a domain decomposition method based on weighted interior penalties for advection diffusion reaction problems we propose a domain decomposition method for advection diffusion reaction equations based on nitsche s transmission conditions the advection dominated case is stabilized using a continuous interior penalty approach based on the jumps in the gradient over element boundaries we prove the convergence of the finite element solutions of the discrete problem to the exact solution and propose a parallelizable iterative method the convergence of the resulting domain decomposition method is proved and this result holds true uniformly with respect to the diffusion parameter the numerical scheme that we propose here can thus be applied straightforwardly to diffusion dominated advection dominated and hyperbolic problems some numerical examples are presented in different flow regimes showing the influence of the stabilization parameter on the performance of the iterative method and we compare our method with some other domain decomposition techniques for advection diffusion equations

a robust nitsche s formulation for interface problems abstract in this work we propose a novel weighting for the interfacial consistency terms arising in a nitsche variational form we demonstrate through numerical analysis and extensive numerical evidence that the choice of the weighting parameter has a great bearing on the stability of the method consequently we propose a weighting that results in an estimate for the stabilization parameter such that the method remains well behaved in varied settings ranging from the configuration of embedded interfaces resulting in arbitrarily small elements to such cases where a large contrast in material properties exists an important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of a elements with arbitrarily small volume fractions b large material heterogeneities and c both large heterogeneities as well as arbitrarily small elements we then highlight the accuracy and efficiency of the proposed formulation through numerical examples focusing particular attention on interfacial quantities of interest

imposing dirichlet boundary conditions in the extended finite element method this paper is devoted to the imposition of dirichlet type conditions within the extended finite element method x fem this method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces imposing neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh observed numerically in earlier papers on the contrary much less work has been devoted to dirichlet boundary conditions for the x fem or the limiting case of stiff boundary conditions in this paper we introduce a strategy to impose dirichlet boundary conditions while preserving the optimal rate of convergence the key aspect is the construction of the correct lagrange multiplier space on the boundary as an application we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding rtm process while avoiding remeshing the case of inner conditions is also discussed as well as two important practical cases material interfaces and phase transformation front capturing copyright 2006 john wiley sons ltd

a quadratic convergence yielding iterative method for the implementation of lavrentiev regularization method for ill posed equations george and elmahdy 2012 considered an iterative method which converges quadratically to the unique solution is a nonlinear monotone operator defined on a real hilbert space x the convergence analysis of the method was based on a majorizing sequence in this paper we are concerned with the problem of expanding the applicability of the method considered by george and elmahdy 2012 by weakening the restrictive conditions imposed on the radius of the convergence ball and also by weakening the popular lipschitz type hypotheses considered in earlier studies such as george and elmahdy 2012 mahale and nair 2009 mathe and perverzev 2003 nair and ravishankar 2008 semenova 2010 and tautanhahn 2002 we show that the adaptive scheme considered by perverzev and schock 2005 for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate in the concluding section the method is applied to numerical solution of the inverse gravimetry problem

cutfem topology optimization of 3d laminar incompressible flow problems abstract this paper studies the characteristics and applicability of the cutfem approach of burman et al 2014 as the core of a robust topology optimization framework for 3d laminar incompressible flow and species transport problems at low reynolds numbers r e 200 cutfem is a methodology for discretizing partial differential equations on complex geometries by immersed boundary techniques in this study the geometry of the fluid domain is described by an explicit level set method where the parameters of a discretized level set function are defined as functions of the optimization variables the fluid behavior is modeled by the incompressible navier stokes equations species transport is modeled by an advection diffusion equation the governing equations are discretized in space by a generalized extended finite element method face oriented ghost penalty terms are added for stability reasons and to improve the conditioning of the system the boundary conditions are enforced weakly via nitsche s method the emergence of isolated volumes of fluid surrounded by solid during the optimization process leads to a singular analysis problem an auxiliary indicator field is modeled to identify these volumes and to impose a constraint on the average pressure numerical results for 3d steady state and transient problems demonstrate that the cutfem analyses are sufficiently accurate and the optimized designs agree well with results from prior studies solved in 2d or by density approaches

