissues and three dimensional specific models have been proposed to represent their behaviour the numerical simulation of the 3d 3d fluid structure interaction fsi coupled problem has high computational costs in terms of required time and memory storage even if many possible solutions have been explored to speed up the resolution of such problem we are far from having a 3d 3d fsi model that can be solved quickly in 3d 3d fsi models two of the main sources of complexity are represented by the domain motion and the coupling between the fluid and the structural part nevertheless in many cases we are interested in the blood flow dynamics in compliant vessels whereas the displacement of the domain is small and the structure dynamics is less relevant in these situations techniques to reduce the complexity of the problem can be used one consists in using transpiration conditions for the fluid model as surrogate for the wall displacement thus allowing problem s solution on a fixed domain another strategy consists in modelling the arterial wall as a thin membrane under specific assumptions figueroa et al 2006 nobile and vergara 2008 instead of using a more realistic but more computationally intensive 3d elastodynamic model using this strategy the dynamics of the vessel motion is embedded in the equation for the blood flow combining the transpiration conditions with the membrane model assumption we obtain an attractive formulation in fact instead of solving two different models on two moving physical domains we solve only a navier stokes system in a fixed fluid domain where the structure model is integrated as a generalized robin condition in this paper we present a general formulation in the boundary conditions which is independent of the time discretization scheme choice and on the stress strain constitutive relation adopted for the vessel wall structure our aim is first to write a formulation of a reduced order model with zero order transpiration conditions for a generic time discretization scheme then to compare a 3d 3d fsi model and a reduced fsi one in two realistic patient specific cases a femoropopliteal bypass and an aorta in particular we are interested in comparing the wall shear stresses in fact this quantity can be used as a risk factor for some pathologies such as atherosclerosis or thrombogenesis more in general we want to assess the accuracy and the computational convenience to use simpler formulations based on reduced order models in particular we show that in the case of small displacements using a 3d 3d fsi linear elastic model or the correspondent reduced order one yields many similar results

kinematic splitting algorithm for fluid structure interaction in hemodynamics abstract in this paper we study a kinematic splitting algorithm for fluid structure interaction problems this algorithm belongs to the class of loosely coupled fluid structure interaction schemes we will present stability analysis for a coupled problem of non newtonian shear dependent fluids in moving domains with viscoelastic boundaries fluid flow is described by the conservation laws with nonlinearities in convective and diffusive terms for simplicity of presentation the structure is modelled by the generalized string equation but the results presented in the paper may be generalized to more complex structure models the arbitrary lagrangian eulerian approach is used in order to take into account moving computational domain numerical experiments including numerical error analysis and comparison of hemodynamic parameters for newtonian and non newtonian fluids demonstrate reliability of the proposed scheme

three dimensional incompressible viscoelasticity in large strains formulation and numerical approximation abstract this paper considers viscoelastic incompressible solids in large deformations a three dimensional thermodynamical model is first proposed it leads to a nonlinear variational problem which is approximated in space by mixed finite elements and in time by a first order implicit scheme this approximation is analyzed on the corresponding linearized problem finally an elimination strategy is described which reduces each step of the discrete problem to the solution of a standard hyperelastic problem to be solved by newton s method

a minimal stabilisation procedure for mixed finite element methods stabilisation methods are often used to circumvent the difficulties associated with the stability of mixed finite element methods stabilisation however also means an excessive amount of dissipation or the loss of nice conservation properties it would thus be desirable to reduce these disadvantages to a minimum we present a general framework not restricted to mixed methods that permits to introduce a minimal stabilising term and hence a minimal perturbation with respect to the original problem to do so we rely on the fact that some part of the problem is stable and should not be modified sections 2 and 3 present the method in an abstract framework section 4 and 5 present two classes of stabilisations for the inf sup condition in mixed problems we present many examples most arising from the discretisation of flow problems section 6 presents examples in which the stabilising terms is introduced to cure coercivity problems

some common fixed point theorems for mappings and multi valued mappings using certain weak conditions of commutativity we prove some common fixed point theorems in complete metrically convex spaces which in turn generalize results due to assad and kirk itoh khan and several others

