a preconditioner for the steady state navier stokes equations we present a new method for solving the sparse linear system of equations arising from the discretization of the linearized steady state navier stokes equations also known as the oseen equations the solver is an iterative method of krylov subspace type for which we devise a preconditioner through a heuristic argument based on the fundamental solution tensor for the oseen operator the preconditioner may also be conceived through a weaker heuristic argument involving differential operators computations indicate that convergence for the preconditioned discrete oseen problem is only mildly dependent on the viscosity inverse reynolds number and most importantly that the number of iterations does not grow as the mesh size is reduced indeed since the preconditioner is motivated through analysis of continuous operators the number of iterations decreases for smaller mesh size which accords with better approximation of these operators

stabilized finite element schemes with lbb stable elements for incompressible flows we study stabilized fe approximations of supg type to the incompressible navier stokes problem revisiting the analysis for the linearized model we show that for conforming lbb stable elements the design of the stabilization parameters for many practical flows differs from that commonly suggested in literature and initially designed for the case of equal order approximation then we analyze a reduced supg scheme often used in practice for lbb stable elements to provide the reduced scheme with appropriate stability estimates we introduce a modified lbb condition which is proved for a family of fe approximations the analysis is given for the linearized equations numerical experiments for some linear and nonlinear benchmark problems support the theoretical results

conservative rezoning remapping for general quadrilateral meshes a classic problem in lagrangian numerical hydrodynamics is the conservative transfer of quantities from an old distorted mesh to a new mesh the same problem arises whenever the mesh is changed as for example in adaptive mesh techniques this transfer of information is an interpolation process which is frequently called rezoning or remapping the general problem of conservative rezoning from one arbitrary mesh to another may be formulated as follows m sub k integral integral integral sub v sub k rho r dv that is we compute the mass m sub k of each cell of the new mesh by integrating the known density distribution in the old mesh over the cell volume v sub k a direct intregation is generally prohibitive we show however that is is possible to convert this integral to a surface integral by the appropriate use of the divergence theorem thus greatly reducing the complexity of the problem for two dimensional generall quadrilateral meshes the resulting method is exact and particularly simple

an analysis of the time integration algorithms for the finite element solutions of incompressible navier stokes equations based on a stabilised formulation abstract this work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the navier stokes equations the stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes particular attention is given to the generalised method and the linear discontinuous in time finite element scheme the time integration schemes are first applied to two model problems represented by a first order differential equation in time and the one dimensional advection diffusion equation and subjected to a detailed mathematical analysis based on the fourier series expansion in order to establish the accuracy and efficiency of the time integration schemes for the navier stokes equations a detailed computational study is performed of two standard numerical examples unsteady flow around a cylinder and flow across a backward facing step it is concluded that the semi discrete generalised method provides a viable alternative to the more sophisticated and expensive space time methods for simulations of unsteady flows of incompressible fluids governed by the navier stokes equations

stability of the fractional step t scheme for the nonstationary navier stokes equations stability and convergence analysis of the fractional step theta scheme for the navier stokes equations coupled with the finite element approximations is presented the solutions of the underlying stokes and burgers problems are described in detail the asymptotic solution obtained by the numerical stabilization is compared with the analytical solution of the lid driven cavity problem

on large start up error of bdf2 abstract the second order backward difference bdf2 method requires solutions at two previous time steps and is often started with the first order backward difference bdf1 method the initial time step is typically taken to be small in order to minimize the effect of the first order error however contrary to expectations the solution computed in this way generates a very large error for stiff problems and the error grows as the initial time step is further reduced as shown in this note the problem is originated from the fact that the variable coefficient bdf2 method combined with any first or higher order one step method is asymptotically equivalent to the trapezoidal method and therefore not l stable based on the study presented it is strongly recommended that bdf2 method be started with a self starting second order implicit runge kutta method which guarantees second order accuracy and l stability at no additional cost

adaptive time stepping for incompressible flow part i scalar advection diffusion even the simplest advection diffusion problems can exhibit multiple time scales this means that robust variable step time integrators are a prerequisite if such problems are to be efficiently solved computationally the performance of the second order trapezoid rule using an explicit adams bashforth method for error control is assessed in this work this combination is particularly well suited to long time integration of advection dominated problems herein it is shown that a stabilized implementation of the trapezoid rule leads to a very effective integrator in other situations specifically diffusion problems with rough initial data and general advection diffusion problems with different physical time scales governing the system evolution

