a hybridizable discontinuous galerkin method for stokes flow abstract in this paper we introduce a hybridizable discontinuous galerkin method for stokes flow the method is devised by using the discontinuous galerkin methodology to discretize a velocity pressure gradient formulation of the stokes system with appropriate choices of the numerical fluxes and by applying a hybridization technique to the resulting discretization one of the main features of this approach is that it reduces the globally coupled unknowns to the numerical trace of the velocity and the mean of the pressure on the element boundaries thereby leading to a significant reduction in the size of the resulting matrix moreover by using an augmented lagrangian method the globally coupled unknowns are further reduced to the numerical trace of the velocity only another important feature is that the approximations of the velocity pressure and gradient converge with the optimal order of k 1 in the l 2 norm when polynomials of degree k 0 are used to represent the approximate variables based on the optimal convergence of the hdg method we apply an element by element postprocessing scheme to obtain a new approximate velocity which converges with order k 2 in the l 2 norm for k 1 the postprocessing performed at the element level is less expensive than the solution procedure numerical results are provided to assess the performance of the method

isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

a superconvergent ldg hybridizable galerkin method for second order elliptic problems we identify and study an ldg hybridizable galerkin method which is not an ldg method for second order elliptic problems in several space dimensions with remarkable convergence properties unlike all other known discontinuous galerkin methods using polynomials of degree k 0 for both the potential as well as the flux the order of convergence in l 2 of both unknowns is k 1 moreover both the approximate potential as well as its numerical trace superconverge in l 2 like norms to suitably chosen projections of the potential with order k 2 this allows the application of element by element postprocessing of the approximate solution which provides an approximation of the potential converging with order k 2 in l 2 the method can be thought to be in between the hybridized version of the raviart thomas and that of the brezzi douglas marini mixed methods

a hybridizable discontinuous galerkin method for modeling fluid structure interaction abstract this work presents a novel application of the hybridizable discontinuous galerkin hdg finite element method to the multi physics simulation of coupled fluid structure interaction fsi problems recent applications of the hdg method have primarily been for single physics problems including both solids and fluids which are necessary building blocks for fsi modeling utilizing these established models hdg formulations for linear elastostatics a nonlinear elastodynamic model and arbitrary lagrangian eulerian navier stokes are derived the elasticity formulations are written in a lagrangian reference frame with the nonlinear formulation restricted to hyperelastic materials with these individual solid and fluid formulations the remaining challenge in fsi modeling is coupling together their disparate mathematics on the fluid solid interface this coupling is presented along with the resultant hdg fsi formulation verification of the component models through the method of manufactured solutions is performed and each model is shown to converge at the expected rate the individual components along with the complete fsi model are then compared to the benchmark problems proposed by turek and hron 1 the solutions from the hdg formulation presented in this work trend towards the benchmark as the spatial polynomial order and the temporal order of integration are increased

partitioning strategies for the interaction of a fluid with a poroelastic material based on a nitsche s coupling approach abstract we develop a computational model to study the interaction of a fluid with a poroelastic material the coupling of stokes and biot equations represents a prototype problem for these phenomena which feature multiple facets on one hand it shares common traits with fluid structure interaction on the other hand it resembles the stokes darcy coupling for these reasons the numerical simulation of the stokes biot coupled system is a challenging task the need of large memory storage and the difficulty to characterize appropriate solvers and related preconditioners for the equations at hand are typical shortcomings of classical discretization methods applied to this problem such as the finite element method for spatial discretization and finite differences for time stepping the application of loosely coupled time advancing schemes mitigates these issues because it allows to solve each equation of the system independently with respect to the others at each time step in this work we develop and thoroughly analyze a loosely coupled scheme for stokes biot equations the scheme is based on nitsche s method for enforcing interface conditions once the interface operators corresponding to the interface conditions have been defined time lagging allows us to build up a loosely coupled scheme with good stability properties the stability of the scheme is guaranteed provided that appropriate stabilization operators are introduced into the variational formulation of each subproblem the error of the resulting method is also analyzed showing that splitting the equations pollutes the optimal approximation properties of the underlying discretization schemes in order to restore good approximation properties while maintaining the computational efficiency of the loosely coupled approach we consider the application of the loosely coupled scheme as a preconditioner for the monolithic approach both theoretical insight and numerical results confirm that this is a promising way to develop efficient solvers for the problem at hand

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

fluid structure interaction for non conforming interfaces based on a dual mortar formulation abstract in the present work the problem of fluid structure interaction fsi with independently space discretized fluid and structure fields is addressed in the context of finite elements to be able to deal with non conforming meshes at the fluid structure interface we propose the integration of a dual mortar method into the general fsi framework this method has lately been used successfully to impose interface constraints in other contexts such as finite deformation contact the main focus is set on monolithic coupling algorithms for fsi here in these cases the dual mortar approach allows for the elimination of the additional lagrange multiplier degrees of freedom from the global system by condensation the resulting system matrices have the same block structure as their counterparts for the conforming case and permit the same numerical treatment partitioned dirichlet neumann coupling is also considered briefly and it is shown that the dual mortar approach permits a numerically efficient mapping between fluid and structure quantities at the interface numerical examples demonstrate the efficiency and robustness of the proposed method we present results for a variety of different element formulations for the fluid and the structure field indicating that the proposed method is not limited to any specific formulation furthermore the applicability of state of the art iterative solvers is considered and the convergence behavior is shown to be comparable to standard simulations with conforming discretizations at the interface

finite element approximations for stokes darcy flow with beavers joseph interface conditions numerical solutions using finite element methods are considered for transient flow in a porous medium coupled to free flow in embedded conduits such situations arise for example for groundwater flows in karst aquifers the coupled flow is modeled by the darcy equation in a porous medium and the stokes equations in the conduit domain on the interface between the matrix and conduit beavers joseph interface conditions instead of the simplified beavers joseph saffman conditions are imposed convergence and error estimates for finite element approximations are obtained numerical experiments illustrate the validity of the theoretical results

