on deflation and singular symmetric positive semi definite matrices for various applications it is well known that the deflated iccg is an efficient method for solving linear systems with invertible coefficient matrix we propose two equivalent variants of this deflated iccg which can also solve linear systems with singular coefficient matrix arising from discretization of the discontinuous poisson equation with neumann boundary conditions it is demonstrated both theoretically and numerically that the resulting methods accelerate the convergence of the iterative process moreover in practice the singular coefficient matrix has often been made invertible by modifying the last element since this can be advantageous for the solver however the drawback is that the condition number becomes worse conditioned we show that this problem can completely be remedied by applying the deflation technique with just one deflation vector

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

staggered grids hybrid dual spectral element method for second order elliptic problems application to high order time splitting methods for navier stokes equations abstract this paper is a presentation of a mixed spectral element method based on staggered grids strategy used for the discretization of second order elliptic problems set under hybrid dual formulation a numerical analysis is performed proving the expected optimality and some numerical tests are described to confirm the predicted convergence rates the second part deals with a high order time splitting spectral scheme for the simulation of unsteady incompressible navier stokes flows we show how to use such a solver as a filtering tool to obtain divergence free discrete velocity some computational examples are reported which quantify the improvements induced by such a filter we have observed a strengthening of the stability and of the accuracy

numerical analysis of a finite element projection based vms turbulence model with wall laws abstract this paper deals with the numerical analysis of a finite element projection based vms turbulence model that includes general non linear wall laws only a single mesh and interpolation operators on a virtual coarser mesh are needed to implement the model we include a projection stabilization of pressure to use the same interpolation for velocity and pressure good accuracy is obtained with benchmark turbulent flow problems on coarse grids that justify the interest of this approach also the model solves smooth flows with optimal accuracy

finite element discretization of the stokes and navier stokes equations with boundary conditions on the pressure we consider the stokes and navier stokes equations with boundary conditions of dirichlet type on the velocity on one part of the boundary and involving the pressure on the rest of the boundary we write the variational formulations of such problems next we propose a finite element discretization of them and perform the a priori and a posteriori analysis of the discrete problem some numerical experiments are presented in order to justify our strategy

inflow outflow with dirichlet boundary conditions for pressure in isph in the present work we propose a new algorithm for open boundary treatment in isph in the literature a few models for open boundary conditions are available but most of them are applied to weakly compressible sph wcsph only in our method the inflow outflow is driven by true dirichlet boundary conditions of the projected pressure field we ensure the dirichlet boundary condition by a particle mirroring technique at the open boundary to compute the pressure field this procedure enables us to handle variable inlet velocities across the open boundary the dirichlet boundary conditions are introduced for the projected pressure matrix we apply an error analysis for a hagenpoiseuille flow driven by a pressure gradient and demonstrate the robustness and accuracy with a flow around a cylinder and an oscillating flow where inlet and outlet conditions periodically change additionally a volume flux controller is presented in combination with variable pressure boundary conditions finally the new open boundary treatment is applied to a bubble formation process during gas injection and validated with experimental results

effect of prey refuge on spatiotemporal dynamics of the reaction diffusion system abstract the present investigation deals with the problem of a two dimensional diffusive predator prey model incorporating prey refuge in this investigation we determine the turing space in the spatial domain and perform extensive numerical simulation from both the mathematical and the biological points of view in order to study the effect of diffusion coefficients and other momentous parameters of the system in turing instability furthermore the stability behaviours of the proposed model in the absence of diffusion about the feasible equilibrium points are explored in particular the exact turing domain is delineated in the parameter space various spatial patterns in spatial domain through diffusion driven instability of the present model are depicted and analysed the results indicate that the effect of prey refuge plays a significant role on the control of pattern formation of the populations

higher order quasi monte carlo integration for bayesian pde inversion abstract we analyze combined quasi monte carlo quadrature and finite element approximations in bayesian estimation of solutions to countably parametric operator equations with holomorphic dependence on the parameters as considered in schillings and schwab 2014 such problems arise in numerical uncertainty quantification and in bayesian inversion of operator equations with distributed uncertain inputs such as uncertain coefficients uncertain domains or uncertain source terms and boundary data we show that the parametric bayesian posterior densities belong to a class of weighted bochner spaces of functions of countably many variables with a particular structure of the qmc quadrature weights up to a problem dependent and possibly large finite dimension s product weights can be used and beyond this dimension weighted spaces with so called spod weights recently introduced in dick et al 2014 are used to describe the solution regularity we establish error bounds for higher order quasi monte carlo quadrature for the bayesian estimation based on dick et al 2016 it implies in particular regularity of the parametric solution and of the countably parametric bayesian posterior density in spod smoothness driven product and order dependent weighted spaces of integrand functions this in turn implies that the quasi monte carlo quadrature methods in dick et al 2014 are applicable to these problem classes with dimension independent convergence rates o n 1 p of n point hoqmc approximated bayesian estimates where 0 p 1 depends only on the sparsity class of the uncertain input in the bayesian estimation fast component by component cbc for short construction gantner and schwab 2016 allow efficient deterministic bayesian estimation with up to 1 0 4 parameters

stabilized finite element method for navier stokes equations with physical boundary conditions this paper deals with the numerical approximation of the 2d and 3d navier stokes equations satisfying nonstandard boundary conditions this lays on the finite element discretisation of the corresponding stokes problem which is achieved through a three fields stabilized mixed formulation a priori and a posteriori error bounds are established for the nonlinear problem ascertaining the convergence of the method finally numerical tests are presented including mesh refinement via error indicators

local boundary value problems for the error in fe approximation of non linear diffusion systems in this work we investigate the a posteriori error estimation for a class of non linear multicomponent diffusion operators which includes the stefan maxwell equations the local error indicators for the global error are based on local boundary value problems which are chosen to approximate either the global residual of the finite element approximation or the global linearized error equation using representative numerical examples it is shown that the error indicators based on the latter approach are more accurate for estimating the global error for this problem class as the problem becomes more non linear and can even produce better adaptive mesh refinement amr in addition we propose a new local error indicator for the error in output functionals that is accurate inexpensive to compute and is suitable for amr as demonstrated by numerical examples

nonlinear discontinuous petrov galerkin methods the discontinuous petrov galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper its lowest order version applies to a nonlinear uniformly convex model example and is equivalently characterized as a mixed formulation a reduced formulation and a weighted nonlinear least squares method quasi optimal a priori and reliable and efficient a posteriori estimates are obtained for the abstract nonlinear dpg framework for the approximation of a regular solution the variational model example allows for a built in guaranteed error control despite inexact solve the subtle uniqueness of discrete minimizers is monitored in numerical examples

exact distribution of estimators of parameters in ornstein uhlenbeck processes abstract the exact distribution of the maximum likelihood estimator of the drift damping parameter in a stationary ornstein uhlenbeck or ar 1 process is investigated tables of the distribution function for different levels are given by the help of maple the comparison with old calculations are discussed