a self optimizing single step algorithm for the numerical integration of parabolic equations abstract an unconditionally stable single step time marching algorithm for parabolic equations is presented in this algorithm time stepping parameters are chosen according to an error condition such that large time step accuracy is achieved and the solution is also free from oscillations it is different from conventional algorithms such as the crank nicolson scheme in that it uses a matrix of time stepping parameters rather than a single scalar value these parameters can be approximated inexpensively this algorithm is easily programmed and comparisons are made with other methods for different cases of a heat conduction problem it is found that the algorithm gives a more accurate solution for all degrees of freedom for the cases considered it is therefore advantageous over the conventional variably weighted euler method which has a constraint that the same value must be used for all degrees of freedom and that values have to be changed by trial and error for different time step sizes and mesh refinements in order to achieve best results

a linearly implicit predictor corrector method for reaction diffusion equations abstract a novel linearly implicit predictor corrector scheme is developed for the numerical solution of reaction diffusion equations iterative processes are avoided by treating the nonlinear reaction terms explicitly while maintaining superior accuracy and stability properties compared to the well known methods and linearly implicit runge kutta methods the proposed method allows the opportunity of solving large systems of reaction diffusion equations by alleviating the necessity of solving the accompanying large linear systems of algebraic equations due to the natural parallelism which surfaces across the system numerical results confirm the enhanced stability accuracy and efficiency of the method when applied to reaction diffusion equations arising in biochemistry and population ecology

higher rank numerical ranges and compression problems we consider higher rank versions of the standard numerical range for matrices a central motivation for this investigation comes from quantum error correction we develop the basic structure theory for the higher rank numerical ranges and give a complete description in the hermitian case we also consider associated projection compression problems

l0 stable split linear multistep formulas for parabolic pdes abstract new types of split linear multistep methods slmms are applied to the numerical solution of parabolic differential equations the formulas derived achieve up to fourth order in time and unlike the crank nicolson method are l0 stable

a boundary integral modification of the galerkin least squares formulation for the stokes problem abstract a new boundary integral term is added to the galerkin least squares gls formulation the new formulation compensates for a lack of consistency noted for the traditional gls method when bilinear interpolation is used on quadrilateral elements exact solutions at the nodes are obtained for poiseuille flow and another test problem and the pressure singularity for the driven cavity is better captured

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

minimal and maximal operator spaces and operator systems in entanglement theory abstract we examine k minimal and k maximal operator spaces and operator systems and investigate their relationships with the separability problem in quantum information theory we show that the matrix norms that define the k minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k positive linear maps and bound entanglement similarly we investigate the k super minimal and k super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k block positive operators and unnormalized states with schmidt number no greater than k respectively we characterize a class of norms on the k super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the schmidt number of a quantum state that generalizes the recently developed separability criterion based on trace contractive maps

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

detecting quantum entanglement we review the criteria for separability and quantum entanglement both in a bipartite as well as a multipartite setting we discuss bell inequalities entanglement witnesses entropic inequalities bound entanglement and several features of multipartite entanglement we indicate how these criteria bear on the experimental detection of quantum entanglement

quantum gate fidelity in terms of choi matrices we provide new results for computing and comparing the quantum gate fidelity of quantum channels via their choi matrices we extend recent work that showed that there exist non dual pairs of quantum channels with equal gate fidelity by providing an explicit characterization of all such channels we use our characterization to show that when the dimension is 2 or 3 under slightly stronger hypotheses the gate fidelity of two channels is equal if and only if their difference equals the difference of some unital map and its dual a fact that has been shown to be false when the dimension is 4 or larger we also present a formula for the minimum gate fidelity of a channel in terms of a well studied norm on a compression of its choi matrix as a consequence several new ways of bounding and approximating the minimum gate fidelity follow including a simple semidefinite program to compute it for qubit channels

invasion percolation and the incipient infinite cluster in 2d we establish two links between two dimensional invasion percolation and kesten s incipient infinite cluster iic we first prove that the kth moment of the number of invaded sites within the box n n n n is of order n2 n k for k 1 where n is the probability that the origin in critical percolation is connected to the boundary of a box of radius n this improves a result of y zhang we show that the size of the invaded region when scaled by n2 n is tight

hp adaptive time integration based on the bdf for viscous flows this paper presents a procedure based on the backward differentiation formulas of order 1 to 5 to obtain efficient time integration of the incompressible navier stokes equations the adaptive algorithm performs both stepsize and order selections to control respectively the solution accuracy and the computational efficiency of the time integration process the stepsize selection h adaptivity is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance the order selection p adaptivity relies on the idea that low accuracy solutions can be computed efficiently by low order time integrators while accurate solutions require high order time integrators to keep computational time low the selection is based on a stability test that detects growing numerical noise and deems a method of order p stable if there is no method of lower order that delivers the same solution accuracy for a larger stepsize hence it guarantees both that 1 the used method of integration operates inside of its stability region and 2 the time integration procedure is computationally efficient the proposed time integration procedure also features a time step rejection and quarantine mechanisms a modified newton method with a predictor and dense output techniques to compute solution at off step points