a stabilized mixed finite element method for darcy flow we develop new stabilized mixed finite element methods for darcy flow stability and an a priori error estimate in the stability norm are established a wide variety of convergent finite elements present themselves unlike the classical galerkin formulation which requires highly specialized elements an interesting feature of the formulation is that there are no mesh dependent parameters numerical tests confirm the theoretical results 2002 elsevier science b v all rights reserved

a hybridizable discontinuous galerkin method for linear elasticity this paper describes the application of the so called hybridizable discontinuous galerkin hdg method to linear elasticity problems the method has three significant features the first is that the only globally coupled degrees of freedom are those of an approximation of the displacement defined solely on the faces of the elements the corresponding stiffness matrix is symmetric positive definite and possesses a block wise sparse structure that allows for a very efficient implementation of the method the second feature is that when polynomials of degree k are used to approximate the displacement and the stress both variables converge with the optimal order of k 1 for any k 0 the third feature is that by using an element by element post processing a new approximate displacement can be obtained that converges at the order of k 2 whenever k 2 numerical experiments are provided to compare the performance of the hdg method with that of the continuous galerkin cg method for problems with smooth solutions and to assess its performance in situations where the cg method is not adequate that is when the material is nearly incompressible and when there is a crack copyright 2009 john wiley sons ltd

algorithms for strong coupling procedures abstract this paper considers algorithms for computing the response of a coupled problem with a partitioned approach one important example treated here is fluid structure interaction fsi often procedures or even software exists to solve each sub problem separately and one wants to couple both this setting seems to allow only the so called weak coupling which is not sufficient for some problems the so called strong coupling is often a totally implicit formulation where the system components are evaluated at the same time level this usually requires an iterative procedure the fsi problem is cast as an abstract differential algebraic equation dae for which the coupling procedures are developed with the partitioned approach one simple and frequently used computational procedure is similar to a block gauss seidel iteration it is shown why this approach may experience difficulties and how they may be circumvented with newton like methods still staying in the partitioned framework the functional and software engineering requirements for the simulation interface are described and analysed some simple coupled example problems demonstrate how the proposed procedures work

an embedded dirichlet formulation for 3d continua this paper presents a new approach for imposing dirichlet conditions weakly on non fitting finite element meshes such conditions also called embedded dirichlet conditions are typically but not exclusively encountered when prescribing dirichlet conditions in the context of the extended finite element method xfem the methods key idea is the use of an additional stress field as the constraining lagrange multiplier function the resulting mixed hybrid formulation is applicable to 1d 2d and 3d problems the method does not require stabilization for the lagrange multiplier unknowns and allows the complete condensation of these unknowns on the element level furthermore only non zero diagonal terms are present in the tangent stiffness which allows the straight forward application of state of the art iterative solvers like algebraic multigrid amg techniques within this paper the method is applied to the linear momentum equation of an elastic continuum and to the transient incompressible navier stokes equations steady and unsteady benchmark computations show excellent agreement with reference values the general formulation presented in this paper can also be applied to other continuous field problems

a face oriented stabilized nitsche type extended variational multiscale method for incompressible two phase flow a novel extended variational multiscale method for incompressible two phase flow is proposed in this approach the level set method which allows for accurately representing complex interface evolutions is combined with an extended finite element method for the fluid field sharp representation of the discontinuities at the interface related to surface tension effects and large material parameter ratios are enabled by this approach to capture the discontinuities jump enrichments are applied for both velocity and pressure field nitsche s method is then used to weakly impose the continuity of the velocity field for a stable formulation on the entire domain residual based variational multiscale terms are supported by appropriate face oriented ghost penalty and fluid stabilization terms in the region of enriched elements both face oriented terms as well as interfacial terms related to nitsche s method are introduced such that it is accounted for viscous and convection dominated transient flows as a result stability and wellconditioned systems are guaranteed independent of the interface location the proposed method is applied to four numerical examples of increasing complexity two dimensional rayleigh taylor instabilities a twodimensional collapsing water column three dimensional rising bubbles as well as a three dimensional bubble coalescence excellent agreement with either analytical solutions or numerical and experimental reference data as well as robustness for all flow regimes is demonstrated for all examples

a taxonomy and comparison of parallel block multi level preconditioners for the incompressible navier stokes equations in recent years considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible navier stokes equations based on preconditioned krylov methods these include physics based methods such as simple and purely algebraic preconditioners based on the approximation of the schur complement all these techniques can be represented as approximate block factorization abf type preconditioners the goal is to decompose the application of the preconditioner into simplified sub systems in which scalable multi level type solvers can be applied in this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the navier stokes system first presented in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations computational methods in applied mechanical engineering 188 2000 505 526 this taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain schur complement operators we then present a parallel computational study that examines the performance of these methods and compares them to an additive schwarz domain decomposition dd algorithm results are presented for two and three dimensional steady state problems for enclosed domains and inflow outflow systems on both structured and unstructured meshes the numerical experiments are performed using mpsalsa a stabilized finite element code

coupling biot and navier stokes equations for modelling fluid poroelastic media interaction the interaction between a fluid and a poroelastic structure is a complex problem that couples the navier stokes equations with the biot system the finite element approximation of this problem is involved due to the fact that both subproblems are indefinite in this work we first design residual based stabilization techniques for the biot system motivated by the variational multiscale approach then we state the monolithic navier stokes biot system with the appropriate transmission conditions at the interface for the solution of the coupled system we adopt both monolithic solvers and heterogeneous domain decomposition strategies different domain decomposition methods are considered and their convergence is analyzed for a simplified problem we compare the efficiency of all the methods on a test problem that exhibits a large added mass effect as it happens in hemodynamics applications