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

high order gradients with the shifted boundary method an embedded enriched mixed formulation for elliptic pdes abstract we propose an extension of the embedded boundary method known as shifted boundary method to elliptic diffusion equations in mixed form e g darcy flow heat diffusion problems with rough coefficients etc our aim is to obtain an improved formulation that for linear finite elements is at least second order accurate for both flux and primary variable when either dirichlet or neumann boundary conditions are applied following previous work of nishikawa and mazaheri in the context of residual distribution methods we consider the mixed form of the diffusion equation i e with darcy type operators and introduce an enrichment of the primary variable this enrichment is obtained exploiting the relation between the primary variable and the flux variable which is explicitly available at nodes in the mixed formulation the proposed enrichment mimics a formally quadratic pressure approximation although only nodal unknowns are stored similar to a linear finite element approximation we consider both continuous and discontinuous finite element approximations and present two approaches a non symmetric enrichment which as in the original references only improves the consistency of the overall method and a symmetric enrichment which enables a full error analysis in the classical finite element context combined with the shifted boundary method these two approaches are extended to high order embedded computations and enable the approximation of both primary and flux gradient variables with second order accuracy independently on the type of boundary conditions applied we also show that the the primary variable is third order accurate when pure dirichlet boundary conditions are embedded

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

interior penalty variational multiscale method for the incompressible navier stokes equation monitoring artificial dissipation in this paper we apply the interior penalty method analyzed in e burman and m a fernndez numer math 107 2007 no 1 39 77 mr2317827 2008c 65317 to flows at high reynolds number as a possible measure of solution quality we propose to monitor the ratio between the artificial dissipation induced by the numerical method and the computed physical dissipation for smooth flows we prove that for our method the artificial dissipation serves as an a posteriori error estimator and also that it vanishes at an optimal rate in the case of flows with multiscale features we discuss a heuristic approach relating the decay of the artificial dissipation to the decay rate of the power spectrum some numerical results in two space dimensions are presented examining the relation between the numerical dissipation and solution quality

a stabilised immersed boundary method on hierarchical b spline grids for fluid rigid body interaction with solid solid contact abstract an accurate efficient and robust numerical scheme is presented for the simulation of the interaction between flexibly supported rigid bodies and incompressible fluid flow with topology changes and solid solid contact the solution of the incompressible navier stokes equations is approximated by employing a stabilised formulation on cartesian grids discretised with hierarchical b splines the solid is modelled as a rigid body and represented by linear segments along its boundary kinematic conditions along the fluid rigid body interface are enforced weakly using nitsche s method while ghost penalty operators are employed to avoid excessive ill conditioning of the system matrix arising from small cut cells a staggered scheme is used for resolving the coupled fluid rigid body interaction the contact between moving or moving and fixed solid bodies is modelled with lagrange multipliers the excellent performance and wide range of applicability of the proposed scheme are demonstrated in a number of benchmark tests as well as industrially relevant model problems the examples cover the galloping phenomena particulate flow hydraulic check valves and a model turbine

entiers friables inegalite de turan kubilius et applications we generalize the classical turan kubilus inequality to friable integers and obtain sharp estimates for the involved constant in large subdomains several applications are derived in probabilitic number theory

a symmetric method for weakly imposing dirichlet boundary conditions in embedded finite element meshes summary in this paper we propose a way to weakly prescribe dirichlet boundary conditions in embedded finite element meshes the key feature of the method is that the algorithmic parameter of the formulation which allows to ensure stability is independent of the numerical approximation relatively small and can be fixed a priori moreover the formulation is symmetric for symmetric problems an additional element discontinuous stress field is used to enforce the boundary conditions in the poisson problem additional terms are required in order to guarantee stability in the convection diffusion equation and the stokes problem the proposed method is then easily extended to the transient navier stokes equations copyright 2012 john wiley sons ltd