a new stabilization of adaptive step trapezoid rule based on finite difference interrupts the adaptive step trapezoid rule tr is a generally effective numerical integrator but it is prone to ringing instability and solution stall we introduce a new stabilization algorithm based on finite difference interrupts fdi that reduces ringing and prevents stall unlike previously reported stabilization schemes our algorithm achieves stability and second order accuracy without incurring significant computational cost or spurious diffusion tr with fdi is at least as stable and accurate as existing methods when solving the prototypical scalar problem we demonstrate that it has better stability accuracy and cost when solving the tightly coupled system describing free surface flows of viscoelastic liquids though we demonstrate tr with fdi in the context of a finite element method of lines it is applicable to any tr based algorithm

least squares finite element formulations for viscous incompressible and compressible fluid flows abstract we present least squares based finite element formulations for the numerical solution of viscous fluid flows governed by the navier stokes equations as an alternate approach to the well known weak form galerkin finite element formulations the use of least squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise and the resulting linear algebraic problem will have a symmetric positive definite coefficient matrix we address the issue of norm equivalence of the least squares functional and its implications on the resulting finite element model in particular we develop an understanding of the compromise that must exist between the optimality and practicality of the finite element model the latter measured in terms of c k regularity across inter element boundaries we show through numerical examples that for the traditional c 0 basis such a compromise is possible when high p levels are used to span the finite element spaces when a low p level solution is desired guidelines are presented to obtain a reliable least squares collocation solution numerical examples are presented to demonstrate the high and low p level approaches these include incompressible flow past two circular cylinders in a side by side arrangement for gap sizes s d 2 0 and 0 85 incompressible flow past a square cylinder and subsonic transonic and supersonic compressible flow past a circular cylinder in addition we present a discontinuous least squares formulation where c k regularity across inter element boundaries is enforced in a weak sense through the least squares functional allowing for h and p type non conformities in the computational domain

a least squares finite element formulation for unsteady incompressible flows with improved velocity pressure coupling in the weak form galerkin formulation for incompressible flows the pressure has a well understood role at all times it may be interpreted as a lagrange multiplier that enforces the divergence free constraint on the velocity field this is not the case in least squares formulations for incompressible flows where the divergence free constraint is enforced in a least squares sense in a variational setting of residual minimization thus the role of the pressure in a least squares formulation is rather vague we find that this lack of velocity pressure coupling in least squares formulations may induce spurious temporal pressure oscillations when using the non stationary form of the equations we present a least squares formulation with improved velocity pressure coupling based on the use of a regularized divergence free constraint a first order system least squares fosls approach based on velocity pressure and vorticity is used to allow the use of practical c0 element expansions in the finite element model we use high order spectral element expansions in space and second and third order time stepping schemes excellent conservation of mass and accuracy of computed pressure metrics are demonstrated in the numerical results

time integration for ale simulations of fluid structure interaction problems stepsize and order selection based on the bdf abstract we present an adaptive algorithm for time integration of fluid structure integration problems the method relies on a fully coupled procedure to solve fsi problems in which a naturally gcl compliant ale formulation for the finite element spatial discretization is used the main originality of the proposed solution procedure is that time integration is performed using automatic order and stepsize selections hp adaptivity based on the backward differentiation formulas bdf the stepsize selection is based on a local error estimate an error controller and a step rejection mechanism it guarantees that the solution precision is within the user targeted tolerance the order selection is based on a stability test and a quarantine mechanism the selection is performed to ensure that no other methods within the family of 0 stable bdf methods would produce a solution of the targeted precision for a larger stepsize and thus a lower computational time to improve efficiency the time integration procedure also relies on a modified newton method and a predictor the time adaptive algorithm behaviors and performances are assessed on the vortex induced translational and rotational vibrations of a square cylinder and on the wake induced vibrations of 3 cylinders in an in line arrangement the algorithm yields substantial cpu time savings compared to constant stepsize and order integration while delivering solutions of prescribed accuracies

a new family of projection schemes for the incompressible navier stokes equations with control of high frequency damping abstract a simple spatially discrete model problem consisting of mass points and dash pots is presented which allows for the assessment of the properties of different projection schemes for the solution of the incompressible navier stokes equations in particular the temporal accuracy the stability and the numerical damping are investigated the present study suggests that it is not possible to formulate a second order accurate projection pressure correction scheme which possesses any high frequency damping motivated by this observation two new families of projection schemes are proposed which are developed from the generalised midpoint rule and from the generalised method respectively and offer control over high frequency damping both schemes are investigated in detail on the basis of the model problem and subsequently implemented in the context of a finite element formulation for the incompressible navier stokes equations comprehensive numerical studies of the flow in a lid driven cavity and the flow around a cylinder are presented the observations made are in agreement with the conclusions drawn from the model problem