scalable parallel methods for monolithic coupling in fluid structure interaction with application to blood flow modeling we introduce and study numerically a scalable parallel finite element solver for the simulation of blood flow in compliant arteries the incompressible navier stokes equations are used to model the fluid and coupled to an incompressible linear elastic model for the blood vessel walls our method features an unstructured dynamic mesh capable of modeling complicated geometries an arbitrary lagrangian eulerian framework that allows for large displacements of the moving fluid domain monolithic coupling between the fluid and structure equations and fully implicit time discretization simulations based on blood vessel geometries derived from patient specific clinical data are performed on large supercomputers using scalable newton krylov algorithms preconditioned with an overlapping restricted additive schwarz method that preconditions the entire fluid structure system together the algorithm is shown to be robust and scalable for a variety of physical parameters scaling to hundreds of processors and millions of unknowns

an implicit high order hybridizable discontinuous galerkin method for the incompressible navier stokes equations we present an implicit high order hybridizable discontinuous galerkin method for the steady state and time dependent incompressible navier stokes equations the method is devised by using the discontinuous galerkin discretization for a velocity gradient pressure velocity formulation of the incompressible navier stokes equations with a special choice of the numerical traces the method possesses several unique features which distinguish itself from other discontinuous galerkin methods first it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries thereby leading to a significant reduction in the degrees of freedom moreover if the augmented lagrangian method is used to solve the linearized system the globally coupled unknowns become the approximate trace of the velocity only second it provides for smooth viscous dominated problems approximations of the velocity pressure and velocity gradient which converge with the optimal order of k 1 in the l2 norm when polynomials of degree k 0 are used for all components of the approximate solution and third it displays superconvergence properties that allow us to use the above mentioned optimal convergence properties to define an element by element postprocessing scheme to compute a new and better approximate velocity indeed this new approximation is exactly divergence free h div conforming and converges with order k 2 for k 1 and with order 1 for k 0 in the l2 norm moreover a novel and systematic way is proposed for imposing boundary conditions for the stress viscous stress vorticity and pressure which are not naturally associated with the weak formulation of the method this can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of reynolds numbers and for various polynomial degrees

a third order boundary value problem with jumping nonlinearities abstract a third order boundary value problem with jumping nonlinearities is studied some results obtained for second order boundary value problems with jumping nonlinearities are shown to hold also for the third order case

on the peroidic bvp for the forced duffing equation with jumping nonlinearity on etudie le probleme suivant u cu f t u e t p p sur 0 t u t u 0 0 u t u 0 0 c est une fonction arbitraire dans l 1 0 t r c r et l application non lineaire f est a saut

an unfitted nitsche method for incompressible fluid structure interaction using overlapping meshes we consider the extension of the nitsche method to the case of fluid structure interaction problems on unfitted meshes we give a stability analysis for the space semi discretized problem and show how this estimate may be used to derive optimal error estimates for smooth solutions irrespectively of the mesh interface intersection we also discuss different strategies for the time discretization using either fully implicit or explicit coupling loosely coupled schemes some numerical examples illustrate the theoretical discussion

regular trace formula and base change lifting introduction the selberg trace formula has become a major tool in the study of automorphic forms on reductive groups although its underlying principle of computing traces of representations by means of orbital integrals is very simple the standard expressions for this important formula are rather complicated this makes applications hard to accomplish the complexity of the expression for the formula may be due to the choice of truncation made in its proof it would be advantageous to have a simple expression for the formula at least for a set of test functions which is sufficiently large for applications the possibility of its existence was suggested to us by some of kazhdan s striking work on the trace formula see e g the density theorem of kl appendix or the study of lifting in k2 here we derive an asymptotic expression of this nature in the simplest case of gl 2 for test functions with a component which is sufficiently regular with respect to all other components we obtain a simple practical form of the trace formula more precisely if fu is a nonarchimedean local field and m 2 1 is an integer we say that a locally constant function f on gu gl 2 fu which is supported on a compact mod center is m regular if it vanishes outside the open closed subset sm zg g g in gu a z in f x with i a 1 r denotes a uniformizer in f x of gu and its normalized orbital integral f g fu a g 4 g fu is the characteristic function of sm in g14 if f is a global field u is a nonarchimedean place of f and fu f is a product over all places v u of f of smooth compactly supported mod center functionsfv on gv such thatfv is the unit elementf v of the hecke algebra for almost all v then we show that there exists mr0 mo fu such that for any m 2r mo and m regular functionfu f m the regular test functionf f m fu vanishes on the conjugacy classes in

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

a unified stabilized method for stokes and darcy s equations we use the lowest possible approximation order piecewise linear continuous velocities and piecewise constant pressures to compute solutions to stokes equation and darcy s equation applying an edge stabilization term to avoid locking we prove that the formulation satisfies the discrete inf sup condition we prove an optimal a priori error estimate for both problems the formulation is then extended to the coupled case using a nitsche type weak formulation allowing for different meshes in the two subdomains finally we present some numerical examples verifying the theoretical predictions and showing the flexibility of the coupled approach

a posteriori error analysis of an augmented mixed finite element method for darcy flow abstract we develop an a posteriori error analysis of residual type of a stabilized mixed finite element method for darcy flow the stabilized formulation is obtained by adding to the standard dual mixed approach suitable residual type terms arising from darcy s law and the mass conservation equation we derive sufficient conditions on the stabilization parameters that guarantee that the augmented variational formulation and the corresponding galerkin scheme are well posed then we obtain a simple a posteriori error estimator and prove that it is reliable and locally efficient finally we provide several numerical experiments that illustrate the theoretical results and support the use of the corresponding adaptive algorithm in practice

