numerical analysis of a finite element volume penalty method we present here some contributions to the numerical analysis of the penalty method in the finite element context we are especially interested in the ability provided by this approach to use cartesian non boundary fitted meshes to solve elliptic problems in complicated domain in the spirit of fictitious domains the initial problem is replaced by a penalized one posed over a simply shaped domain which covers the original one this method relies on two parameters namely h space discretization parameter and varepsilon penalty parameter we propose here a general strategy to estimate the error in both parameters and we present how it can be applied to various situations we pay special attention to a scalar version of the rigid motion constraint for fluid particle flows

the fixed mesh ale approach for the numerical approximation of flows in moving domains in this paper we propose a method to approximate flow problems in moving domains using always a given grid for the spatial discretization and therefore the formulation to be presented falls within the category of fixed grid methods even though the imposition of boundary conditions is a key ingredient that is very often used to classify the fixed grid method our approach can be applied together with any technique to impose approximately boundary conditions although we also describe the one we actually favor our main concern is to properly account for the advection of information as the domain boundary evolves to achieve this we use an arbitrary lagrangian eulerian framework the distinctive feature being that at each time step results are projected onto a fixed background mesh that is where the problem is actually solved

a cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces we present a finite volume method for the solution of the two dimensional elliptic equation s x u x f x with variable discontinuous coefficients and solution discontinuities on irregular domains the method uses bilinear ansatz functions on cartesian grids for the solution u x resulting in a compact nine point stencil the resulting linear problem has been solved with a standard multigrid solver singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two step asymptotic approach the method achieves second order of accuracy in the l and l2 norm

optimal isoparametric finite elements and error estimates for domains involving curved boundaries we describe a practical procedure for the triangulation of arbitrary n dimensional regular bounded domains by isoparametric simplicial elements preserving the optimal order of accuracy along similar lines we prove general error estimates for the finite element solution of second order elliptic problems various kinds of nonhomogeneous boundary conditions can be dealt with such as dirichlet neumann and robin boundary conditions

a penalization method to take into account obstacles in incompressible viscous flows from the navier stokes brinkman model a penalization method has been derived by several authors to compute incompressible navier stokes equations around obstacles in this paper convergence theorems and error estimates are derived for two kinds of penalization the first one corresponds to l 2 penalization inducing a darcy equation in the solid body the second one corresponds to a h 1 penalization and induces a brinkman equation in the body numerical tests are performed to confirm the efficiency and accuracy of the method

a comparison of fictitious domain methods appropriate for spectral hp element discretisations abstract the fictitious domain finite cell and fat boundary method which are commonly adopted for fluid structure interaction fsi problems have been considered and implemented in a spectral hp element code for a 1d test problem the finite cell and fat boundary method are extended for the general case of a deformable solid in a fluid and analysed furthermore a new implicit variation of the fat boundary method is proposed the 1d framework allows for a clear and detailed analysis of the different approaches and to highlight the similarities and differences between the methods exponential p convergence which is typical for smooth solutions discretised with spectral hp elements is demonstrated for all methods except the classic fictitious domain formulation this overview can be considered as a starting point to study the capability advantages and disadvantages of these methods for fluid structure interaction problems in a more realistic setting and in higher dimensions

quick and accurate q parameterization in viscoelastic wave modeling we introduce a procedure for including the attenuation factor q in a consistent manner in seismic modeling and show 3d examples the q fitting over a chosen frequency band involves two algorithms the first creates starting values of relaxation times and the second does nonlinear inversion using the results of the first as initial values the resulting q function gives a good approximation to a constant q over the chosen frequency band the algorithm is combined with a finite difference f d code that includes topographies in 3d seismic media the velocity stress formulation for viscoelastic wave modeling is used with an arbitrary number of relaxation mechanisms to model a desired q behavior these equations are discretized by high order f ds in the interior of the medium and we gradually reduce the f d order to two at the stress free surface where we implement our free surface boundary conditions the seismic f d algorithm is applied to a marine seismic experiment with and without viscoelasticity to emphasize the importance of including physical attenuation and dispersion in seismic modeling their inclusion even for marine surveys is clearly important for lossy ocean bottoms our procedure for more accurate modeling of physical dispersion and attenuation may increase future motivation to include viscoelasticity in seismic inversion