efficient preconditioning of the linearized navier stokes equations for incompressible flow we outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of navier stokes equations we describe a very general strategy for preconditioning that has two basic building blocks a multigrid v cycle for the scalar convection diffusion operator and a multigrid v cycle for a pressure poisson operator we present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid robust with respect to the time step and only deteriorates very slowly as the reynolds number is increased

segregated runge kutta methods for the incompressible navier stokes equations in this work we propose runge kutta time integration schemes for the incompressible navier stokes equations with two salient properties first velocity and pressure computations are segregated at the time integration level without the need to perform additional fractional step techniques that spoil high orders of accuracy second the proposed methods keep the same order of accuracy for both velocities and pressures the segregated runge kutta methods are motivated as an implicit explicit runge kutta time integration of the projected navier stokes system onto the discrete divergence free space and its re statement in a velocity pressure setting using a discrete pressure poisson equation we have analysed the preservation of the discrete divergence constraint for segregated runge kutta methods and their relation in their fully explicit version with existing half explicit methods we have performed a detailed numerical experimentation for a wide set of schemes from first to third order including implicit and imex integration of viscous and convective terms for incompressible laminar and turbulent flows further segregated runge kutta schemes with adaptive time stepping are proposed

a better consistency for low order stabilized finite element methods abstract the standard implementation of stabilized finite element methods with a piece wise function space of order lower than the highest derivative present in the partial differential equation often suffers from a weak consistency that can lead to reduced accuracy the popularity of these low order elements motivates the development of a new stabilization operator which globally reconstructs the derivatives not present in the local element function space this new method is seen to engender a stronger consistency leading to better convergence and improved accuracy applications to the navier stokes equations are given which illustrate the improvement at a negligible additional cost

finite element stabilization parameters computed from element matrices and vectors abstract we propose new ways of computing the stabilization parameters used in the stabilized finite element methods such as the streamline upwind petrov galerkin supg and pressure stabilizing petrov galerkin pspg formulations the parameters are computed based on the element level matrices and vectors which automatically take into account the local length scales advection field and the reynolds number we describe how we compute these parameters in the context of first a time dependent advection diffusion equation and then the navier stokes equations of unsteady incompressible flows

composite three dimensional grids generated by elliptic systems a technique is presented for constructing a boundary conforming grid throughout a general threedimensional flow region as a composite of subregion grids each subregion grid is generated numerically by solving a nonlinear system of elliptic equations the boundary values represent nodal points in a quasi twodimensional grid that covers the curved surface bounding the subregion and are generated numerically by a modified elliptic system a primary feature of the technique is that the composite three dimensional grid remains both continuous and smooth across the surface of juncture between any two adjoining subregions sample results are given for a wing body combination

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

implicit finite incompressible elastodynamics with linear finite elements a stabilized method in rate form abstract we propose a stabilization method for linear tetrahedral finite elements suitable for the implicit time integration of the equations of nearly and fully incompressible nonlinear elastodynamics in particular we derive and discuss a generalized framework for stabilization and implicit time integration that can comprehensively be applied to the class of all isotropic hyperelastic models in this sense the presented development can be considered an important extension and complement to the stabilization approach proposed by the authors in previous work which was instead focused on explicit time integration and simple neo hookean models for nearly incompressible elasticity with the goal of computational efficiency we also present a two step block gauss seidel strategy for the time update of displacements velocities and pressures specifically a mixed system of equations for the velocity and pressure is updated implicitly in a first stage and the displacements are updated explicitly in a second stage the proposed mixed formulation is then embedded in newton type strategies for the nonlinear solution of the equations of motion various implicit time integration strategies are considered and particularly we focus on high frequency dissipation time integrators which are preferable in transient mechanics applications an extensive set of numerical computations with linear tetrahedral elements is presented to demonstrate the performance of the proposed approach

fast iterative solvers for buoyancy driven flow problems we outline a new class of robust and efficient methods for solving the navier stokes equations with a boussinesq model for buoyancy driven flow we describe a general solution strategy that has two basic building blocks an implicit time integrator using a stabilized trapezoid rule with an explicit adams bashforth method for error control and a robust krylov subspace solver for the spatially discretized system we present numerical experiments illustrating the efficiency of the chosen preconditioning schemes with respect to the discretization parameters

split linear multistep methods for the numerical integration of stiff differential systems a new approach to the problem of numerically integrating stiff differential systems is described in this approach a linear multistep method the basic method is split into a kind of predictor corrector scheme where the predictor is also implicit if this splitting is done in an appropriate manner the modified method has considerably better stability properties than the basic method as a result splitting methods are particularly useful for problems where conventional integration methods experience stability difficulties in particular some highly stable split linear multistep methods based on backward differentiation formulae are derived and a highly stable variable step implementation is proposed