block preconditioners based on approximate commutators this paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible navier stokes equations we consider the pressure convection diffusion preconditioners proposed by kay loghin and wathen siam j sci comput 24 2002 pp 237 256 and silvester elman kay and wathen j comput appl math 128 2001 pp 261 279 numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall efficacy of this methodology a drawback however is that it requires the construction of a convection diffusion operator denoted f p projected onto the discrete pressure space this means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues something often held only by the developers of the model as an alternative we consider automatic ways of computing f p based on purely algebraic considerations the new methods are closely related to the bfbt preconditioner of elman siam j sci comput 20 1999 pp 1299 1316 we use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convection diffusion operators together with methods for computing sparse approximate inverses to generate the required matrix f p automatically we demonstrate that with this strategy the favorable convergence properties of the preconditioning methodology are retained

a new face oriented stabilized xfem approach for 2d and 3d incompressible navier stokes equations abstract we develop a new stabilized xfem based fixed grid approach for the transient incompressible navier stokes equations using cut elements our framework is based on a new flexible strategy to handle several degrees of freedom dofs at nodes required for complex shaped embedded meshes as they occur for fixed grid fluid structure interaction problems boundary conditions on embedded boundaries are imposed weakly using a nitsche type approach ghost penalty terms for velocity and pressure are added for stability reasons and to improve the conditioning of the system matrix the idea of ghost penalties previously developed for stokes problems is extended to the incompressible navier stokes equations by using face oriented fluid stabilizations for both viscous and convective dominated flows furthermore the need for additional terms to enforce boundary conditions for non viscous flows is addressed a detailed numerical study of our stabilized fluid formulation including studies of nitsche s parameter and the ghost penalty parameter shows optimal error convergence and a good system conditioning in the viscous and the convective dominated cases compared to other fixed grid fluid formulations our method is much more accurate and less sensitive to the location of the interface regarding the accuracy and the sensitivity with respect to the interface position a clear improvement could be observed for the viscous and pressure fluxes as well as for the imposition of the boundary condition results of a transient two dimensional high reynolds number flow over a cylinder and the flow over a thick plate in 3d confirm the applicability of the proposed method

isogeometric mortar methods the application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically for the lagrange multiplier two choices of uniformly stable spaces are presented both of them are spline spaces but of a different degree in one case we consider an equal order pairing for which a cross point modification based on a local degree reduction is required in the other case the degree of the dual space is reduced by two compared to the primal this pairing is proven to be inf sup stable without any necessary cross point modification several numerical examples confirm the theoretical results and illustrate additional aspects

inexact accurate partitioned algorithms for fluid structure interaction problems with finite elasticity in haemodynamics in this paper we consider the numerical solution of the three dimensional fluid structure interaction problem in haemodynamics in the case of real geometries physiological data and finite elasticity vessel deformations we study some new inexact schemes obtained from semi implicit approximations which treat exactly the physical interface conditions while performing just one or few iterations for the management of the interface position and of the fluid and structure non linearities we show that such schemes allow to improve the efficiency while preserving the accuracy of the related exact implicit scheme to do this we consider both a simple analytical test case and two real cases of clinical interest in haemodynamics we also provide an error analysis for a simple differential model problem when a bdf method is considered for the time discretization and only few newton iterations are performed at each temporal instant

on the symmetric square unstable local transfer we prove the fundamental lemma for spherical functions with respect to the natural induction lifting frompgl 2 topgl 3 which appears as the unstable counterpart of the stable symmetric square lifting fromsl 2 topgl 3 see iv for an introduction to this project and vi for the final results thus spherical functions onpgl 2 andpgl 3 which correspond to each other by satisfying an elementary representation theoretic relation are shown to have matching orbital integrals the proof of this local statement is based on an application of the global trace formula

a weakly compressible hybridizable discontinuous galerkin formulation for fluid structure interaction problems abstract a scheme for the solution of fluid structure interaction fsi problems with weakly compressible flows is proposed in this work a novel hybridizable discontinuous galerkin hdg method is derived for the discretization of the fluid equations while the standard continuous galerkin cg approach is adopted for the structural problem the chosen hdg solver combines robustness of discontinuous galerkin dg approaches in advection dominated flows with higher order accuracy and efficient implementations two coupling strategies are examined in this contribution namely a partitioned dirichlet neumann scheme in the context of hybrid hdg cg discretizations and a monolithic approach based on nitsche s method exploiting the definition of the numerical flux and the trace of the solution to impose the coupling conditions numerical experiments show optimal convergence of the hdg and cg primal and mixed variables and superconvergence of the postprocessed fluid velocity the robustness and the efficiency of the proposed weakly compressible formulation in comparison to a fully incompressible one are also highlighted on a selection of two and three dimensional fsi benchmark problems

an implicit high order hybridizable discontinuous galerkin method for nonlinear convection diffusion equations in this paper we present hybridizable discontinuous galerkin methods for the numerical solution of steady and time dependent nonlinear convection diffusion equations the methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary applying the newton raphson procedure and the hybridization technique we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every newton iteration the high number of globally coupled degrees of freedom in the discontinuous galerkin approximation is therefore significantly reduced we then extend the method to time dependent problems by approximating the time derivative by means of backward difference formulae when the time marching method is p 1 th order accurate and when polynomials of degree p 0 are used to represent the scalar variable each component of the flux and the approximate trace we observe that the approximations for the scalar variable and the flux converge with the optimal order of p 1 in the l 2 norm finally we apply element by element postprocessing schemes to obtain new approximations of the flux and the scalar variable the new approximate flux which has a continuous interelement normal component is shown to converge with order p 1 in the l 2 norm the new approximate scalar variable is shown to converge with order p 2 in the l 2 norm the postprocessing is performed at the element level and is thus much less expensive than the solution procedure for the time dependent case the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required extensive numerical results are provided to demonstrate the performance of the present method