a numerical method for solving incompressible flow problems with a surface of discontinuity a numerical method for solving problems in which a moving surface of discontinuity separates regions of incompressible flow is presented the method developed is notable in that it does not introduce any artificial smoothing of the change in fluid properties across the surface of discontinuity this results in an increase in accuracy relative to methods which introduce smoothing effects the method was also shown to be fairly versatile problems describing a free surface an immiscible fluid interface and a premixed flame discontinuity were solved there is a limitation however in that the method appears to be most suitable for application to inviscid problems the reason for this limitation and possible approaches toward resolving it are discussed

moyennes de certaines fonctions multiplicatives sur les entiers friables 2 nous evaluons les moyennes indiquees dans le titre sous des hypotheses analytiques concernant la serie de dirichlet associee

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

robust imposition of dirichlet boundary conditions on embedded surfaces summary we develop both stable and stabilized methods for imposing dirichlet constraints on embedded three dimensional surfaces in finite elements the stable method makes use of the vital vertex algorithm of bechet et al bechet et al 2009 int j numer meth engng 78 8 931 954 albeit with a modified set of discontinuous basis functions for the lagrange multiplier field on the embedded surface the stabilized method extends the work of dolbow and harari dolbow and harari 2009 int j numer meth engng 78 2 229 252 to three dimensional surfaces algorithmic and implementational details of both methods are provided several three dimensional benchmark problems are studied to compare and contrast the accuracy of the two approaches the results indicate that both methods yield optimal rates of convergence in various quantities of interest with the primary differences being in the surface flux the utility of the domain integral for extracting accurate surface fluxes is demonstrated for both techniques published by john wiley sons ltd copyright c 2000 john wiley sons ltd

a new approach to wall modeling in les of incompressible flow via function enrichment a novel approach to wall modeling for the incompressible navier stokes equations including flows of moderate and large reynolds numbers is presented the basic idea is that a problem tailored function space allows prediction of turbulent boundary layer gradients with very coarse meshes the proposed function space consists of a standard polynomial function space plus an enrichment which is constructed using spalding s law of the wall the enrichment function is not enforced but allowed in a consistent way and the overall methodology is much more general and also enables other enrichment functions the proposed method is closely related to detached eddy simulation as near wall turbulence is modeled statistically and large eddies are resolved in the bulk flow interpreted in terms of a three scale separation within the variational multiscale method the standard scale resolves large eddies and the enrichment scale represents boundary layer turbulence in an averaged sense the potential of the scheme is shown applying it to turbulent channel flow of friction reynolds numbers from re 590 and up to 5 000 flow over periodic constrictions at the reynolds numbers re h 10 595 and 19 000 as well as backward facing step flow at re h 5 000 all with extremely coarse meshes excellent agreement with experimental and dns data is observed with the first grid point located at up to y 1 500 and especially under adverse pressure gradients as well as in separated flows

numerical simulation of premixed combustion using an enriched finite element method in this paper we present a novel discretization technique for the simulation of premixed combustion based on a locally enriched finite element method fem use is made of the g function approach to premixed combustion in which the domain is divided into two parts one part containing the burned and another containing the unburned gases a level set or g function is used to define the flame interface separating burned from unburned gases the extended finite element method x fem is employed which allows for velocity and pressure fields that are discontinuous across the flame interface lagrange multipliers are used to enforce the correct essential interface conditions in the form of jump conditions across the embedded flame interface a persisting problem with the use of lagrange multipliers in x fem has been the discretization of the lagrange multipliers in this paper the distributed lagrange multiplier technique is adopted we will provide results from a spatial convergence analysis showing good convergence however a small modification of the interface is required to ensure a unique solution finally results are presented from the application of the method to the problems of moving flame fronts the darrieus landau instability and a piloted bunsen burner flame