consistent supg method for transient transport problems stability and convergence we consider the time space discretization of the transient advection equation discretization in space is performed by the streamline upwind petrov galerkin method and in time we use an a stable finite difference operator the formulation is strongly consistent in the sense that the time derivative is included in the stabilization term uniform stability of the general formulation is proved under a regularity condition on data or a moderate inverse cfl condition that allows for optimal choices of the discretization parameters both the backward euler method bdf1 the crank nicolson scheme and the second order backward differentiation formula bdf2 enter the framework and quasi optimal convergence is proved for these schemes

relations between invasion percolation and critical percolation in two dimensions we study invasion percolation in two dimensions we compare connectivity properties of the origin s invaded region to those of a the critical percolation cluster of the origin and b the incipient infinite cluster to exhibit similarities we show that for any k 1 the k point function of the first socalled pond has the same asymptotic behavior as the probability that k points are in the critical cluster of the origin more prominent though are the differences we show that there are infinitely many ponds that contain many large disjoint pc open clusters further for k 1 we compute the exact decay rate of the distribution of the radius of the kth pond and see that it differs from that of the radius of the critical cluster of the origin we finish by showing that the invasion percolation measure and the incipient infinite cluster measure are mutually singular

a stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation abstract a stabilized finite element formulation for incompressible viscous flows is derived the starting point are the modified navier stokes equations incorporating naturally the necessary stabilization terms via a finite increment calculus fic procedure application of the standard finite element galerkin method to the modified differential equations leads to a stabilized discrete system of equations overcoming the numerical instabilities emanating from the advective terms and those due to the lack of compatibility between approximate velocity and pressure fields the fic method also provides a natural explanation for the stabilization terms appearing in all equations for both the navier stokes and the simpler stokes equations transient solution schemes with enhanced stabilization properties are also proposed finally a procedure for computing the stabilization parameters is presented

a family of norms with applications in quantum information theory we consider a family of vector and operator norms defined by the schmidt decomposition theorem for quantum states we use these norms to tackle two fundamental problems in quantum information theory the classification problem for k positive linear maps and entanglement witnesses and the existence problem for nonpositive partial transpose bound entangled states we begin with an analysis of the norms showing that the vector norms can be explicitly calculated and we derive several inequalities in order to bound the operator norms and compute them in special cases we then use the norms to establish what appears to be the most general spectral test for k positivity currently available showing how it implies several other known tests as well as some new ones building on this work we frame the nonpositive partial transpose bound entangled problem as a concrete problem on a specific limit specifically that a particular entangled werner state is bound entangled if and only if a certain norm inequality ho

coupling stabilized finite element methods with finite difference time integration for advection diffusion reaction problems abstract we present some numerical schemes for the unsteady advection diffusion reaction linear problem in one space dimension we investigate two possible different ways of combining the discretization in time and in space where the sequence of the discretizations is interchanged discretization in time is performed by using the crank nicolson finite difference scheme while for the space discretization we consider three classical stabilized finite element schemes and the more recent link cutting bubble strategy proposed in f brezzi g hauke l d marini g sangalli link cutting bubbles for the stabilization of convection diffusion reaction problems math models methods appl sci 13 2003 445 461 numerical experiments are presented to assess and valuate the capabilities of the proposed methods an l 1 error analysis of the link cutting bubble strategy for solving the steady problem is included

an adaptive remeshing strategy for flows with moving boundaries and fluid structure interaction the primary objective of this work is to extend the capability of the arbitrary lagrangian eulerian ale based strategy for solving fluid structure interaction problems this is driven by the fact that the ale mesh movement techniques will not be able to treat problems in which fluid structure interface experiences large motion in addition for certain problems the need arises to capture accurately flow features such as a region with high gradients of the solution variables this can be achieved by incorporating an adaptive remeshing procedure into the solution strategy as in our previous works comput methods appl mech eng 2006 195 1633 1666 comput methods appl mech eng 2006 195 5754 5779 here the fluid flow is governed by the incompressible navier stokes equations and modelled by using stabilized low order velocity pressure finite elements the motion of the fluid domain is accounted for by an arbitrary lagrangian eulerian ale strategy the flexible structure is represented by means of appropriate standard finite element formulations while the motion of the rigid body is described by rigid body dynamics for temporal discretization of both fluid and solid bodies the discrete implicit generalized method is employed the resulting strongly coupled set of non linear equations is then solved by means of a novel partitioned solution procedure which is based on the newton raphson methodology and incorporates full linearization of the overall incremental problem within the adaptive solution strategy the quality of fluid mesh and the solution quality indicator are evaluated regularly and compared against the appropriate remeshing criteria to decide whether a remeshing step is required the adaptive remeshing procedure follows closely the standard computational procedure in which the adaptive remeshing process produces a mesh that can capture salient features of the flow field for the problems under consideration in this work the motion of the fluid boundary very often results in boundaries with very high curvatures and a fluid domain that contain areas with small cross sections to be able to generate meshes that give result with acceptable accuracy these local geometrical features need to be included in determining the element density distribution the numerical examples demonstrate the robustness and efficiency of the methodology copyright 2007 john wiley sons ltd