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

implicit formulation of material point method for analysis of incompressible materials abstract material point method mpm is a particle based numerical technique with application to solid mechanics problems involving large deformations recently mpm has been successfully applied in many engineering applications including solid and solid fluid interaction problems due to its capability to model complex history dependent material behaviours however the conventional explicit mpm approach exhibits numerical limitations in incompressible or near incompressible material behaviour problems we study an implicit treatment of the pressure in mpm algorithm to simulate material incompressibility avoiding artificial pressure oscillations and significantly small time steps present in the explicit mpm approach applying chorin s projection method to the velocity pressure coupled equations an elliptic equation is solved for the pressure which is then used to solve the divergence free velocity field the results of dam break problem show no spurious pressure oscillations and large time steps compared to the traditional explicit mpm the ability of the present implicit mpm formulation to model incompressible materials will be an additional advantage in simulating materials with complex constitutive equations which exhibit a time dependent memory

comparisons of weakly compressible and truly incompressible algorithms for the sph mesh free particle method in the smoothed particle hydrodynamics sph discretisation method for the navier stokes equations the most widespread method to solve for pressure and mass conservation is the weakly compressible assumption wcsph this includes hydraulics applications and leads to some drawbacks such as severe artificial pressure fluctuations and a limitation to very small time steps related to the wcsph mach number and explicit method this paper presents comparisons of a semi implicit and truly incompressible sph isph algorithm with the classical wcsph method showing how some of the problems encountered in wcsph have been resolved by using isph to simulate incompressible flows mathematical models are presented before describing sph formalism several standard boundary conditions are introduced and special attention is given to tracking the surface particles the lid driven cavity flow re 400 and 1000 is performed as a benchmarking test a bluff body test case a square cylinder in a closed channel re d 20 and 100 based on the cylinder diameter shows that pressure fields extracted from wcsph are very unreliable whereas isph predict pressures and forces in closer agreement with classical finite volume cfd methods dam breaking cases with dry or wet beds downstream are then presented to highlight free surface flow and rapid dynamics effects the wcsph and isph results are generally verified with reference data from experiment and or another numerical method all the comparisons show improvement with isph and good agreement in general

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

consistency stability a priori and a posteriori errors for petrov galerkin methods applied to nonlinear problems in an abstract framework we present a formalism which specifies the notions of consistency and stability of petrov galerkin methods used to approximate nonlinear problems which are in many practical situations strongly nonlinear elliptic problems this formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems this theory is illustrated with the example div k u u c u f in a two dimensional domain with dirichlet boundary conditions

numerical investigation of newtonian and non newtonian multiphase flows using isph method we have presented a multiphase incompressible smoothed particle hydrodynamics method with an improved interface treatment procedure to demonstrate the effectiveness of the interface treatment which can handle multiphase flow problems with high density and viscosity ratios we have modeled several challenging two phase flow problems namely single vortex flow square droplet deformation droplet deformation in shear flow and finally the newtonian bubble rising in viscous and viscoelastic liquids the proposed interface treatment includes the usage of i different smoothing functions in this work cubic spline kernel function for discretizing equations associated with the calculation of the surface tension force while the quintic spline for the discretization of governing equations and the relevant boundary conditions and ii a new discretization scheme for calculating the pressure gradient it is shown that with the application of the improved interface treatment it becomes possible to model multiphase flow problems with the density and viscosity ratios up to 1000 and 100 respectively while using standard projection method the utilization of cubic spline for the continuum surface force model significantly improves the quality of the calculated interface thereby eliminating the interphase particle penetrations and in turn leading to the calculation of more accurate velocity and pressure fields

analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems a fully discrete analysis of the finite element heterogeneous multiscale method for a class of nonlinear elliptic homogenization problems of nonmonotone type is proposed in contrast to previous results obtained for such problems in dimension d leq2 for the h 1 norm and for a semi discrete formulation w e p ming and p zhang j amer math soc 18 2005 no 1 121 156 we obtain optimal convergence results for dimension d leq3 and for a fully discrete method which takes into account the microscale discretization in addition our results are also valid for quadrilateral finite elements optimal a priori error estimates are obtained for the h 1 and l 2 norms improved estimates are obtained for the resonance error and the newton method used to compute the solution is shown to converge numerical experiments confirm the theoretical convergence rates and illustrate the behavior of the numerical method for various nonlinear problems

modeling low reynolds number incompressible flows using sph the method of smoothed particle hydrodynamics sph is extended to model incompressible flows of low reynolds number for such flows modification of the standard sph formalism is required to minimize errors associated with the use of a quasi incompressible equation of state treatment of viscosity state equation kernel interpolation and boundary conditions are described simulations using the method show close agreement with series solutions for couette and poiseuille flows furthermore comparison with finite element solutions for flow past a regular lattice of cylinders shows close agreement for the velocity and pressure fields the sph results exhibit small pressure fluctuations near curved boundaries further improvements to the boundary conditions may be possible which will reduce these errors a similar method to that used here may permit the simulation of other flows at low reynolds numbers using sph further development will be needed for cases involving free surfaces or substantially different equations of state

adaptive wavelet schemes for nonlinear variational problems we develop and analyze wavelet based adaptive schemes for nonlinear variational problems we derive estimates for convergence rates and corresponding work counts that turn out to be asymptotically optimal our approach is based on a new paradigm that has been put forward recently for a class of linear problems the original problem is transformed first into an equivalent one which is well posed in the euclidean metric ell 2 then conceptually one seeks iteration schemes for the infinite dimensional problem that exhibits at least a fixed error reduction per step this iteration is then realized approximately through an adaptive application of the involved operators with suitable dynamically updated accuracy tolerances the main conceptual ingredients center around nonlinear tree approximation and the sparse evaluation of nonlinear mappings of wavelet expansions we prove asymptotically optimal complexity for adaptive realizations of first order iterations and of newton s method