finite element formulations for hyperelastic transversely isotropic biphasic soft tissues abstract this paper presents a finite element model for the three dimensional 3 d nonlinear analysis of soft hydrated tissues such as articular cartilage in diarthrodial joints under physiologically relevant loading conditions a biphasic continuum description is used to represent the soft tissue as a two phase mixture of incompressible inviscid fluid and a hyperelastic transversely isotropic solid alternate mixed penalty and velocity pressure finite element formulations are used to solve the nonlinear biphasic governing equations including the effects of a strain dependent permeability and a hyperelastic solid phase under finite deformation the resulting first order nonlinear system of equations are discretized in time using an implicit finite difference scheme and solved using the newton raphson method a significant contribution of this work is the implementation and testing of a biphasic description with a transversely isotropic hyperelastic solid phase this description considers a helmholtz free energy function of five invariants of the cauchy green deformation tensor and the preferred direction of the material allowing for asymmetric behavior in tension and compression an exponential form is suggested and a set of material parameters is identified to represent the response of soft tissues in ranges of deformation and stress observed experimentally after demonstrating the behavior of this constitutive model in simple tension and compression a sample problem of unconfined compression is used to further validate the finite element implementation

dual quadratic mortar finite element methods for 3d finite deformation contact mortar finite element methods allow for a flexible and efficient coupling of arbitrary nonconforming interface meshes and are by now quite well established in nonlinear contact analysis in this pa

on the symmetric square twisted trace formula abstract we compute a trace formula for a test function on pgl 3 a twisted by the outer automorphism g j t g 1 j j 0 1 1 1 0 the resulting formula is then compared with trace formulae for h h 0 0 sl 2 and h 1 pgl 2 and matching functions f 0 and f 1 thereof we obtain a trace formula identity which plays a key role in our study of the symmetric square lifting from h a to g a our formulae are remarkably simple due to the introduction of a new concept of a regular function this eliminates the singular and weighted integrals in the trace formulae

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

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

modular vs non modular preconditioners for fluid structure systems with large added mass effect in this article we address the numerical simulation of fluid structure interaction fsi problems featuring large added mass effect we analyze different preconditioners for the coupled system matrix obtained after space time discretization and linearization of the fsi problem the classical dirichlet neumann preconditioner has the advantage of modularity because it allows to reuse existing fluid and structure codes with minimum effort simple interface communication unfortunately its performance is very poor in case of large added mass effects alternatively we consider two non modular approaches the first one consists in preconditioning the coupled system with a suitable diagonal scaling combined with an ilut preconditioner the system is then solved by a krylov method the drawback of this procedure is that the combination of fluid and structure codes to solve the coupled system is not straightforward the second non modular approach we consider is a splitting technique based on an inexact block lu factorization of the linear fsi system the resulting algorithm computes the fluid velocity separately from the coupled pressure structure system at each iteration reducing the computational cost independently of the preconditioner the efficiency of semi implicit algorithms i e those that treat geometric and fluid nonlinearities in an explicit way is highlighted and their performance compared to the one of implicit algorithms all the methods are tested on three dimensional blood vessel systems the algorithm combining the non modular ilut preconditioner with krylov methods proved to be the fastest

studies of refinement and continuity in isogeometric structural analysis we investigate the effects of smoothness of basis functions on solution accuracy within the isogeometric analysis framework we consider two simple one dimensional structural eigenvalue problems and two static shell boundary value problems modeled with trivariate nurbs solids we also develop a local refinement strategy that we utilize in one of the shell analyses we find that increased smoothness that is the k method leads to a significant increase in accuracy for the problems of structural vibrations over the classical c0c0 continuous p method whereas a judicious insertion of c0c0 continuous surfaces about singularities in a mesh otherwise generated by the k method usually outperforms a mesh in which all basis functions attain their maximum level of smoothness we conclude that the potential for the k method is high but smoothness is an issue that is not well understood due to the historical dominance of c0c0 continuous finite elements and therefore further studies are warranted

solution techniques for the fully discretized equations in computation of fluid structure interactions with the space time formulations we provide an overview of the solution techniques we have developed for the fully discretized equations encountered at every time step in computation of fluid structure interactions with the space time techniques these coupled nonlinear equations are generated from the finite element discretization of the governing equations for the fluid mechanics structural mechanics and the motion of the fluid mechanics mesh the fluid mechanics equations are discretized with the deforming spatial domain stabilized space time formulation the mesh motion is governed by the equations of elasticity with the smaller elements stiffened in the finite element formulation the coupled fully discretized equations are solved with the block iterative quasi direct and direct coupling methods we present numerical examples with incompressible flows and membrane and cable structures 2005 elsevier b v all rights reserved

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

on the symmetric square orbital integrals j tg 1j for g in g the elements fi 8 of g f are called stably a conjugate if there is h in g f resp g with h a h1 conjugacy and stable conjugacy is defined analogously for h f and iii f on omitting a in sect 1 we show that the map n described in 1 5 below is a well defined bijection called the norm map from the set of stable a conjugacy classes in g f to the set of stable conjugacy classes in it f we also define a surjection n1 from the subset of this set described in 1 2 3 4 to the set of stable conjugacy classes in hi f in the case 4 n1 actually relates conjugacy classes let be an element of g f the set of eigenvalues of a is of the form 2 1 2 1 1 if 6a 1 put n 5 1 and ni 1 2 if 6a 6 is a non trivial unipotent let n6 and ni be the non trivial unipotent classes 3 if the eigenvalues 2 1 2 1 of 6a 6 are distinct let n6 be the class in h f determined by the eigenvalues 2 2 1 and ni6 the class in hi f with eigenvalues 4 1

remarks and generalizations related to a recent multiplicity result of a lazer and p mckenna we will show here that this result has a natural generalization to the case that u 5 ai to state the result we consider the model problem lu au autel 2 where 0 0 and set t 0 we have a nonlinear eigenvalue problem lu yu au 4 the following proposition is known cf fueik 3 dancer 2 and also lo where the result appears in the formulation given below