adaptive time stepping for incompressible flow part ii navier stokes equations we outline a new class of robust and efficient methods for solving the navier stokes equations we describe a general solution strategy that has two basic building blocks an implicit time integrator using a stabilized trapezoid rule with an explicit adams bashforth method for error control and a robust krylov subspace solver for the spatially discretized system we present numerical experiments illustrating the potential of our approach

algebraic grid generation three methods are described for transforming grids in bounded two and three dimensional physical domains into a uniform grid in a rectangular computational domain the methods are based on mathematical interpolation functions and do not require the solution of differential equations or the use of complex variables they are simply referred to as algebraic methods and are called transfinite interpolation the multisurface method and the two boundary technique the primary advantage of the methods is that they provide explicit control of physical grid shape and physical grid spacing secondly they require relatively few computations consequently the application of inter active computer graphics in conjunction with the methods is advocated for rapid generation of grids the basic mathematical structure of each method is described and particular attention is given to surface representation parameter variable generation and control function generation physical boundary topology and grid derivative requirements are presented

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

aladins an algebraic splitting time adaptive solver for the incompressible navier stokes equations we address a time adaptive solver specifically devised for the incompressible navier stokes ins equations one of the challenging issues in this context is the identification of a reliable a posteriori error estimator typical strategies are based on the combination of the solutions computed either with two different time steps or two schemes with different accuracy in this paper we move from the pressure correction algebraic factorizations formerly proposed by saleri veneziani 2005 these schemes feature an intrinsic hierarchical nature such that an accurate solution for the pressure is obtained by computing intermediate low order guesses the difference between the two estimates provide a natural a posteriori estimator after introducing the incremental formulation of the pressure correction schemes we address the properties of this approach including extensive implementation details numerical results presented refer to 2d and 3d unstructured problems with a particular emphasis on cardiovascular problems which are expected to largely benefit from time adaptive solvers in memory of f saleri 1965 2007

analysis of the block gauss seidel solution procedure for a strongly coupled model problem with reference to fluid structure interaction the block gauss seidel procedure is widely used for the resolution of the strong coupling in the computer simulation of fluid structure interaction based on a simple model problem this work presents a detailed analysis of the convergence behaviour of the method in particular the model problem is used to highlight some aspects that arise in the context of the application of the block gauss seidel method to fsi problems thus the effects of the time integration schemes chosen of relaxation techniques of physical constraints and non linearities on the convergence of the iterations are investigated

scaling limit of the invasion percolation cluster on a regular tree we prove existence of the scaling limit of the invasion percolation cluster ipc on a regular tree the limit is a random real tree with a single end the contour and height functions of the limit are described as certain diusive stochastic processes this convergence allows us to recover and make precise certain asymptotic results for the ipc in particular we relate the limit of the rescaled level sets of the ipc to the local time of the scaled height function

stabilized shock hydrodynamics i a lagrangian method abstract a new supg stabilized formulation for lagrangian hydrodynamics of materials satisfying mie gruneisen equation of state is proposed it allows the use of simplex type triangular tetrahedral meshes as well as the more commonly used brick type quadrilateral hexahedral meshes the proposed method yields a globally conservative formulation in which equal order interpolation p1 or q1 isoparametric finite elements is applied to velocities displacements and pressure as a direct consequence and in contrast to traditional cell centered multidimensional hydrocode implementations the proposed formulation allows a natural representation of the pressure gradient on element interiors the supg stabilization involves additional design requirements specific to the lagrangian formulation a discontinuity capturing operator in the form of a noh type viscosity with artificial heat flux is used to preserve stability and smoothness of the solution in shock regions a set of challenging shock hydrodynamics benchmark tests for the euler equations of gas dynamics in one and two space dimensions is presented in the two dimensional case computations performed on quadrilateral and triangular grids are analyzed and compared these results indicate that the new formulation is a promising technology for hydrocode applications

application of a variational method for generating adaptive grids the application of variational methods for generating adaptive grids is not as straightforward as one is led to believe proper scaling suitable weight functions and appropriate clustering on boundaries must be employed to obtain a satisfactory grid this work which is based on the framework developed by brackbill and saltzman provides simple methods for determining scaling and investigates possible options for selecting the weight function and clustering points on the boundaries the concepts developed here are applied to two two dimensional problems a model problem based on burger s equation which contains two length scales and transonic flow past airfoils using the euler equations