mixed mimetic spectral element method for stokes flow a pointwise divergence free solution in this paper we apply the recently developed mimetic discretization method to the mixed formulation of the stokes problem in terms of vorticity velocity and pressure the mimetic discretization presented in this paper and in kreeft et al 51 is a higher order method for curvilinear quadrilaterals and hexahedrals fundamental is the underlying structure of oriented geometric objects the relation between these objects through the boundary operator and how this defines the exterior derivative representing the grad curl and div through the generalized stokes theorem the mimetic method presented here uses the language of differential k forms with k cochains as their discrete counterpart and the relations between them in terms of the mimetic operators reduction reconstruction and projection the reconstruction consists of the recently developed mimetic spectral interpolation functions the most important result of the mimetic framework is the commutation between differentiation at the continuous level with that on the finite dimensional and discrete level as a result operators like gradient curl and divergence are discretized exactly for stokes flow this implies a pointwise divergence free solution this is confirmed using a set of test cases on both cartesian and curvilinear meshes it will be shown that the method converges optimally for all admissible boundary conditions

a fourth order accurate method for the incompressible navier stokes equations on overlapping grids a method is described to solve the time dependent incompressible navier stokes equations with finite differences on curvilinear overlapping grids in two or three space dimensions the scheme is fourth order accurate in space and uses the momentum equations for the velocity coupled to a poisson equation for the pressure the boundary condition for the pressure is taken as u 0 extra numerical boundary conditions are chosen to make the scheme accurate and stable the velocity is advanced explicitly in time any standard time stepping scheme such as runge kutta can be used the poisson equation is solved using direct or iterative sparse matrix solvers or by the multigrid algorithm computational results in two and three space dimensions are given

numerical solutions for nonlinear elliptic problems based on first order system we present a theoretical analysis of numerical solutions for nonlinear elliptic problems our analysis is based on the abstract approximation theory for branches of nonsingular solutions developed by brezzi rappaz and raviart brr in most cases of finite element analysis the same spaces are used for the domain and range of the nonlinear operator concerning the application of brr theory this results in a loss of accuracy in the error estimates while numerical experiments show optimal convergence rates the main contribution of this paper is a theoretical analysis for the optimal convergence rates this is achieved by choosing different spaces for the domain and range of the nonlinear operator used in the application of brr theory previously this idea was used for an analysis of petrov galerkin formulation for the brr theory our analysis is developed for the first order least squares finite element methods and the mixed galerkin methods based on first order systems of equations

unified semi analytical wall boundary conditions applied to 2 d incompressible sph this work aims at improving the 2 d incompressible sph model isph by adapting it to the unified semi analytical wall boundary conditions proposed by ferrand et al 10 the isph algorithm considered is as proposed by lind et al 25 based on the projection method with a divergence free velocity field and using a stabilising procedure based on particle shifting however we consider an extension of this model to reynolds averaged navier stokes equations based on the k e turbulent closure model as done in 10 the discrete sph operators are modified by the new description of the wall boundary conditions in particular a boundary term appears in the laplacian operator which makes it possible to accurately impose a von neumann pressure wall boundary condition that corresponds to impermeability the shifting and free surface detection algorithms have also been adapted to the new boundary conditions moreover a new way to compute the wall renormalisation factor in the frame of the unified semi analytical boundary conditions is proposed in order to decrease the computational time we present several verifications to the present approach including a lid driven cavity a water column collapsing on a wedge and a periodic schematic fish pass our results are compared to finite volumes methods using volume of fluids in the case of free surface flows we briefly investigate the convergence of the method and prove its ability to model complex free surface and turbulent flows the results are generally improved when compared to a weakly compressible sph model with the same boundary conditions especially in terms of pressure prediction

spectral discretization of the vorticity velocity and pressure formulation of the stokes problem we consider the stokes problem in a square or a cube provided with nonstandard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity we write a variational formulation of this problem with three independent unknowns the vorticity the velocity and the pressure next we propose a discretization by spectral methods which relies on this formulation and since it leads to an inf sup condition on the pressure in a natural way we prove optimal error estimates for the three unknowns we present numerical experiments which are in perfect coherence with the analysis

a posteriori existence in numerical computations a typical a posteriori error analysis for numerical methods for differential equations consists of two elements the stability of the continuous problem and an a posteriori computed residual estimate if the problem is nonlinear then this is possible only if it is strongly monotone in this paper we will estimate the stability constant a posteriori as well in order to apply the inverse function theorem to derive the existence of solutions from the numerical computation and thus obtain rigorous a posteriori error estimates for nonmonotone equations

a priori and a posteriori error analysis of a mixed scheme for the brinkman problem this paper deals with the analysis of new mixed finite element methods for the brinkman equations formulated in terms of velocity vorticity and pressure employing the babuska brezzi theory it is proved that the resulting continuous and discrete variational formulations are well posed in particular we show that raviart thomas elements of order k ge 0 k 0 for the approximation of the velocity field piecewise continuous polynomials of degree k 1 k 1 for the vorticity and piecewise polynomials of degree k for the pressure yield unique solvability of the discrete problem on the other hand we also show that families of finite elements based on brezzi douglas marini elements of order k 1 k 1 for the approximation of velocity piecewise continuous polynomials of degree k 2 k 2 for the vorticity and piecewise polynomials of degree k for the pressure ensure the well posedness of the associated galerkin scheme we note that these families provide exactly divergence free approximations of the velocity field we establish a priori error estimates in the natural norms with constants independent of the viscosity and we carry out the reliability and efficiency analysis of a residual based a posteriori error estimator finally we report several numerical experiments illustrating the behaviour of the proposed schemes and confirming our theoretical results on unstructured meshes additional examples of cases not covered by our theory are also presented

an adaptive finite element algorithm for a two dimensional stationary stefan like problem an efficient adaptive algorithm is presented for a stationary regularized stefan problem in 2d the adaptive criteria relies upon a posteriori estimates based on the residual equation since the problem we are studying is a non linear diffusion convection problem these error estimates are relatively crude in the sense that the constant relating the true error to the error indicator is unknown thus instead of trying to build a mesh such that the true error is below a given tolerance the local error indicator is equidistributed in a way such that the final number of triangles is close to a desired value at each iteration of the adaptive algorithm the new triangulation is obtained from the previous one by adding or deleting vertices according to the error indicator and a global mesh regeneration is performed using a delaunay mesh generator numerical experiments for two practical situations show the efficiency and the robustness of our approach