a semi lagrangean time integration approach for extended finite element methods many computational problems incorporate discontinuities that evolve in time the extendend finite element method xfem is able to represent discontinuities sharply on fixed arbitrary meshes but numerical difficulties arise if these discontinuities move in time we point out that this issue is crucial for interface problems with strongly discontinuous fields on fixed grids a method using semi lagrangean techniques is proposed to adequately handle time integration based on finite difference schemes in the context of the xfem the basic idea is to adapt previous numerical solutions to the current interface position by tracking back virtual lagrangean particles to their previous positions where an appropriate solution can be extrapo lated from a smooth field convergence properties of the proposed method in time and space are thoroughly studied for two one dimensional model problems finally the method is applied to the particularly challenging problem of premixed combustion where the discontinuity appears at the flame front separating the burnt from the unburnt gases a two dimensional and a three dimensional expanding flame demonstrates that the method is sufficiently accurate to retain the properties of the overall nitsche type formulation for interface problems with embedded strong discontinuities

an extended residual based variational multiscale method for two phase flow including surface tension abstract in this study an extended residual based variational multiscale method is proposed for two phase flow including surface tension the extended residual based variational multiscale method combines a residual based form of the variational multiscale method and the extended finite element method xfem by extending the solution spaces it is possible to reproduce discontinuities of the solution fields inside elements intersected by the interface in particular we propose a quasi static enrichment to reproduce time dependent discontinuities kink enrichments of both velocity and pressure as well as kink enrichments of velocity combined with jump enrichments of pressure are considered here to capture the interface between the phases on a fixed grid a level set approach is used a residual based variational multiscale method is employed for computing both flow and interface motion the presented method is tested for various two phase flow examples exhibiting small and large density and viscosity ratios with and without surface tension a two phase couette flow a rayleigh taylor instability a sloshing tank and a three dimensional rising bubble to the best of our knowledge these are the first simulation results for representative time dependent three dimensional two phase flow problems using an extended finite element method stable and accurate results are obtained for all test examples

the shifted boundary method for embedded domain computations part i poisson and stokes problems abstract we propose a new finite element method for embedded domain computations which falls in the category of surrogate approximate boundary algorithms the key feature of the proposed approach is the idea of shifting the location where boundary conditions are applied from the true to the surrogate boundary and to appropriately modify the shifted boundary conditions enforced weakly in order to preserve optimal convergence rates of the numerical solution this process yields a method which in our view is simple efficient and also robust since it is not affected by the small cut cell problem although general in nature here we apply this new concept to the poisson and stokes problems we present in particular the full analysis of stability and convergence for the case of the poisson operator and numerical tests for both the poisson and stokes equations for geometries of progressively higher complexity

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

skeleton stabilized immersogeometric analysis for incompressible viscous flow problems a skeleton stabilized immersogeometric analysis technique is proposed for incompressible viscous flow problems with moderate reynolds numbers the proposed formulation fits within the framework of the finite cell method where essential boundary conditions are imposed weakly using a nitsche type method the key idea of the proposed formulation is to stabilize the jumps of high order derivatives of variables over the skeleton of the background mesh the formulation allows the use of identical finite dimensional spaces for the approximation of the pressure and velocity fields in immersed domains the stability issues observed for inf sup stable discretizations of immersed incompressible flow problems are avoided with this formulation for b spline basis functions of degree formula presented with highest regularity only the derivative of order formula presented has to be controlled which requires specification of only a single stabilization parameter for the pressure field the stokes and navier stokes equations are studied numerically in two and three dimensions using various immersed test cases oscillation free solutions and high order optimal convergence rates can be obtained the formulation is shown to be stable even in limit cases where almost every elements of the physical domain is cut and hence it does not require the existence of interior cells in terms of the sparsity pattern the algebraic system has a considerably smaller stencil than counterpart approaches based on lagrange basis functions this important property makes the proposed skeleton stabilized technique computationally practical to demonstrate the stability and robustness of the method we perform a simulation of fluid flow through a porous medium of which the geometry is directly extracted from 3d formula presented scan data

a new fictitious domain approach inspired by the extended finite element method the purpose of this paper is to present a new fictitious domain approach inspired by the extended finite element method introduced by moes dolbow and belytschko in internat j numer methods engrg 46 1999 pp 131 150 an optimal method is obtained thanks to an additional stabilization technique some a priori estimates are established and numerical experiments illustrate different aspects of the method the presentation is made on a simple poisson problem with mixed neumann and dirichlet boundary conditions the extension to other problems or boundary conditions is quite straightforward