a new finite element formulation for computational fluid dynamics x the compressible euler and navier stokes equations a space time element method is presented for solving the compressible euler and navier stokes equations the proposed formulation includes the variational equation predictor multi corrector algorithms and boundary conditions the variational equation is based on the time discontinuous galerkin method in which the physical entropy variables are employed a least squares operator and a discontinuity capturing operator are added resulting in a high order accurate and unconditionally stable method implicit explicit predictor multi corrector algorithms applicable to steady as well as unsteady problems are presented techniques are developed to enhance their efficiency implementation of boundary conditions is addressed in particular a technique is introduced to satisfy nonlinear essential boundary conditions and a consistent method is presented to calculate boundary fluxes numerical results are presented to demonstrate the performance of the method

invasion percolation on regular trees we consider invasion percolation on a rooted regular tree for the infinite cluster invaded from the root we identify the scaling behavior of its r point function for any r 2 and of its volume both at a given height and below a given height we find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation the scaling functions differ thus somewhat surprisingly the two clusters behave differently in fact we prove that their laws are mutually singular in addition we derive scaling estimates for simple random walk on the cluster starting from the root we show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster far above the root the two clusters have the same law locally but not globally a key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster these weights decay toward the critical value for ordinary percolation but only slowly and this slow decay causes the scaling behavior to differ from that of the incipient infinite cluster

a fictitious domain distributed lagrange multiplier based fluid structure interaction scheme with hierarchical b spline grids abstract we present a numerical scheme for fluid structure interaction based on hierarchical b spline grids and fictitious domain distributed lagrange multipliers the incompressible navier stokes equations are solved over a cartesian grid discretised with b splines the fluid grid near the immersed solids is refined locally using hierarchical b splines the immersed solid is modelled as geometrically exact beam discretised with standard linear lagrange shape functions the kinematic constraint at the fluid solid interface is enforced with distributed lagrange multipliers the unconditionally stable and second order accurate generalised method is used for integration in time for both the fluid and solid domains a fully implicit and fully coupled solution scheme is developed by using the newton raphson method to solve the non linear system of equations obtained with galerkin weak formulation first the spatial and temporal convergence of the proposed scheme is assessed by studying steady and unsteady flow past a fixed cylinder then the scheme is applied to several benchmark problems to demonstrate the efficiency and robustness of the proposed scheme the results obtained with the present scheme are compared with the reference values

space time coupled spectral hp least squares finite element formulation for the incompressible navier stokes equations we consider least squares finite element models for the numerical solution of the non stationary navier stokes equations governing viscous incompressible fluid flows the paper presents a formulation where the effects of space and time are coupled resulting in a true space time least squares minimization procedure as opposed to a space time decoupled formulation where a least squares minimization procedure is performed in space at each time step the formulation is first presented for the linear advection diffusion equation and then extended to the navier stokes equations the formulation has no time step stability restrictions and is spectrally accurate in both space and time to allow the use of practical c0 element expansions in the resulting finite element model the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity as an additional independent variable and the least squares method is used to develop the finite element model of the governing equations high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method in matrix free form spectral convergence of the l2 least squares functional and l2 error norms in space time is verified using a smooth solution to the two dimensional non stationary incompressible navier stokes equations numerical results are presented for impulsively started lid driven cavity flow oscillatory lid driven cavity flow transient flow over a backward facing step and flow around a circular cylinder the results demonstrate the predictive capability and robustness of the proposed formulation even though the space time coupled formulation is emphasized we also present the formulation and numerical results for least squares space time decoupled finite element models the numerical results show that the space time coupled formulation has superior predictive capabilities for flows demanding high space time resolution exemplified here by the transient flow over a backward facing step

spectral conditions for positive maps we provide partial classification of positive linear maps in matrix algebras which is based on a family of spectral conditions this construction generalizes the celebrated choi example of a map which is positive but not completely positive it is shown how the spectral conditions enable one to construct linear maps on tensor products of matrix algebras which are positive but only on a convex subset of separable elements such maps provide basic tools to study quantum entanglement in multipartite systems