boundary conditions involving pressure for the stokes problem and applications in computational hemodynamics pressure driven flows typically occur in hydraulic networks e g oil ducts water supply biological flows microfluidic channels etc however stokes and navier stokes problems are most often studied in a framework where dirichlet type boundary conditions on the velocity field are imposed thanks to the simpler settings from the theoretical and numerical points of view in this work we propose a novel formulation of the stokes system with pressure boundary condition together with no tangential flow on a part of the boundary in a standard stokes functional framework using lagrange multipliers to enforce the latter constraint on velocity more precisely we carry out i a complete analysis of the formulation from the continuous to discrete level in two and three dimensions ii the description of our solution strategy iii a verification of the convergence properties with an analytic solution and finally iv three dimensional simulations of blood ow in the cerebral venous network that are in line with in vivo measurements and the presentation of some performance metrics with respect to our solution strategy

hypothesis testing for nearly nonstationary autoregressive models abstract in this paper a new asymptotic theory is developed for nearly nonstationary autoregressive processes of first and second order a general model is investigated and a hypothesis testing method is proposed for it in important special cases the limit distribution of the test statistic can be expressed in terms of functionals of simple diffusion processes

adaptive finite elements for a linear parabolic problem a posteriori error estimates for the heat equation in two space dimensions are presented a classical discretization is used euler backward in time and continuous piecewise linear triangular finite elements in space the error is bounded above and below by an explicit error estimator based on the residual numerical results are presented for uniform triangulations and constant time steps the quality of our error estimator is discussed an adaptive algorithm is then proposed successive delaunay triangulations are generated so that the estimated relative error is close to a preset tolerance again numerical results demonstrate the efficiency of our approach c 1998 elsevier science s a all rights reserved

incompressible finite element methods for navier stokes equations with nonstandard boundary conditions in r 3 application des methodes a elements finis incompressibles aux equations de navier stokes avec des conditions aux limites non standards dans r 3 utilisation du potentiel vecteur de la vorticite et de la pression estimations d erreurs

some aspects of adaptive grid technology related to boundary and interior layers we consider the use of adaptive mesh strategies for solution of problems exhibiting boundary and interior layer solutions as the presence of these layer structures suggests reliable and accurate solution of this class of problems using finite difference finite volume or finite element schemes requires grading the mesh into the layers and due attention to the associated algorithms when the nature and structure of the layer is known mesh grading can be achieved during the grid generation by specifying an appropriate grading function however in many applications the location and nature of the layer behavior is not known in advance consequently adaptive mesh techniques that employ feedback from intermediate grid solutions are an appealing approach in this paper we provide a brief overview of the main adaptive grid strategies in the context of problems with layers associated error indicators that guide the refinement feedback control grid optimization process are also covered and there is a brief commentary on the supporting data structure requirements some current issues concerning the use of stabilization in conjunction with adaptive mesh refinement amr the question of pollution effects in computation of local error indicators the influence of nonlinearities and the design of meshes for targeted optimization of specific quantities are considered the application of amr for layer problems is illustrated by means of case studies from semiconductor device transport drift diffusion nonlinear reaction diffusion layers due to surface capillary effects and shockwaves in compressible gas dynamics

an optimal adaptive wavelet method for first order system least squares in this paper it is shown that any well posed 2nd order pde can be reformulated as a well posed first order least squares system this system will be solved by an adaptive wavelet solver in optimal computational complexity the applications that are considered are second order elliptic pdes with general inhomogeneous boundary conditions and the stationary navier stokes equations

finite element approximations of nonlinear eigenvalue problems in quantum physics in this paper we study finite element approximations of a class of nonlinear eigenvalue problems arising from quantum physics we derive both a priori and a posteriori finite element error estimates and obtain optimal convergence rates for both linear and quadratic finite element approximations in particular we analyze the convergence and complexity of an adaptive finite element method in our analysis we utilize certain relationship between the finite element eigenvalue problem and the associated finite element boundary value approximations we also present several numerical examples in quantum physics that support our theory

phase field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing this paper describes a novel numerical algorithm for simulating interfacial dynamics of non newtonian fluids the interface between two immiscible fluids is treated as a thin mixing layer across which physical properties vary steeply but continuously the property and evolution of the interfacial layer is governed by a phase field variable that obeys a cahn hilliard type of convection diffusion equation this circumvents the task of directly tracking the interface and produces the correct interfacial tension from the free energy stored in the mixing layer viscoelasticity and other types of constitutive equations can be incorporated easily into the variational phase field framework the greatest challenge of this approach is in resolving the sharp gradients at the interface this is achieved by using an efficient adaptive meshing scheme governed by the phase field variable the finite element scheme easily accommodates complex flow geometry and the adaptive meshing makes it possible to simulate large scale two phase systems of complex fluids in two dimensional and axisymmetric three dimensional implementations the numerical toolkit is applied here to drop deformation in shear and elongational flows rise of drops and retraction of drops and torii some of these solutions serve as validation of the method and illustrate its key features while others explore novel physics of viscoelasticity in the deformation and evolution of interfaces

drift bifurcation of dissipative solitons due to a change of shape experiment and theory abstract the subject of this paper is self organized solitary current filaments being observed experimentally in a planar dc gas discharge system with high ohmic barrier we report on the dynamical behavior of these objects which we refer to as dissipative solitons an application of stochastic data analysis methods enables the separation of intrinsic dynamics from stochastic contributions using the externally applied voltage as control parameter we observe a drift bifurcation that is accompanied by a change of shape of the bifurcating dissipative soliton in order to interpret the experimental finding a theoretical model in form of a three component reaction diffusion system is considered allowing for a qualitative understanding of the obtained results we discuss an analytical expression for the velocity of the dissipative soliton as a function of an appropriate control parameter and demonstrate that the change of the shape can induce the drift bifurcation this analytical finding is supported by numerical simulations of the model equations

spectral discretization of the axisymmetric vorticity velocity and pressure formulation of the navier stokes problem we deal in this work with the nonlinear navier stokes equations set in a three dimensional axisymmetric bounded domain the boundary conditions that we consider are given on the normal component of the velocity and the tangential component of the vorticity such conditions occur in a large number of flows and we are led to write a vorticity velocity pressure formulation under assumptions on the data of the problem the three dimensional problem is reduced in a two dimensional one for the discretization we use the spectral methods which are well adapted here we prove the well posedness of the obtained formulations and we derive optimal error estimates on the three unknowns the results of the numerical experiments for known functions and a given data corresponding to a poiseuille type flow are coherent with the theoretical ones