galerkin finite element methods with symmetric pressure stabilization for the transient stokes equations stability and convergence analysis we consider the stability and convergence analysis of pressure stabilized finite element approximations of the transient stokes equation the analysis is valid for a class of symmetric pressure stabilization operators but also for standard inf sup stable velocity pressure spaces without stabilization provided the initial data are chosen as a specific method dependent ritz projection we get unconditional stability and optimal convergence for both pressure and velocity approximations in natural norms for arbitrary interpolations of the initial data a condition between the space and time discretization parameters has to be verified in order to guarantee pressure stability

incremental displacement correction schemes for incompressible fluid structure interaction in this paper we introduce a class of incremental displacement correction schemes for the explicit coupling of a thin structure with an incompressible fluid these methods enforce a specific robin neumann explicit treatment of the interface coupling we provide a general stability and convergence analysis that covers both the incremental and the non incremental variants their stability properties are independent of the added mass effect the superior accuracy of the incremental schemes with respect to the original non incremental variant is highlighted by the error estimates and then confirmed in a benchmark by numerical experiments

time domain boundary elements for dynamic contact problems abstract this article considers a unilateral contact problem for the wave equation the problem is reduced to a variational inequality for the dirichlet to neumann operator for the wave equation on the boundary which is solved in a saddle point formulation using boundary elements in the time domain as a model problem also a variational inequality for the single layer operator is considered a priori estimates are obtained for galerkin approximations both to the variational inequality and the mixed formulation in the case of a flat contact area where the existence of solutions to the continuous problem is known numerical experiments demonstrate the performance of the proposed mixed method they indicate the stability and convergence beyond flat geometries

stabilization of explicit coupling in fluid structure interaction involving fluid incompressibility abstract in this work we propose a stabilized explicit coupling scheme for fluid structure interaction problems involving a viscous incompressible fluid the coupled discrete formulation is based on nitsche s method with a time penalty term giving l 2 control on the fluid pressure variations at the interface for a linear model problem we prove that the scheme is stable in the energy norm irrespectively of the so called added mass effect namely the fluid structure density ratio and the geometry of the domain numerical experiments in the linear and non linear case show that optimal time accuracy can be obtained by performing one defect correction iteration

fixed points of generalized contractive multi valued mappings in a recent paper n mizoguchi and w takahashi gave a positive answer to the conjecture of s reich concerning the existence of fixed points of multi valued mappings that satisfy a certain contractive condition in this paper we provide an alternative and somewhat more straightforward proof for the theorem of mizoguchi and takahashi also the problems associated with fixed points of weakly contractive multi valued mappings are studied finally we make a few comments that improve other results from their paper j math anal appl 141 1989 177 188

nitsche s method combined with space time finite elements for ale fluid structure interaction problems we propose a weak method for handling the fluid structure interface in finite element fluid structure interaction based on nitsche s method abh math univ hamburg 36 1971 9 we assume transient incompressible newtonian flow and for the structure undamped linear elasticity for the time discretization we use the time continuous energy conserving galerkin method for the structure and for the fluid we employ the time discontinuous galerkin method this means that the velocity becomes piecewise constant on each timestep for the fluid matching the time derivative of the displacements in the solid which is also piecewise constant over a time step we formulate the method and report some numerical examples using space time oriented elements for the fluid in order to mimic lagrangian or ale type simulations

a finite element formulation for the stokes problem allowing equal velocity pressure interpolation abstract in this paper we study a variational formulation of the stokes problem that accommodates the use of equal velocity pressure finite element interpolations the motivation of this method relies on the analysis of a class of fractional step methods for the navier stokes equations for which it is known that equal interpolations yield good numerical results the reason for this turns out to be the difference between two discrete laplacian operators computed in a different manner the formulation of the stokes problem considered here aims to reproduce this effect from the analysis of the finite element approximation of the problem we obtain stability and optimal error estimates using velocity pressure interpolations satisfying a compatibility condition much weaker than the inf sup condition of the standard formulation in particular this condition is fulfilled by the most common equal order interpolations