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

on strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method we consider a problem stemming from recent models of phase transitions in stimulus responsive hydrogels wherein a sharp interface separates swelled and collapsed phases extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly fit the interface geometry are emphasized attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface we show that as the reciprocal interfacial mobility vanishes it plays the role of a penalty parameter enforcing a pure dirichlet constraint eventually triggering oscillations in the interfacial velocity we also examine alternative formulations employing a lagrange multiplier to enforce this constraint it is shown that the most convenient choice of basis for the lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms suggesting a lack of stability in the discrete formulation under such conditions neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux fortunately the adaptation and use of local domain integral methodologies considerably improves the flux evaluations several example problems are presented to compare and contrast the various techniques and methods

a finite element variational multiscale method for incompressible flow in this paper we present a numerical scheme prove stability existence of solutions uniqueness and convergence of the incompressible navier stokes equations it has the advantage of being defined from strictly algebraic considerations a significant feature of the present method is that the structure of the stabilization term based on the multiscale enrichment and derived from the navier stokes problem itself ample numerical experiments are carried out to confirm the theory and illustrate the effectiveness of the scheme on incompressible fluid

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

the extended finite element method for two phase and free surface flows a systematic study in immiscible two phase flows jumps or kinks are present in the velocity and pressure fields across the interfaces of the two fluids the extended finite element method xfem is able to reproduce such discontinuities within elements robust and accurate interface capturing schemes with no restrictions on the interface topology are thereby enabled this paper investigates different enrichment schemes and time integration schemes within the xfem test cases with and without surface tension on moving or stationary meshes are studied and compared to interface tracking results when possible a particularly useful setting is extracted which is recommended for two phase flows an extension of this formulation for the simulation of free surface flows and of floating objects is proposed

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

the shifted boundary method for embedded domain computations part ii linear advection diffusion and incompressible navier stokes equations abstract we propose a new embedded finite element method for the linear advection diffusion equation and the laminar and turbulent incompressible navier stokes equations the proposed method belongs to the class of surrogate approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary accordingly boundary conditions enforced weakly are appropriately modified to preserve optimal error convergence rates we include the full analysis of stability and convergence of the method in the linear advection diffusion equation and a battery of tests for the case of laminar and turbulent incompressible navier stokes equations we also discuss the conservation properties of the proposed method in all cases

a stabilized non conforming finite element method for incompressible flow in this paper we extend the recently introduced edge stabilization method to the case of non conforming finite element approximations of the linearized navier stokes equation to get stability also

unfitted extended finite elements for composite grids abstract we consider an extended finite elements method to handle the case of composite independent unstructured grids that lead to unfitted meshes in particular we address the case of two overlapped meshes a background and a foreground one where the thickness of the latter is smaller than the elements of the background mesh this situation may lead to elements split into several portions thus generating polyhedral elements we detail the corresponding discrete formulation for the poisson problem with discontinuous coefficients we also provide some technical details for the 3d implementation finally we provide some numerical examples with the aim of showing the effectiveness of the proposed formulation