an analysis of upwind schemes for the navier stokes equations we state a general result on approximation of branches of nonsingular solutions of nonlinear problems the abstract theory is then applied to the study of two mixed finite element schemes for the navier stokes equations using an upwind discretization of the convective terms

accuracy and stability in incompressible sph isph based on the projection method and a new approach the stability and accuracy of three methods which enforce either a divergence free velocity field density invariance or their combination are tested here through the standard taylor green and spin down vortex problems while various approaches to incompressible sph isph have been proposed in the past decade the present paper is restricted to the projection method for the pressure and velocity coupling it is shown that the divergence free isph method cannot maintain stability in certain situations although it is accurate before instability sets in the density invariant isph method is stable but inaccurate with random noise like disturbances the combined isph combining advantages in divergence free isph and density invariant isph can maintain accuracy and stability although at a higher computational cost redistribution of particles on a fixed uniform mesh is also shown to be effective but the attraction of a mesh free method is lost a new divergence free isph approach is proposed here which maintains accuracy and stability while remaining mesh free without increasing computational cost by slightly shifting particles away from streamlines although the necessary interpolation of hydrodynamic characteristics means the formulation ceases to be strictly conservative this avoids the highly anisotropic particle spacing which eventually triggers instability importantly pressure fields are free from spurious oscillations up to the highest reynolds numbers tested

spectral discretization of the vorticity velocity and pressure formulation of the navier stokes equations we consider the navier stokes equations in a two or three dimensional domain provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity we write a variational formulation of this problem with three independent unknowns the vorticity the velocity and the pressure and prove the existence of a solution for this problem next we propose a discretization by spectral methods which relies on this formulation in the two dimensional case we prove quasi optimal error estimates for the three unknowns we conclude with some numerical experiments nous considerons les equations de navier stokes dans un domaine biou tri dimensionnel munies de conditions aux limites non usuelles portant sur la composante normale de la vitesse et la ou les composantes tangentielles du tourbillon nous ecrivons une formulation variationnelle de ce probleme qui comporte trois inconnues independantes le tourbillon la vitesse et la pression nous prouvons que ce probleme admet au moins une solution nous proposons une discretisation par methodes spectrales construite a partir de cette formulation dans le cas bidimensionnel nous etablissons des majorations quasi optimales de l erreur pour les trois inconnues nous concluons par quelques experiences numeriques

traveling spike autosolitons in the gray scott model abstract we developed singular perturbation techniques based on the strong separation of time and length scales to construct the solutions in the form of the traveling spike autosolitons self sustained solitary waves in the gray scott model of an autocatalytic reaction we found that when the inhibitor diffusion is sufficiently slow the ultrafast traveling spike autosolitons are realized in a wide range of parameters when the diffusion of the inhibitor is sufficiently fast the slower traveling spike autosolitons with the diffusion precursor are realized we asymptotically calculated the main parameters such as speed and amplitude of these autosolitons as well as the regions of their existence in the gray scott model we also showed that in certain parameter regions the traveling spike autosolitons coexist with static

pressure boundary conditions for computing incompressible flows with sph in smoothed particle hydrodynamics sph methods for fluid flow incompressibility may be imposed by a projection method with an artificial homogeneous neumann boundary condition for the pressure poisson equation this is often inconsistent with physical conditions at solid walls and inflow and outflow boundaries for this reason open boundary flows have rarely been computed using sph in this work we demonstrate that the artificial pressure boundary condition produces a numerical boundary layer that compromises the solution near boundaries we resolve this problem by utilizing a rotational pressure correction scheme with a consistent pressure boundary condition that relates the normal pressure gradient to the local vorticity we show that this scheme computes the pressure and velocity accurately near open boundaries and solid objects and extends the scope of sph simulation beyond the usual periodic boundary conditions

stability of pressure boundary conditions for stokes and navier stokes equations abstract the stability of a finite difference discretization of the time dependent incompressible navier stokes equations in velocity pressure formulation is studied in paticular we compare the stability for different pressure boundary conditions in a semiimplicit time integration scheme where only the viscous term is treated implicitly the stability is studied in three different ways by a normal mode analysis by numerical computation of the amplification factors and by direct numerical simulation of the governing equations all three approaches identify the same pressure boundary condition as the best alternative this condition implicitly enforces the normal derivative of the divergence to be zero on the boundary by coupling the normal derivative of the pressure to the normal component of the curl of the vorticity using this boundary condition we demonstrate that the time step is determined only by the convective term

error indicators for the navier stokes equations in stream function and vorticity formulation this paper deals with a posteriori estimates for the finite element solution of the stokes problem in stream function and vorticity formulation for two different discretizations we propose error indicators and we prove estimates in order to compare them with the local error in a second step these results are extended to the navier stokes equations

new statistical investigations of the ornstein uhlenbeck process an asymptotic analysis is presented for estimation in the three parameter ornstein uhlenbeck process where the parameters are the local mean the drift and the variance we are interested in the case when the damping parameter or t is nearly zero the asymptotic sufficient statistics can be related to noncentral 12 distribution the maximum likelihood estimate of the parameter vector is a solution of a rather complicated system of equations we describe the methods for solving maximum likelihood equations classical and robust estimators are determined for parameters it is shown that the lower confidence limit of the drift or damping parameter is equal to zero with positive probability when it is near to zero

an augmented mixed finite element method for the vorticity velocity pressure formulation of the stokes equations this paper deals with the numerical approximation of the stationary two dimensional stokes equations formulated in terms of vorticity velocity and pressure with non standard boundary conditions here by introducing a galerkin least squares term we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem we propose two families of mixed finite elements to solve the discrete problem in the first family the unknowns are approximated by piecewise continuous and quadratic elements brezzi douglas marini and piecewise constant finite elements respectively while in the second family the unknowns are approximated by piecewise linear and continuous raviart thomas and piecewise constant finite elements respectively the wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the babuska brezzi theory we establish a priori error estimates in the natural norms and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes additional examples of cases not covered by our theory are also presented

an improved sph method towards higher order convergence this paper evaluates various formulations of the sph method for solving the euler equations convergence and stability aspects are discussed and tested taking into account subtleties induced by the presence of a free surface the coherence between continuity and momentum equations is considered using a variational study the use of renormalization to improve the accuracy of the simulations is investigated and discussed a new renormalization based formulation involving wide accuracy improvements of the scheme is introduced the classical sph and renormalized approaches are compared to the new method using simple test cases thus outlining the efficiency of this new improved sph method finally the so called tensile instability is shown to be prevented by this enhanced sph method through accuracy increases in the gradient approximations