a cartesian grid method for modeling multiple moving objects in 2d incompressible viscous flow we present an efficient method for solving 2d incompressible viscous flows around multiple moving objects our method employs an underlying regular cartesian grid to solve the system using a streamfunction vorticity formulation and with discontinuities representing the embedded objects the no penentration condition for the moving geometry is satisfied by superposing a homogenous solution to the poisson s equation for the streamfunction the no slip condition is satisfied by generating vorticity on the surfaces of the objects both the initial poisson solution and the evaluation of the homogenous solution require embedding irregular discontinuities in a fast poisson solver computation time is dictated by the time required to do a fast poisson solution plus solve an integral form of laplace s equation there is no significant increase in computational cost if the geometry of the embedded objects is variable and moving relative to the underlying grid we test the method against the canonical example of flow past a cylinder and obtained new results on the flow and forces of two cylinders moving relative to each other

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

the exponential accuracy of fourier and chebyshev differencing methods it is shown that when differencing analytic functions using the pseudospectral fourier or chebyshev methods the error committed decays to zero at an exponential rate

on the order of accuracy of the immersed boundary method higher order convergence rates for sufficiently smooth problems the immersed boundary method is both a mathematical formulation and a numerical scheme for problems involving the interaction of a viscous incompressible fluid and a visco elastic structure in m c lai simulations of the flow past an array of circular cylinders as a test of the immersed boundary method ph d thesis courant institute of mathematical sciences new york university 1998 m c lai c s peskin an immersed boundary method with formal second order accuracy and reduced numerical viscosity j comput phys 160 2000 705 719 lai and peskin introduced a formally second order accurate immersed boundary method but the convergence properties of their algorithm have only been examined computationally for problems with nonsmooth solutions consequently in practice only first order convergence rates have been observed in the present work we describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid structure interaction problem involving an immersed viscoelastic shell of finite thickness studied over a broad range of reynolds numbers we consider two sets of material properties for the viscoelastic structure including a case where the material properties of the coupled system are discontinuous at the fluid structure interface for both sets of material properties the true solutions appear to possess sufficient smoothness for the method to converge at a second order rate for fully resolved computations

analysis of the fully discrete fat boundary method the fat boundary method is a method of the fictitious domain class which was proposed to solve elliptic problems in complex geometries with non conforming meshes it has been designed to recover optimal convergence at any order despite of the non conformity of the mesh and without any change in the discrete laplace operator on the simple shape domain we propose here a detailed proof of this high order convergence and propose some numerical tests to illustrate the actual behaviour of the method

proteus a direct forcing method in the simulations of particulate flows a new and efficient direct numerical method for the simulation of particulate flows is introduced the method combines desired elements of the immersed boundary method the direct forcing method and the lattice boltzmann method adding a forcing term in the momentum equation enforces the no slip condition on the boundary of a moving particle by applying the direct forcing scheme proteus eliminates the need for the determination of free parameters such as the stiffness coefficient in the penalty scheme or the two relaxation parameters in the adaptive forcing scheme the method presents a significant improvement over the previously introduced immersed boundary lattice boltzmann method ib lbm where the forcing term was computed using a penalty method and a user defined parameter the method allows the enforcement of the rigid body motion of a particle in a more efficient way compared to the bounce back scheme used in the conventional lbm the direct forcing method provides a smoother computational boundary for particles and is capable of achieving results at higher reynolds number flows by using a set of lagrangian points to track the boundary of a particle proteus eliminates any need for the determination of the boundary nodes that are prescribed by the bounce back scheme at every time step it also makes computations for particles of irregular shapes simpler and more efficient proteus has been developed in two as well as three dimensions this new method has been validated by comparing its results with those from experimental measurements for a single sphere settling in an enclosure under gravity as a demonstration of the efficiency and capabilities of the present method the settling of a large number 1232 of spherical particles is simulated in a narrow box under two different boundary conditions it is found that when the no slip boundary condition is imposed at the front and rear sides of the box the particles motion is significantly hindered under the periodic boundary conditions the particles move faster the simulations show that the sedimentation characteristics in a box with periodic boundary conditions at the two sides are very close to those found in the sedimentation of two dimensional circular particles

the number of composition factors of order p in completely reducible groups of characteristic p abstract let q be a power of a prime p and let g be a completely reducible subgroup of gl d q we prove that the number of composition factors of g that have prime order p is at most e q d 1 p 1 where e q is a function of q satisfying 1 e q 3 2 for every q we give examples showing this bound is sharp infinitely often