an asymptotically compatible formulation for local to nonlocal coupling problems without overlapping regions abstract in this paper we design and analyze an explicit partitioned procedure for a 2d dynamic local to nonlocal ltn coupling problem based on a new nonlocal robin type transmission condition the nonlocal subproblem is modeled by the nonlocal heat equation with a finite horizon parameter characterizing the range of nonlocal interactions and the local subproblem is described by the classical heat equation we consider a heterogeneous system where the local and nonlocal subproblems present different physical properties and employ no overlapping region between the two subdomains we first propose a new generalization of classical local neumann type condition by converting the local flux to a correction term in the nonlocal model and show that the proposed neumann type boundary formulation recovers the local case as o 2 in the l norm we then extend the nonlocal neumann type boundary condition to a robin type boundary condition and develop a local to nonlocal coupling formulation with robin dirichlet transmission conditions to stabilize the explicit coupling procedure and to achieve asymptotic compatibility the choice of the coefficient in the robin condition is obtained via amplification factor analysis for the discretized system with coarse grids employing a high order meshfree discretization method in the nonlocal solver and a linear finite element method in the local solver the selection of optimal robin coefficients are verified with numerical experiments on heterogeneous and complicated domains with the developed optimal coupling strategy we numerically demonstrate the coupling framework s asymptotic convergence to the local limit with an o o h rate when there is a fixed ratio between the horizon size and the spatial discretization size h

an efficient interface capturing method for a large collection of interacting bodies immersed in a fluid an efficient method to capture an arbitrary number of fluid structure interfaces in a level set framework is built following ideas introduced for contour capturing in image analysis using only three label maps and two distance functions we succeed in locating and evolving the bodies independently in the whole domain and get the distance between the closest bodies in order to apply a collision force whatever the number of cells is the method is applied to rigid solid bodies in order to compare to the results available in the literature in that case a global penalization model uses the label maps to follow the solid bodies all together without a separate computation of each body velocity numerical simulations are performed in two and three dimensions an application to immersed vesicles is also proposed and shows the capability and efficiency of the method to handle numerical contacts between elastic bodies at low resolution two dimensional simulations of vesicles under various flow conditions are presented

higher order discontinuous galerkin time stepping and local projection stabilization techniques for the transient stokes problem abstract we introduce and analyze discontinuous galerkin time discretizations coupled with continuous finite element methods based on equal order interpolation in space for velocity and pressure in transient stokes problems spatial stability of the pressure is ensured by adding a stabilization term based on local projection we present error estimates for the semi discrete problem after discretization in space only and for the fully discrete problem the fully discrete pressure shows an instability in the limit of small time step length numerical tests are presented which confirm our theoretical results including the pressure instability

complete group classification of a class of nonlinear wave equations preliminary group classification became a prominent tool in the symmetry analysis of differential equations due to the paper by ibragimov torrisi and valenti j math phys 32 2988 2995 1991 10 1063 1 529042 in this paper the partial preliminary group classification of a class of nonlinear wave equations was carried out via the classification of one dimensional lie symmetry extensions related to a fixed finite dimensional subalgebra of the infinite dimensional equivalence algebra of the class under consideration in the present paper we implement the complete group classification of the same class up to both usual and general point equivalence using the algebraic method of group classification this includes the complete preliminary group classification of the class and finding those lie symmetry extensions which are not associated with subalgebras of the equivalence algebra the complete preliminary group classification is based on listing all inequivalent subalgebras of the whole infinite dimensi

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

on equivalence of generalized multi valued contractions and nadler s fixed point theorem we consider two generalizations of nadler s theorem one proved by mizoguchi and takahashi in response to the reich conjecture and another theorem proved by kaneko we show that due to the additional conditions of these theorems the given multi valued map reduces to a multi valued contraction map we prove this result by showing that the orbit of the multi valued map is bounded under the contractive conditions of the two generalizations

fixed point theorems for multivalued mappings on complete metric spaces on obtient une version multivaluee du theoreme de point fixe de caristi qui est equivalent au principe e variationnel du a ekeland on utilise ce resultat pour obtenir un theoreme qui deduit le theoreme de nadler et un theoreme de point fixe commun pour une application a 1 valeur et une application multivaluee