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

a posteriori analysis of the finite element discretization of some parabolic equations nous considerons la discretisation d equations paraboliques soit lineaires soit semi lineaires par un schema d euler implicite en temps et par elements finis en espace l idee de cet article est de construire des indicateurs d erreur lies a l approximation en temps et en espace et de prouver leur equivalence avec l erreur dans le but de travailler avec des pas de temps adaptatifs et des maillages d elements finis adaptes a la solution

gaussian likelihood estimation for nearly nonstationary ar 1 processes abstract an asymptotic analysis is presented for estimation in the three parameter first order autoregressive model where the parameters are the mean autoregressive coefficient and variance of the shocks the nearly nonstationary asymptotic model is considered wherein the autoregressive coefficient tends to 1 as sample size tends to infinity three different estimators are considered the exact gaussian maximum likelihood estimator the conditional maximum likelihood or least squares estimator and some naive estimators it is shown that the estimators converge in distribution to analogous estimators for a continuous time ornstein uhlenbeck process simulation results show that the mle has smaller asymptotic mean squared error than the other two and that the conditional maximum likelihood estimator gives a very poor estimator of the process mean keywords likelihood estimation autoregressive processes nearly nonstationary time series ornstein uhlenbeck process

finite difference schemes for incompressible flow based on local pressure boundary conditions in this paper we discuss the derivation and use of local pressure boundary conditions for finite difference schemes for the unsteady incompressible navier stokes equations in the velocity pressure formulation their use is especially well suited for the computation of moderate to large reynolds number flows we explore the similarities between the implementation and use of local pressure boundary conditions and local vorticity boundary conditions in the design of numerical schemes for incompressible flow in 2d in their respective formulations when these local numerical boundary conditions are coupled with a fully explicit convectively stable time stepping procedure the resulting methods are simple to implement and highly efficient unlike the vorticity formulation the use of the local pressure boundary condition approach is readily applicable to 3d flows the simplicity of the local pressure boundary condition approach and its easy adaptation to more general flow settings make the resulting scheme an attractive alternative to the more popular methods for solving the navier stokes equations in the velocity pressure formulation we present numerical results of a second order finite difference scheme on a nonstaggered grid using local pressure boundary conditions stability and accuracy of the scheme applied to stokes flow is demonstrated using normal mode analysis also described is the extension of the method to variable density flows

scenarios of domain pattern formation in a reaction diffusion system we performed an extensive numerical study of a two dimensional reaction diffusion system of the activator inhibitor type in which domain patterns can form we showed that both multidomain and labyrinthine patterns may form spontaneously as a result of the turing instability in the stable homogeneous system with the fast inhibitor one can excite both localized and extended patterns by applying a localized stimulus depending on the parameters and the excitation level of the system stripes spots wriggled stripes or labyrinthine patterns form the labyrinthine patterns may be both connected and disconnected in the stable homogeneous system with the slow inhibitor one can excite self replicating spots breathing patterns autowaves and turbulence the parameter regions in which different types of patterns are realized are explained on the basis of the asymptotic theory of instabilities for patterns with sharp interfaces developed by us in phys rev e 53 3101 1996 the dynamics of the patterns observed in our simulations is very similar to that of the patterns forming in the ferrocyanide iodate sulfite reaction textcopyright 1996 the american physical society

stability of the static spike autosolitons in the gray scott model we performed an asymptotic linear stability analysis of the static spike autosolitons ass self sustained solitary inhomogeneous states in the gray scott model of an autocatalytic chemical reaction we found that in one dimension these ass destabilize with respect to pulsations or the onset of traveling motion when the inhibitor is slow enough in higher dimensions the one dimensional static spike ass are always unstable with respect to corrugation and wriggling the higher dimensional radially symmetric static spike ass may destabilize with respect to the radially nonsymmetric fluctuations leading to their splitting when the inhibitor is fast or with respect to pulsations when the inhibitor is slow

the linearization method for the numerical analysis of finite element solutions to quasi linear elliptic partial differential equations linearization methods have been used in the numerical analysis of finite element solutions to nonlinear partial differential equations pdes for quite a long time frequently essential properties such as differentiability of the nonlinear operator and boundedness and invertibility properties for linearized operators are assumed without deeper considerations in this work we describe the main difficulties and build a framework which is able to circumvent them allowing for the consideration of larger classes of problems a priori error estimates are developed using that framework and a model problem is presented which illustrates the main results of this paper

adaptive streamline diffusion methods for compressible flow using conservation variables we consider the streamline diffusion finite element method applied to compressible flow using conservation variables we propose some adaptive algorithms and present related numerical results

a scalable consistent second order sph solver for unsteady low reynolds number flows abstract smoothed particle hydrodynamics sph has successfully been used to study a variety of cases involving nearly inviscid flows where conservation properties allow for good physical approximation despite poor theoretical approximation properties of differential operators when used to study unsteady low reynolds number flow with large dissipation conservation alone cannot ensure quality of approximation and the traditional approach is inconsistent an alternative formulation has recently become popular making use of an approximate splitting scheme to ensure a divergence free velocity field however this scheme relies on an inconsistent discretization of the laplacian that diverges as particles become disordered under flow we present an incremental pressure correction scheme and combination of existing differential operator renormalizations that are able to achieve second order accuracy in time and space a brief review of sph approximation theory is provided to highlight the necessity of these renormalizations in implementing an approximate factorization scheme we demonstrate that when fast algebraic multigrid preconditioners are used to solve the resulting linear systems the scheme results in a consistent approximation that is scalable and amenable to parallelization several validation cases are presented for which a speedup of several orders of magnitude is achieved over traditional sph approaches finally an interface has been developed between the particle library lammps and the sparse linear algebra libraries in trilinos providing a massively parallel 3d sph capability scaling results for up to 134 million particles on 32 768 cores are presented along with a demonstration of the capability to simulate complex 3d geometries these results show that the added complexity of applying the necessary consistency corrections actually provides a factor of four speed up per linear solver iteration versus the uncorrected case despite the additional cost of constructing the corrections the resulting library provides a method that is consistent efficient and second order in both space and time while maintaining the flexibility of classical sph for single phase flows