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

finite element methods with symmetric stabilization for the transient convection diffusion reaction equation we consider implicit and semi implicit time stepping methods for finite element approximations of singularly perturbed parabolic problems or hyperbolic problems we are interested in problems where the advection dominates and stability is obtained using a symmetric weakly consistent stabilization operator in the finite element method several a stable time discretizations are analyzed and shown to lead to unconditionally stable and optimally convergent schemes in particular we show that the contribution from the stabilization leading to an extended matrix pattern may be extrapolated from previous time steps and hence handled explicitly without loss of stability and accuracy a fully explicit treatment of the stabilization term is obtained under a cfl condition

a stabilized nitsche overlapping mesh method for the stokes problem we develop a nitsche based formulation for a general class of stabilized finite element methods for the stokes problem posed on a pair of overlapping non matching meshes by extending the least squares stabilization to the overlap region we prove that the method is stable consistent and optimally convergent to avoid an ill conditioned linear algebra system the scheme is augmented by a least squares term measuring the discontinuity of the solution in the overlap region of the two meshes as a consequence we may prove an estimate for the condition number of the resulting stiffness matrix that is independent of the location of the interface finally we present numerical examples in three spatial dimensions illustrating and confirming the theoretical results

quadrature schemes for arbitrary convex concave volumes and integration of weak form in enriched partition of unity methods abstract quadrature schemes are constructed based on moment fitting equations to integrate polynomials over arbitrary convex concave volumes that arise among others in enriched partition of unity finite element methods epum the building block of the scheme involves the divergence theorem of multivariable calculus which is used to integrate the base functions an efficient and robust point distribution method is proposed and the quadrature weights at the corresponding points are obtained by solving a least squares problem the method is applied initially to integrate given polynomial functions over complex volumes and further to simulate simple three dimensional fluid dynamic problems which involve very complex volumes when solved with epum accuracy of the present quadrature construction scheme is demonstrated by comparing the results with the available exact numerical solutions and efficiency of the method is proved by comparing the computational time with that of the widely used tessellation method

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

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

a nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity in this note we propose a finite element method for incompressible or compressible elasticity problems with discontinuous modulus of elasticity or if compressible poisson s ratio the problem is written on mixed form using p1 continuous displacements and elementwise p0 pressures leading to the possibility of eliminating the pressure beforehand in the compressible case in the incompressible case the method is augmented by a stabilization term penalizing the pressure jumps we show a priori error estimates under certain regularity hypothesis in particular we prove that if the exact solution is sufficiently smooth in each subdomain then the convergence order is optimal

stabilized method of imposing dirichlet boundary conditions using a recovered stress field abstract this paper proposes a new formulation to impose dirichlet boundary conditions on immersed boundary cartesian finite element meshes the method uses a recovered stress field calculated by superconvergent patch recovery to stabilize the lagrange multiplier formulation of the problem the optimal convergence of the method and the convergence of the proposed iterative procedure are demonstrated the proposed method is also suitable for problems with non linear material behavior some numerical examples are included to confirm the theoretical results

on the adaptive selection of the parameter in regularization of ill posed problems we study the possibility of using the structure of the regularization error for a posteriori choice of the regularization parameter as a result a rather general form of a selection criterion is proposed and its relation to the heuristical quasi optimality principle of tikhonov and glasko z vychisl mat mat fiz 4 1964 pp 564 571 and to an adaptation scheme proposed in a statistical context by lepskii theory probab appl 36 1990 pp 454 466 is discussed the advantages of the proposed criterion are illustrated by using such examples as self regularization of the trapezoidal rule for noisy abel type integral equations lavrentiev regularization for nonlinear ill posed problems and an inverse problem of the two dimensional profile reconstruction

nitsche xfem with streamline diffusion stabilization for a two phase mass transport problem we consider an unsteady convection diffusion equation which models the transport of a dissolved species in two phase incompressible flow problems the so called henry interface condition leads to a jump condition for the concentration at the interface between the two phases in a hansbo and p hansbo comput methods appl mech engrg 191 2002 pp 5537 5552 for the purely elliptic stationary case an extended finite element method xfem is combined with a nitsche type method and optimal error bounds are derived these results were extended to the unsteady case in a reusken and t nguyen j fourier anal appl 15 2009 pp 663 683 in the latter paper convection terms are also considered but assumed to be small in many two phase flow applications however convection is the dominant transport mechanism hence there is a need for a stable numerical method for the case of a convection dominated transport equation in this paper we address this topic and study the streamline diffusion sta