fluid structure algorithms based on steklov poincare operators in this paper we review some classical algorithms for fluid structure interaction problems and we propose an alternative viewpoint mutuated from the domain decomposition theory this approach yields preconditioned richardson iterations on the steklov poincare nonlinear equation at the fluid structure interface

nitsche s method for general boundary conditions we introduce a method for treating general boundary conditions in the finite element method generalizing an approach due to nitsche 1971 for approximating dirichlet boundary conditions we use poisson s equations as a model problem and prove a priori and a posteriori error estimates the method is also compared with the traditional galerkin method the theoretical results are verified numerically

cutfem discretizing geometry and partial differential equations we discuss recent advances on robust unfitted finite element methods on cut meshes these methods are designed to facilitate computations on complex geometries obtained for example from computer

on the symmetric square applications of a trace formula in this paper we prove the existence of the symmetric square lifting of admissible and of automorphic representations from the group sl 2 to the group pgl 3 complete local results are obtained relating the character of an sl 2 packet with the twisted character of self contragredient pgl 3 modules our global results relate packets of cuspidal representations of sl 2 with a square integrable component and self contragredient automorphic pgl 3 modules with a component coming from a square integrable one the sharp results which concern sl 2 rather than gl 2 are afforded by the usage of the trace formula the surjectivity and injectivity of the correspondence implies that any self contragredient automorphic pgl 3 module as above is a lift and that the space of cuspidal sl 2 modules with a square integrable component admits multiplicity one theorem and rigidity strong multiplicity one theorem for packets and not for individual representations

a generalized α method for integrating the filtered navier stokes equations with a stabilized finite element method abstract a generalized method is developed and analyzed for linear first order systems the method is then extended to the filtered navier stokes equations within the context of a stabilized finite element method the formulation is studied through the application to laminar flow past a circular cylinder and turbulent flow past a long transverse groove the method is formulated to obtain a second order accurate family of time integrators whose high frequency amplification factor is the sole free parameter such an approach allows the replication of midpoint rule zero damping gear s method maximal damping or anything in between

an artificial compressibility flux for the discontinuous galerkin solution of the incompressible navier stokes equations discontinuous galerkin dg methods have proved to be well suited for the construction of robust high order numerical schemes on unstructured and possibly nonconforming grids for a variety of problems their application to the incompressible navier stokes ins equations has also been recently considered although the subject is far from being fully explored in this work we propose a new approach for the dg numerical solution of the ins equations written in conservation form the inviscid numerical fluxes both in the continuity and in the momentum equation are computed using the values of velocity and pressure provided by the exact solution of the riemann problem associated with a local artificial compressibility perturbation of the equations unlike in most of the existing methods artificial compressibility is here introduced only at the interface flux level therefore resulting in a consistent discretization of the ins equations irrespectively of the amount of artificial compressibility introduced the discretization of the viscous term follows the well established dg scheme named br2 the performance and the accuracy of the method are demonstrated by computing the kovasznay flow and the two dimensional lid driven cavity flow for a wide range of reynolds numbers and for various degrees of polynomial approximation

a stabilized mixed finite element method for darcy flow based on a multiscale decomposition of the solution recently masud and hughes proposed a stabilized mixed finite element formulation for darcy flow an interesting feature of this formulation is that there are no mesh dependent parameters in the present work we provide a derivation of this formulation based on a multiscale decomposition of the solution we also extend the work of masud and hughes to three dimensional problems and show the convergence rates for various three dimensional finite elements we also show that this formulation passes three dimensional constant flow patch tests for distorted element geometries i e elements with non constant jacobian robustness of this formulation is illustrated by performing numerical simulations on complex geometries

on the number of solutions of two classes of sturm liouville boundary value problems abstract we employ global bifurcation theory of p h rabinowitz together with sturm comparison theorem to establish multiplicity results for some classes of sturm liouville boundary value problems

a poroelastic immersed boundary method with applications to cell biology the immersed boundary method is a widely used mixed eulerian lagrangian framework for simulating the motion of elastic structures immersed in viscous fluids in the traditional immersed boundary method the fluid and structure move with the same velocity field in this work a model based on the immersed boundary method is presented for simulating poroelastic media in which the fluid permeates a porous elastic structure of small volume fraction that moves with its own velocity field two distinct methods for calculating elastic stresses are presented and compared the methods are validated on a radially symmetric test problem by comparing with a finite difference solution of the classical equations of poroelasticity finally two applications of the modeling framework to cell biology are provided cellular blebbing and cell crawling it is shown that in both examples poroelastic effects are necessary to explain the relevant mechanics

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

stabilized continuous and discontinuous galerkin techniques for darcy flow we design stabilized methods based on the variational multiscale decomposition of darcy s problem a model for the subscales is designed by using a heuristic fourier analysis this model involves a characteristic length scale that can go from the element size to the diameter of the domain leading to stabilized methods with different stability and convergence properties these sta bilized methods mimic the different possible functional settings of the continuous problem the optimal method depends on the velocity and pressure approximation order they also involve a subgrid projector that can be either the identity when applied to finite element residuals or can have an image orthogonal to the finite element space in pa rticular we have designed a new stabilized method that allows the use of piecewise constant pressures we consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations all these methods have been analyzed proving stability and convergence results in some cases duality arguments have been used to obtain error bounds in the l 2 norm

unified stabilized finite element formulations for the stokes and the darcy problems in this paper we propose stabilized finite element methods for both stokes and darcy s problems that accommodate any interpolation of velocities and pressures apart from the interest of this fact the important issue is that we are able to deal with both problems at the same time in a completely unified manner in spite of the fact that the functional setting is different concerning the stabilization formulation we discuss the effect of the choice of the length scale appearing in the expression of the stabilization parameters both in what refers to stability and to accuracy this choice is shown to be crucial in the case of darcy s problem as an additional feature of this work we treat two types of stabilized formulations showing that they have a very similar behavior

on the symmetric square definitions and lemmas we define the symmetric square lifting for admissible and automorphic representations from the group h h 0 sl 2 to the group g pgl 3 and derive its basic properties this lifting is defined by means of shintani character relations the definition is suggested by the computation of orbital integrals stable and unstable in our on the symmetric square orbital integrals math ann 279 1987 173 193 it is compatible with dual group homomorphisms 0 h g and 1 h 1 g where h 1 pgl 2 the lifting is proven for induced trivial and special representations and both spherical functions and orthogonality relations of characters are studied