on the numerical approximation of unstableminimal surfaces with polygonal boundaries this work is concerned with the approximation and the numerical computation of polygonal minimal surfaces in bbb r q q ge 2 polygonal minimal surfaces correspond to the critical points of shiffman s function theta since this function is analytic polygonal minimal surfaces can be characterized by means of the second derivative of theta we present a finite element approximation of quasiminimal surfaces together with an error estimate in this way we obtain discrete approximations theta h of theta and f h of nabla theta in particular we prove that the discrete functions converge uniformly on certain compact subsets this will be the main tool for proving existence and convergence of discrete minimal surfaces in neighbourhoods of non degenerate minimal surfaces in the numerical part of this paper we compute numerical approximations of polygonal minimal surfaces by use of newton s method applied to f h

error bars and quadratically convergent methods for the numerical simulation of the hartree fock equations this paper presents an a posteriori error analysis of the discretization methods used in computational quantum chemistry on the hartree fock equations upper and lower bounds for the energy are obtained from any discrete approximation strategy of the solution and the estimator proposed is shown to possess further approximation virtues

nearly unstable multidimensional ar processes abstract the limiting distribution of the likelihood ratio is given for the hypothesis of unstability in case of nearly unstable multidimensional gaussian ar 1 processes

spectral element discretization of the maxwell equations on considere un probleme variationnel equivalent aux equations de l electromagnetisme avec conditions aux limites absorbantes sur une partie de la frontiere qu on prouve etre bien pose puis on propose une discretisation de ce probleme par schema aux differences finies en temps et elements spectraux en espace et on etablit des estimations d erreur entre solutions exacte et approchee

an application of the kantorovich theoremto nonlinear finite element analysis finite element solutions of strongly nonlinear elliptic boundary value problems are considered in this paper using the kantorovich theorem we show that if the frechet derivative of the nonlinear operator defined by the boundary value problem is an isomorphism at an exact solution then there exists a locally unique finite element solution near the exact solution moreover several a priori error estimates are obtained

a mathematical framework for new fault detection schemes in nonlinear stochastic continuous time dynamical systems in this work a mathematical unifying framework for designing new fault detection schemes in nonlinear stochastic continuous time dynamical systems is developed these schemes are based on a stochastic process called the residual which reflects the system behavior and whose changes are to be detected a quickest detection scheme for the residual is proposed which is based on the computed likelihood ratios for time varying statistical changes in the ornstein uhlenbeck process several expressions are provided depending on a priori knowledge of the fault which can be employed in a proposed cusum type approximated scheme this general setting gathers different existing fault detection schemes within a unifying framework and allows for the definition of new ones a comparative simulation example illustrates the behavior of the proposed schemes

a posteriori estimators for nonlinear elliptic partial differential equations many works have reported results concerning the mathematical analysis of the performance of a posteriori error estimators for the approximation error of finite element discrete solutions to linear elliptic partial differential equations for each estimator there is a set of restrictions defined in such a way that the analysis of its performance is made possible usually the available estimators may be classified into two types i e the implicit estimators based on the solution of a local problem and the explicit estimators based on some suitable norm of the residual in a dual space regarding the performance an estimator is called asymptotically exact if it is a higher order perturbation of a norm of the exact error nowadays one may say that there is a larger understanding about the behavior of estimators for linear problems than for nonlinear problems the situation is even worse when the nonlinearities involve the highest derivatives occurring in the pde being considered strongly nonlinear pdes in this work we establish conditions under which those estimators originally developed for linear problems may be used for strongly nonlinear problems and how that could be done we also show that under some suitable hypothesis the estimators will be asymptotically exact whenever they are asymptotically exact for linear problems those results allow anyone to use the knowledge about estimators developed for linear problems in order to build new reliable and robust estimators for nonlinear problems

error analysis of finite element approximations of the inverse mean curvature flow arising from the general relativity this paper proposes and analyzes a finite element method for a nonlinear singular elliptic equation arising from the black hole theory in the general relativity the nonlinear equation which was derived and analyzed by huisken and ilmanen in j diff geom 59 353 437 represents a level set formulation for the inverse mean curvature flow describing the evolution of a hypersurface whose normal velocity equals the reciprocal of its mean curvature we first propose a finite element method for a regularized flow which involves a small parameter a rigorous analysis is presented to study well posedness and convergence of the scheme under certain mesh constraints and optimal rates of convergence are verified we then prove uniform convergence of the finite element solution to the unique weak solution of the nonlinear singular elliptic equation as the mesh size h and the regularization parameter both tend to zero computational results are provided to show the efficiency of the proposed finite element method and to numerically validate the jumping out phenomenon of the weak solution of the inverse mean curvature flow numerical studies are presented to evidence the existence of a polynomial scaling law between the mesh size h and the regularization parameter for optimal convergence of the proposed scheme finally a numerical convergence study for another approach recently proposed by r moser the inverse mean curvature flow and p harmonic functions preprint u bath 2005 for approximating the inverse mean curvature flow via p harmonic functions is also included

semi group theory for the stokes operator with navier type boundary conditions on l p spaces in this article we consider the stokes problem with navier type boundary conditions on a domain omega not necessarily simply connected since under these conditions the stokes problem has a non trivial kernel we also study the solutions lying in the orthogonal of that kernel we prove the analyticity of several semigroups generated by the stokes operator considered in different functional spaces we obtain strong weak and very weak solutions for the time dependent stokes problem with the navier type boundary condition under different hypotheses on the initial data u0 and external force f then we study the fractional and pure imaginary powers of several operators related with our stokes operators using the fractional powers we prove maximal regularity results for the homogeneous stokes problem on the other hand using the boundedness of the pure imaginary powers we deduce maximal l p l q regularity for the inhomogeneous stokes problem

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

open boundary conditions for isph and their application to micro flow open boundary conditions for incompressible smoothed particle hydrodynamics isph are rare for stable simulations with open boundary conditions one needs to specify all boundary conditions correctly in the pressure force as well as in the linear equation system for pressure calculation especially for homogeneous or non homogeneous dirichlet boundary conditions for pressure there exist several possibilities but only a few lead to stable results however this isn t trivial for open boundary conditions we introduce a new approach for open boundary conditions for isph to enable stable simulations in contrast to existing models for weakly compressible sph we can specify open pressure boundary conditions because in isph pressure can be calculated independently of the density the presented approach is based on the mirror particle approach already introduced for solid wall boundary conditions here we divide the mirror axis in several segments with time dependent positions we validate the presented approach for the example of poiseuille flow and flow around a cylinder at different reynolds numbers and show that we get good agreement with references then we demonstrate that the approach can be applied to free surface flows finally we apply the new approach to micro flow through a random porous medium with a different number of in and outlets and demonstrate its benefits