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

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

on time integration in the xfem the extended finite element method xfem is often used in applications that involve moving interfaces examples are the propagation of cracks or the movement of interfaces in two phase problems this work focuses on time integration in the xfem the performance of the discontinuous galerkin method in time space time finite elements fes and time stepping schemes are analyzed by convergence studies for different model problems it is shown that space time fe achieve optimal convergence rates special care is required for time stepping in the xfem due to the time dependence of the enrichment functions in each time step the enrichment functions have to be evaluated at different time levels this has important consequences in the quadrature used for the integration of the weak form a time stepping scheme that leads to optimal or only slightly sub optimal convergence rates is systematically constructed in this work copyright 2009 john wiley sons ltd

an extended finite element method lagrange multiplier based approach for fluid structure interaction abstract this paper presents a new fixed grid fluid structure interaction scheme that can be applied to the interaction of most general structures with incompressible flow it is based on an extended finite element method xfem based strategy the extended eulerian fluid field and the lagrangian structural field are partitioned and iteratively coupled using lagrange multiplier techniques for non matching grids the approach allows the simulation of the interaction of thin and bulky structures exhibiting large deformations finally qualitative examples and a benchmark computation demonstrate key features and accuracy of the method

a penalty free nonsymmetric nitsche type method for the weak imposition of boundary conditions in this paper we show that the nonsymmetric version of nitsche s method for the weak imposition of boundary conditions is stable without penalty term for nonconforming elements we prove the same result for the symmetric formulation as well we prove optimal h 1 error estimates and l 2 error estimates that are suboptimal with half an order in h both the pure diffusion and the convection diffusion problems are discussed

continuous interior penalty finite element method for the time dependent navier stokes equations space discretization and convergence this paper focuses on the numerical analysis of a finite element method with stabilization for the unsteady incompressible navier stokes equations incompressibility and convective effects are both stabilized adding an interior penalty term giving l 2 control of the jump of the gradient of the approximate solution over the internal faces using continuous equal order finite elements for both velocities and pressures in a space semi discretized formulation we prove convergence of the approximate solution the error estimates hold irrespective of the reynolds number and hence also for the incompressible euler equations provided the exact solution is smooth

the shifted boundary method for hyperbolic systems embedded domain computations of linear waves and shallow water flows abstract we propose a new computational approach for embedded boundary simulations of hyperbolic systems and in particular the linear wave equations and the nonlinear shallow water equations the proposed approach belongs to the class of surrogate approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary accordingly boundary conditions enforced weakly are appropriately modified to preserve optimal error convergence rates this framework is applied here in the setting of a stabilized finite element method even though other spatial discretization techniques could have been employed accuracy stability and robustness of the proposed method are tested by means of an extensive set of computational experiments for the acoustic wave propagation equations and shallow water equations comparisons with standard weak boundary conditions imposed on body fitted grids which conform to the geometry of the computational domain boundaries are also presented

an unfitted formulation for the interaction of an incompressible fluid with a thick structure via an xfem dg approach a numerical procedure that combines an extended finite element formulation and a discontinuous galerkin technique is presented with the final aim of providing an effective tool for the simulation of three dimensional 3d fluid structure interaction problems in this work we consider a thick structure immersed in a fluid we describe the numerical models and discuss the specific implementation issues arising in three dimensions finally 3d numerical results are provided to show the effectiveness of the approach

on integers free of large prime factors the number l x y of integers y has been given satisfactory estimates in the regions y exp log log x 5 3 6 in the intermediate range only very crude estimates have been obtained so far we close this gap and give an expression which approximates l x y uniformly for x y 2 within a factor 1 o log y log x log y y as an application we derive a simple formula for ti cx y fl x y where 1 c y we also prove a short interval estimate for l x y