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

physalis a new o n method for the numerical simulation of disperse systems potential flow of spheres this paper presents a new approach to the direct numerical simulation of potential problems with many spherical internal boundaries e g many spheres in potential flow the basic idea is to use a local analytic representation valid near the particle and to match it to an external field calculated by a standard finite difference or finite element method in this way the geometric complexity arising from the irregular relation between the particle boundary and the underlying mesh is avoided and fast solvers can be used the results suggest that the computational effort increases less than proportionally to the number of particles and additionally that meshes that would be excessively coarse as measured in terms of particle radius in a conventional calculation can be used without significant loss of accuracy in separate if preliminary work the same approach has been extended to the simulation of viscous flow about spheres and cylinders at finite reynolds numbers

smoothed profile method for particulate flows error analysis and simulations abstract we re formulate and analyze a new method for particulate flows the so called smoothed profile method spm first proposed in y nakayama r yamamoto simulation method to resolve hydrodynamic interactions in colloidal dispersions phys rev e 71 2005 036707 which uses a fixed computational mesh without conformation to the geometry of the particles the method represents the particles by certain smoothed profiles to construct a body force term added into the navier stokes equations spm imposes accurately and efficiently the proper conditions at the particle fluid interface in particular while the original method employs a fully explicit time integration scheme we develop a high order semi implicit splitting scheme which we implement in the context of spectral hp element discretization first we show that the modeling error of spm has a non monotonic dependence on the time step size t it is a function of t where is the kinematic viscosity of fluid and is the interface thickness of the smoothed profile subsequently we present several steady and unsteady simulations including flow past 3d complex shaped particles and compare against direct numerical simulations and the force coupling method fcm

a second order method for three dimensional particle simulation this paper describes a numerical method for the direct numerical simulation of navier stokes flows with one or more solid spheres the particles may be fixed or mobile and they may have different radii the basic idea of the method stems from the observation that due to the no slip condition in the reference frame of each particle the velocity near the particle boundary is very small so that the stokes equations constitute an excellent approximation to the full navier stokes problem the general analytic solution of the stokes equations can then be used to transfer the no slip condition from the particle surface to the adjacent grid nodes in this way the geometric complexity arising from the irregular relation between the particle boundary and the underlying mesh is avoided and fast solvers can be used the method is validated by a detailed comparison with spectral solutions for the flow past a sphere at reynolds numbers of 50 and 100 the existence in these situations of a stokes region near the particle is explicitly demonstrated other numerical experiments to show the performance of the code are also described to illustrate the power and efficiency of the method a simulation of decaying homogeneous turbulence in a cell containing 100 moveable spheres is described as implemented here the method can only be applied to simple body shapes such as spheres and cylinders extensions to more general situations are mentioned in the last section

interior estimates for the wavelet galerkin method in this paper we derive an interior estimate for the galerkin method with wavelet type basis such an estimate follows from interior galerkin equations which are common to a class of methods used in the solution of elliptic boundary value problems we show that the error in an interior domain omega 0 can be estimated with the best order of accuracy possible provided the solution u is sufficiently regular in a slightly larger domain and that an estimate of the same order exists for the error in a weaker norm measuring the effects from outside the domain omega 0 examples of the application of such an estimate are given for different problems

convergence analysis of the q1 finite element method for elliptic problems with non boundary fitted meshes summary the aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain s boundary this problematic of the last century e g the finite difference methodology returns to topical studies with the huge development of domain embedding fictitious domain or cartesiangrid methods these methods use regular structured meshes most often cartesian for non aligned domains although non boundary fitted approaches become more and more applied very few studies are devoted to theoretical error estimates in this paper the convergence of a q1 nonconforming finite element method is analyzed for secondorder elliptic problems with dirichlet robin or neumann boundary conditions the finite element method uses standard q1 rectangular finite elements as the finite element approximate space is not contained in the original solution space this method is referred to as nonconforming a stairstep boundary defined from the cartesian mesh approximates the original domain s boundary the convergence analysis of the finite element method for such a kind of non boundary fitted stairstepped approximation is no treated in the literature the study of dirichlet problems is based on similar techniques as those classically used with boundary fitted linear triangular finite elements the estimates obtained for robin problems are novel and use some more technical arguments the rate of convergence is proved to be in o h 1 2 for the h 1 norm for all general boundary conditions and classical duality arguments allow to obtain an o h error estimate in the l 2 norm for dirichlet problems numerical results obtained with fictitious domain techniques that impose original boundary conditions on a non boundary fitted approximate immersed interface are presented these results confirm the theoretical rates of convergence copyright c 2007 john wiley sons ltd