a stabilized nitsche cut finite element method for the oseen problem abstract we provide the numerical analysis for a nitsche based cut finite element formulation for the oseen problem which has been originally presented for the incompressible navier stokes equations by schott and wall 2014 and allows the boundary of the domain to cut through the elements of an easy to generate background mesh the formulation is based on the continuous interior penalty cip method of burman et al 2006 which penalizes jumps of velocity and pressure gradients over inter element faces to counteract instabilities arising for high local reynolds numbers and the use of equal order interpolation spaces for the velocity and pressure since the mesh does not fit the boundary dirichlet boundary conditions are imposed weakly by a stabilized nitsche type approach the addition of cip like ghost penalties in the boundary zone allows to prove that our method is inf supstable and to derive optimal order a priori error estimates in an energy type norm irrespective of how the boundary cuts the underlying mesh all applied stabilization techniques are developed with particular emphasis on low and high reynolds numbers two and three dimensional numerical examples corroborate the theoretical findings finally the proposed method is applied to solve the transient incompressible navier stokes equations on a complex geometry

edge stabilization for the generalized stokes problem a continuous interior penalty method in this note we introduce and analyze a stabilized finite element method for the generalized stokes equation stability is obtained by adding a least squares penalization of the gradient jumps acro

splitting methods based on algebraic factorization for fluid structure interaction we discuss in this paper the numerical approximation of fluid structure interaction fsi problems dealing with strong added mass effect we propose new semi implicit algorithms based on inexact block lu factorization of the linear system obtained after the space time discretization and linearization of the fsi problem as a result the fluid velocity is computed separately from the coupled pressure structure velocity system at each iteration reducing the computational cost we investigate explicit implicit decomposition through algebraic splitting techniques originally designed for the fsi problem this approach leads to two different families of methods which extend to fsi the algebraic pressure correction method and the yosida method two schemes that were previously adopted for pure fluid problems furthermore we have considered the inexact factorization of the fluid structure system as a preconditioner the numerical properties of these methods have been tested on a model problem representing a blood vessel system

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows this note investigates the so called arti cial added mass e ect which is responsible for devastating instabilities within sequentially staggered fluid structure interaction fsi simulations where incompressible fluids are considered a discrete representation of the added mass operatorma is given and instability conditions are evaluated for different temporal discretisation schemes it is proven that for every sequentially staggered scheme and given spatial discretisation of a problem a mass ratio between fluid and structural mass density can be found at which the coupled system becomes unstable the analysis is quite general and does not depend upon the particular spatial discretisation schemes used however here special attention is put on stabilised finite elements employed on the fluid partition numerical investigations further highlight the results

optimal low order finite element methods for incompressible flow abstract in this paper we consider the characterization of optimal stabilisation parameters associated with mixed finite element approximations of stokes flow we show how the optimal parameter in terms of the approximating power of the method is related to the parameter value which minimises the condition number of the associated schur complement system for the pressure field we illustrate the tightness of our abstract theoretical results in the cases of the q 1 p 0 quadrilateral and p 1 p 1 triangular elements

fully decoupled time marching schemes for incompressible fluid thin walled structure interaction in this paper we introduce a class of fully decoupled time marching schemes velocity pressure displacement splitting for the coupling of an incompressible fluid with a thin walled elastic or viscoelastic structure the time splitting combines a projection method in the fluid with a specific robin neumann treatment of the interface coupling a priori energy estimates guaranteeing unconditional stability are established for some of the schemes the accuracy and performance of the methods proposed are illustrated by a thorough numerical study

a second order in time approximation of fluid structure interaction problem we propose and analyze a novel second order in time partitioned method for the interaction between an incompressible viscous fluid and a thin elastic structure the proposed numerical method is based on the crank nicolson discretization scheme which is used to decouple the system into a fluid subproblem and a structure subproblem the scheme is loosely coupled and therefore at every time step each subproblem is solved only once energy and error estimates for a fully discretized scheme using finite element spatial discretization are derived we prove that the scheme is stable under a cfl condition second order convergent in time and optimally convergent in space numerical examples support the theoretically obtained results and demonstrate the applicability of the method to realistic simulations of blood flow