a posteriori error estimates and adaptive finite elements for a nonlinear parabolic problem related to solidification a posteriori error estimates are derived for a nonlinear parabolic problem arising from the isothermal solidification of a binary alloy space discretization with continuous piecewise linear finite elements is considered the l 2 in time h 1 in space error is bounded above and below by an error estimator based on the equation residual numerical results show that the effectivity index is close to one an adaptive finite element algorithm is proposed and a solutal dendrite is computed c 2002 elsevier science b v all rights reserved

an isoparametric spectral element method for solution of the navier stokes equations in complex geometry abstract an isoparametric spectral element technique is presented for solution of the two dimensional navier stokes equations in arbitrary curvy geometries the spatial discretization is described and the method is illustrated by solution of poisson s equation in a cylindrical annulus a fractional step navier stokes solver based on a discretely consistent poisson equation for the pressure is then presented and exponential convergence in space is demonstrated for the case of separated flow between eccentric rotating cylinders the technique is made efficient by the use of explicit collocation rather than galerkin operators for the convective terms and use of static condensation to solve the elliptic equations resulting from the stokes problem the algorithm is amenable to a high degree of parallelism in both multiprocessor and vector pipeline environments

pattern formation by boundary forcing in convectively unstable oscillatory media with and without differential transport convectively unstable open reactive flows of oscillatory media whose phase is fixed or periodically modulated at the inflow boundary are known to result in stationary and traveling waves respectively the latter are implicated in biological segmentation the boundary controlled pattern selection by this flow distributed oscillator fdo mechanism has been generalized to include differential flow difi and differential diffusion turing modes our present goal is to clarify the relationships among these mechanisms in the general case where there is differential flow as well as differential diffusion to do so we analyze the dispersion relation for linear perturbations in the presence of periodic boundary forcing and show how the solutions are affected by differential transport we find that the difi and fdo modes are closely related and lie in the same frequency range while the turing mechanism gives rise to a distinct set of unstable modes in a separate frequency range finally we substantiate the linear analysis by nonlinear simulations and touch upon the issue of competition of spatial modes

a priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems the effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied under similar assumption on the quadrature formula as for linear problems optimal error estimates in the l 2 and the h 1 norms are proved the numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh the analysis is valid for both simplicial and rectangular finite elements of arbitrary order numerical experiments corroborate the theoretical convergence rates

sparse grid reduced basis bayesian inversion we extend the reduced basis rb accelerated bayesian inversion methods for affine parametric linear operator equations which are considered in 16 17 to non affine nonlinear parametric operator equations we generalize the analysis of sparsity of parametric forward solution maps in 20 and of bayesian inversion in 48 49 to the fully discrete setting including petrov galerkin high fidelity hifi discretization of the forward maps we develop adaptive stochastic collocation based reduction methods for the efficient computation of reduced bases on the parametric solution manifold the nonaffinity and nonlinearity with respect to w r t the distributed uncertain parameters and the unknown solution is collocated specifically by the so called empirical interpolation method eim for the corresponding bayesian inversion problems computational efficiency is enhanced in two ways first expectations w r t the posterior are computed by adaptive quadratures with dimension independent convergence rates proposed in 49 the present work generalizes 49 to account for the impact of the pg discretization in the forward maps on the convergence rates of the quantities of interest qoi for short second we propose to perform the bayesian estimation only w r t a parsimonious rb approximation of the posterior density based on the approximation results in 49 the infinite dimensional parametric deterministic forward map and operator admit n term rb and eim approximations which converge at rates which depend only on the sparsity of the parametric forward map in several numerical experiments the proposed algorithms exhibit dimension independent convergence rates which equal at least the currently known rate estimates for n term approximation we propose to accelerate bayesian estimation by first offline construction of reduced basis surrogates of the bayesian posterior density the parsimonious surrogates can then be employed for online data assimilation and for bayesian estimation they also open a perspective for optimal experimental design

sph with the multiple boundary tangent method in this article we present an improved solid boundary treatment formulation for the smoothed particle hydrodynamics sph method benchmark simulations using previously reported boundary treatments can suffer from particle penetration and may produce results that numerically blow up near solid boundaries as well current sph boundary approaches do not properly treat curved boundaries in complicated flow domains these drawbacks have been remedied in a new boundary treatment method presented in this article called the multiple boundary tangent mbt approach in this article we present two important benchmark problems to validate the developed algorithm and show that the multiple boundary tangent treatment produces results that agree with known numerical and experimental solutions the two benchmark problems chosen are the lid driven cavity problem and flow over a cylinder the sph solutions using the mbt approach and the results from literature are in very good agreement these solutions involved solid boundaries but the approach presented herein should be extendable to time evolving free surface boundaries copyright 2008 john wiley sons ltd

two dimensional sph simulations of wedge water entries this paper presents a study based on the smoothed particles hydrodynamics sph method aiming at an accurate numerical simulation of solid fluid coupling in a free surface flow context the sph scheme is first described and discussed through its formulations then a new technic based on a particle sampling method and designed to evaluate fluid pressure on solid boundaries is introduced this method is then extended to the capture of freely moving body dynamics in a fluid solid coupling approach this study involves a spatially varying resolution based on the so called variable smoothing length technique for which a new formulation of the equations is proposed two distinct test cases of wedge water entry are presented in order to validate this new method pressure prediction is first compared with analytical and experimental results evolution in time of the body dynamics is compared with experimental results in both cases and the pressure field on the solid boundaries is studied and discussed on the first impact case

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

a posteriori error estimation for elasto plastic problems based on duality theory abstract in this paper we introduce a new approach to a posteriori error estimation for elasto plastic problems based on the duality theory of the calculus of variations we show that in spite of the prevailing view duality methods provide a viable way for obtaining computable a posteriori error estimates for nonlinear boundary value problems without directly solving the dual problem rigorous mathematical analysis leads to what we call duality error estimators consisting of two parts the error in the constitutive law and the error in the equilibrium equations this representation is both physically meaningful and computationally important the duality error estimators hold for any conforming approximation of the exact solution regardless of whether or not they satisfy the galerkin orthogonality condition in particular they encompass the familiar smoothening or gradient averaging techniques commonly used in practice we prove that the duality error estimators are equivalent to the error when the approximate solution is the exact solution of the discretised problem moreover in the case of linear elasticity the known residual type error estimators can be obtained from the duality error estimators numerical results for a model elasto plastic problem show the accuracy of the duality error estimators