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

nonautonomous boundary value problems with jumping nonlinearities soit f 0 1 r r une fonction continue continument differentiable par rapport au second argument sous l hypothese que les limites lim s f x s s x lim s f x s x x existent et sont uniformes sur 0 1 on etudie le probleme aux valeurs limites u f x u th u 0 u 1 0 ou t est un nombre reel et h une fonction continue pour f fixee on etudie le nombre de solutions

a superconvergent hybridisable discontinuous galerkin method for linear elasticity this is the peer reviewed version of the following article sevilla r giacomini m karkoulias a huerta a a superconvergent hybridisable discontinuous galerkin method for linear elasticity international journal for numerical methods in engineering 12 octubre 2018 vol 116 num 2 p 91 116 which has been published in final form at https doi org 10 1002 nme 5916 this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

facsi a block parallel preconditioner for fluid structure interaction in hemodynamics modeling fluid structure interaction fsi in the vascular system is mandatory to reliably compute mechanical indicators in vessels undergoing large deformations in order to cope with the computational complexity of the coupled 3d fsi problem after discretizations in space and time a parallel solution is often mandatory in this paper we propose a new block parallel preconditioner for the coupled linearized fsi system obtained after space and time discretization we name it facsi to indicate that it exploits the factorized form of the linearized fsi matrix the use of static condensation to formally eliminate the interface degrees of freedom of the fluid equations and the use of a simple preconditioner for saddle point problems facsi is built upon a block gauss seidel factorization of the fsi jacobian matrix and it uses ad hoc preconditioners for each physical component of the coupled problem namely the fluid the structure and the geometry in the fluid subproblem after operating static condensation of the interface fluid variables we use a simple preconditioner on the reduced fluid matrix moreover to efficiently deal with a large number of processes facsi exploits efficient single field preconditioners e g based on domain decomposition or the multigrid method we measure the parallel performances of facsi on a benchmark cylindrical geometry and on a problem of physiological interest namely the blood flow through a patient specific femoropopliteal bypass we analyze the dependence of the number of linear solver iterations on the cores count scalability of the preconditioner and on the mesh size optimality c 2016 elsevier inc all rights reserved

two finite element approaches for darcy and darcy brinkman flow through deformable porous media mixed method vs nurbs based isogeometric continuity abstract porous media problems with coupled finite deformations and flow are still quite challenging if large deformations occur a non homogeneous porosity field is almost inevitable as the porosity can vary due to deformation or due to flow and pore pressure from a numerical point of view this significantly changes the mathematical description of the problem in this contribution we will analyze the consequences of the occurrence of porosity gradients on the finite element formulations of saturated two phase porous media for darcy and darcy brinkman flow we will present two approaches which are capable of fulfilling the numerical requirements for convergence the first will be an approach based on nurbs shape functions the second is a mixed formulation including the porosity as primary solution field in addition we will propose a method to include no penetration or no slip constraints into the monolithic system using a dual lagrange multiplier method the potential of the proposed methods for solving porous flow structure interaction undergoing large deformations will be illustrated by numerical examples

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

the immersed fictitious element method for fluid structure interaction volumetric consistency compressibility and thin members a weak form and an implementation are given for fluid structure interaction by the immersed fictitious element method for compressible fluids the weak form is applicable to models where the fluid is described by eulerian coordinates while the solid is described by lagrangian coordinates which suits their intrinsic characteristics a unique feature of the method is the treatment of the fictitious fluid by a lagrangian description which simplifies the interface conditions methods for enforcing volumetric consistency between the fluid and solid and treating thin members are given although a compressible viscous fluid is considered here the new developments can be applied to incompressible fluids

a temporal consistent monolithic approach to fluid structure interaction enabling single field predictors we present a monolithic approach to large deformation fluid structure interaction fsi problems that allows for choosing fully implicit single step and single stage time integration schemes in the structure and fluid field independently and hence is tailored to the needs of the individual field the independent choice of time integration schemes is achieved by temporal consistent interpolation of the interface traction to reduce computational costs we introduce the possibility of field specific predictors in both structure and fluid field these predictors act on the single fields only possible violations of the interface coupling conditions during the predictor step are dealt with within the monolithic solution procedure we present full detail of such a generalized monolithic solution procedure which is fully consistent in its non conforming temporal and spatial discretization the incorporated mortar approach allows for non matching spatial discretizations of the fluid and solid domain at the fsi interface and is fully integrated in the resulting monolithic system of equations the method is applied to a variety of numerical examples thereby temporal convergence rates the special role of essential boundary conditions at the fluid structure interface and the positive effect of predictors are demonstrated and discussed emphasis is put on the comparison of different time integration schemes in fluid and structure field for what the achieved freedom of choice of time integrators is fully exploited

efficient quadrature for nurbs based isogeometric analysis abstract we initiate the study of efficient quadrature rules for nurbs based isogeometric analysis a rule of thumb emerges the half point rule indicating that optimal rules involve a number of points roughly equal to half the number of degrees of freedom or equivalently half the number of basis functions of the space under consideration the half point rule is independent of the polynomial order of the basis efficient rules require taking into account the precise smoothness of basis functions across element boundaries several rules of practical interest are obtained and a numerical procedure for determining efficient rules is presented we compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics the new rules represent improvements over those used previously in isogeometric analysis