continuous interior penalty hp finite element methods for advection and advection diffusion equations a continuous interior penalty hp finite element method that penalizes the jump of the discrete solution across mesh interfaces is introduced error estimates are obtained for first order and advection dominated transport operators the analysis relies on three technical results that are of independent interest an hp inverse trace inequality a local discontinuous to continuous hp interpolation result and hp error estimates for continuous l2 orthogonal projections

finite dimensional realization of a quadratic convergence yielding iterative regularization method for ill posed equations with monotone operators recently jidesh et al 2015 considered a quadratic convergence yielding iterative method for obtaining approximate solution to nonlinear ill posed operator equation f x y where f d f x x is a monotone operator and x is a real hilbert space in this paper we consider the finite dimensional realization of the method considered in jidesh et al 2015 numerical example justifies our theoretical results

variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows we present an les type variational multiscale theory of turbulence our approach derives completely from the incompressible navier stokes equations and does not employ any ad hoc devices such as eddy viscosities we tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows in the calculations we employed linear quadratic and cubic nurbs a dispersion analysis of simple model problems revealed nurbs elements to be superior to classical finite elements in approximating advective and diffusive processes which play a significant role in turbulence computations the numerical results are very good and confirm the viability of the theoretical framework

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

sur l inegalite de turan kubilius friable the exact value of the friable turan kubilius constant is determined when u log x log y tends to a finite limit for the set of y friable integers not exceeding x

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

imposing dirichlet boundary conditions with nitsche s method and spline based finite elements a key challenge while employing non interpolatory basis functions in finite element methods is the robust imposition of dirichlet boundary conditions the current work studies the weak enforcement of such conditions for b spline basis functions with application to both second and fourth order problems this is achieved using concepts borrowed from nitsche s method which is a stabilized method for imposing constraints on surfaces conditions for the stability of the system of equations are derived for each class of problem stability parameters in the nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the dirichlet boundary the approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not through several numerical examples the approach is shown to yield optimal rates of convergence copyright 2010 john wiley sons ltd

projection based reduced order models for a cut finite element method in parametrized domains abstract this work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method such methods and in particular the cutfem method are attractive in the generation of projection based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes the combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems avoiding remeshing as well as the reference domain formulation often used in the reduced order modeling for boundary fitted finite element formulations the resulting novel methodology is presented on linear elliptic and stokes problems together with several test cases to assess its capability the role of a proper extension and transport of embedded solutions to a common background is analyzed in detail

an adaptive fixed mesh ale method for free surface flows in this work we present a fixed mesh ale method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background domain this mesh is successively refined and unrefined at each time step in order to focus the computational effort on the spatial regions where it is required some of the main ingredients of the formulation are the use of an arbitrary lagrangian eulerian formulation for computing temporal derivatives the use of stabilization terms for stabilizing convection stabilizing the lack of compatibility between velocity and pressure interpolation spaces and stabilizing the ill conditioning introduced by the cuts on the background finite element mesh and the coupling of the algorithm with an adaptive mesh refinement procedure suitable for running on distributed memory environments algorithmic steps for the projection between meshes are presented together with the algebraic fractional step approach used for improving the condition number of the linear systems to be solved the method is tested in several numerical examples the expected convergence rates both in space and time are observed smooth solution fields for both velocity and pressure are obtained as a result of the contribution of the stabilization terms finally a good agreement between the numerical results and the reference experimental data is obtained

an analysis of lavrentiev regularization method and newton type process for nonlinear ill posed problems abstract in this paper we consider the lavrentiev regularization method and a modified newton method for obtaining stable approximate solution to nonlinear ill posed operator equations f x y where f d f x x is a nonlinear monotone operator or f x 0 is nonnegative selfadjoint operator defined on a real hilbert space x we assume that only a noisy data y x with y y are available further we assume that frechet derivative f of f satisfies center type lipschitz condition a priori choice of regularization parameter is presented we proved that under a general source condition on x 0 x the error x x n between the regularized approximation x n x 0 x 0 and the solution x is of optimal order in the concluding section the algorithm is applied to numerical solution of the inverse gravimetry problem