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

a nitsche based cut finite element method for the coupling of incompressible fluid flow with poroelasticity abstract the focus of this contribution is the numerical treatment of interface coupled problems concerning the interaction of incompressible fluid flow and permeable elastic structures the main emphasis is on extending the range of applicability of available formulations on especially three aspects these aspects are the incorporation of a more general poroelasticity formulation the use of the cut finite element method cutfem to allow for large interface motion and topological changes of the fluid domain and the application of a novel nitsche based approach to incorporate the beavers joseph saffmann tangential interface condition this last aspect allows one to extend the practicable range of applicability of the proposed formulation down to very low porosities and permeabilities which is important in several examples in application different aspects of the presented formulation are analyzed in a numerical example including spatial convergence the sensitivity of the solution to the nitsche penalty parameters varying porosities and permeabilities and a varying beavers joseph interface model constant finally a numerical example analyzing the fluid induced bending of a poroelastic beam provides evidence of the general applicability of the presented approach

volumetric coupling approaches for multiphysics simulations on non matching meshes summary in finite element analysis of volume coupled multiphysics different meshes for the involved physical fields are often highly desirable in terms of solution accuracy and computational costs we present a general methodology for volumetric coupling of different meshes within a monolithic solution scheme a straightforward collocation approach is compared to a mortar based method for nodal information transfer for the latter dual shape functions based on the biorthogonality concept are used to build the projection matrices thus further reducing the evaluation costs we give a detailed explanation of the integration scheme and the construction of dual shape functions for general first order and second order langrangian finite elements within the mortar method as well as an analysis of the conservation properties of the projection operators moreover possible incompatibilities due to different geometric approximations of curved boundaries are discussed numerical examples demonstrate the flexibility of the presented mortar approach for arbitrary finite element combinations in two and three dimensions and its applicability to different multiphysics coupling scenarios copyright 2016 john wiley sons ltd

estimation of penalty parameters for symmetric interior penalty galerkin methods this paper presents computable lower bounds of the penalty parameters for stable and convergent symmetric interior penalty galerkin methods in particular we derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements numerical examples in all dimensions and for different polynomial degrees are presented we show the numerical effects of loss of coercivity

an accurate robust and easy to implement method for integration over arbitrary polyhedra application to embedded interface methods we present an accurate method for the numerical integration of polynomials over arbitrary polyhedra using the divergence theorem the method transforms the domain integral into integrals evaluated over the facets of the polyhedra the necessity of performing symbolic computation during such transformation is eliminated by using one dimensional gauss quadrature rule the facet integrals are computed with the help of quadratures available for triangles and quadrilaterals numerical examples in which the proposed method is used to integrate the weak form of the navier stokes equations in an embedded interface method eim are presented the results show that our method is as accurate and generalized as the most widely used volume decomposition based methods moreover since the method involves neither volume decomposition nor symbolic computations it is much easier for computer implementation also the present method is more efficient than other available integration methods based on the divergence theorem efficiency of the method is also compared with the volume decomposition based methods and moment fitting methods to our knowledge this is the first article that compares both accuracy and computational efficiency of methods relying on volume decomposition and those based on the divergence theorem

unified computational framework for the efficient solution of n field coupled problems with monolithic schemes abstract in this paper we propose and evaluate the performance of a unified computational framework for preconditioning systems of linear equations resulting from the solution of coupled problems with monolithic schemes the framework is composed by promising application specific preconditioners presented previously in the literature with the common feature that they are able to be implemented for a generic coupled problem involving an arbitrary number of fields and to be used to solve a variety of applications the first selected preconditioner is based on a generic block gauss seidel iteration for uncoupling the fields and standard algebraic multigrid amg methods for solving the resulting uncoupled problems the second preconditioner is based on the semi implicit method for pressure linked equations simple which is extended here to deal with an arbitrary number of fields and also results in uncoupled problems that can be solved with standard amg finally a more sophisticated preconditioner is considered which enforces the coupling at all amg levels in contrast to the other two techniques which resolve the coupling only at the finest level our purpose is to show that these methods perform satisfactory in quite different scenarios apart from their original applications to this end we consider three very different coupled problems thermo structure interaction fluid structure interaction and a complex model of the human lung numerical results show that these general purpose methods are efficient and scalable in this range of applications

a high order semi explicit discontinuous galerkin solver for 3d incompressible flow with application to dns and les of turbulent channel flow abstract we present an efficient discontinuous galerkin scheme for simulation of the incompressible navier stokes equations including laminar and turbulent flow we consider a semi explicit high order velocity correction method for time integration as well as nodal equal order discretizations for velocity and pressure the non linear convective term is treated explicitly while a linear system is solved for the pressure poisson equation and the viscous term the key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under resolved high reynolds number flows as well as small time steps this penalty method is similar to the grad div stabilization widely used in continuous finite elements we further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations the solver is specifically designed for large scale computations through matrix free linear solvers including efficient preconditioning strategies and tensor product elements which have allowed us to scale this code up to 34 4 billion degrees of freedom and 147 456 cpu cores we validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large eddy simulation of turbulent channel flow at r e 180 as well as 590

robin robin preconditioned krylov methods for fluid structure interaction problems in this work we propose a robin robin preconditioner combined with krylov iterations for the solution of the interface system arising in fluid structure interaction fsi problems it can be seen as a partitioned fsi procedure and in this respect it generalizes the ideas introduced in s badia f nobile c vergara j comput phys 227 2008 7027 7051 we analyze the convergence of gmres iterations with the robin robin preconditioner on a model problem and compare its efficiency with some existing algorithms the method is shown to be very efficient for many challenging fluid structure interaction problems such as those characterized by a large added mass effect or by enclosed fluids in particular the possibility to solve balloon type problems without any special treatment makes this algorithm very appealing compared to the computationally intensive existing approaches c 2009 elsevier b v all rights reserved