flows in forward deformable roll coating gaps roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter a continuous liquid layer and to apply it to a continuous flexible substrate of the two rolls that make a forward roll coating gap one is often covered by a layer of deformable elastomer thin films can be obtained without the risk of clashing two hard rolls liquid carried into the converging side of the gap can develop high enough pressure to deform the resilient cover which changes the gap geometry and thus alters the velocity and pressure fields the complete understanding of the flow in this situation is vital to the optimization of this widely used coating method however this elastohydrodynamic action is not well understood the situation is similar to what is called the soft elastohydrodynamic lubrication regime soft ehl however the range of minimum distance between the rotating rolls roll speed and therefore flow rate through the gap in the roll coating process is one to three orders of magnitude larger than the typical values reported in previous work on soft ehl earlier works on deformable roll coating analyzed the action with both the lubrication approximation and the full navier stokes solution and different one dimensional models of roll cover deformation in order to test the accuracy of the past approaches and to evaluate the relationship between the empirical constant used in the one dimensional model to the relevant physical parameters a complete two dimensional formulation has to be employed for both the liquid flow and the solid deformation in this work the flow between a rigid and a deformable rotating roll was examined by solving the complete navier stokes system coupled with a non linear plane strain model of the roll cover deformation the approximate and computationally cheaper approach is evaluated in which the compliant wall is represented by an array of radially oriented hookean springs the equation system was solved by the galerkin finite element method the resulting set of non linear algebraic equations of the fully coupled problem was solved by newton s method with initialization by pseudo arc length continuation as parameters were varied results show how roll deformation affects the total flow rate and forces on the rolls and illustrate how a deformable roll can be used to obtain thin coated layers with much less sensitivity to roll runout than those obtained with rigid rolls

exponential growth of ponds in invasion percolation on regular trees in invasion percolation the edges of successively maximal weight the outlets divide the invasion cluster into a chain of ponds separated by outlets on the regular tree the ponds are shown to grow exponentially with law of large numbers central limit theorem and large deviation results the tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster with a logarithmic correction

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

fast and accurate simulations of air cooled structures there are fields of engineering for which the cad based navier stokes solvers are too expensive architecture and medicine for blood flows are two such examples mesh generation and adaptation is also a bottleneck because the users are not expected to have the know how we report here on a navier stokes solver for incompressible temperature and time dependent flows dedicated to architectural applications the building blocks are not new a finite element method with time implicit pressure projection steps and mesh adaptativity but putting them together in an easy to use and efficient 3d code is the challenge which motivates this paper for non engineering applications the user interface is a big problem in an earlier attempt we designed freefem3d based on a fictitious domain discretization thus avoiding boundary fitted mesh however it turned out that the display of the solutions requires a boundary fitted mesh it is possible to generate a feasible surface mesh for graphics but it is much more difficult to generate a feasible surface mesh for fem in this project the user interface is taken from freefem3d then with a marching cube algorithm we produce a graphic only feasible mesh finally a surface mesh adapted to a fem computation is constructed with an adaptation module and the result is used as input to a delaunay volumic mesh generator the solver is optimized and parallelized all modules are the authors work three applications are presented for which the data preparation takes less than a day and results are obtained overnight on a pc cluster one of the application is presented in details it is an air cooling system for a canister containing radio nucleides

fourier analysis of semi discrete and space time stabilized methods for the advective diffusive reactive equation i supg abstract in this paper the work initiated in part one and two is extended to the transient subgrid scale gradient subgrid scale sgs gsgs stabilized method temporal accuracy and stability of semi discrete and time discontinuous space time versions of the method are examined for transient advection diffusion reaction problems also for the above methods predictor multi corrector algorithms are analyzed with special emphasis on mass and source lumping which has been shown to enhance both stability and accuracy of explicit stabilized methods it will be shown that the parent supg sgs and sgs gsgs methods perform very similarly and that explicit predictor multi corrector mass source lumping practically retains the accuracy of the parent method while gaining stability

comparison of some finite element methods for solving the diffusion convection reaction equation abstract in this paper we describe several finite element methods for solving the diffusion convection reaction equation none of them is new although the presentation is non standard in an effort to emphasize the similarities and differences between them in particular it is shown that the classical supg method is very similar to an explicit version of the characteristic galerkin method whereas the taylor galerkin method has a stabilization effect similar to a sub grid scale model which is in turn related to the introduction of bubble functions

a stabilised immersed boundary method on hierarchical b spline grids abstract in this work an immersed boundary finite element method is proposed which is based on a hierarchically refined cartesian b spline grid and employs the non symmetric and penalty free version of nitsche s method to enforce the boundary conditions the strategy allows for h and p refinement and employs a so called ghost penalty term to stabilise the cut cells an effective procedure based on hierarchical subdivision and sub cell merging which avoids excessive numbers of quadrature points is used for the integration of the cut cells a basic laplace problem is used to demonstrate the effectiveness of the cut cell stabilisation and of the penalty free nitsche method as well as their impact on accuracy the methodology is also applied to the incompressible navier stokes equations where the supg pspg stabilisation is employed simulations of the lid driven cavity flow and the flow around a cylinder at low reynolds number show the good performance of the methodology excessive ill conditioning of the system matrix is robustly avoided without jeopardising the accuracy at the immersed boundaries or in the field

a computational framework for free surface fluid flows accounting for surface tension abstract this work is concerned with the simulation of incompressible newtonian fluid flow problems with free surfaces in the presence of surface tension the computational framework presented is based on the stabilised velocity pressure finite element method the movement and deformation of the computational domain are accounted for by employing the arbitrary lagrangian eulerian ale description of the fluid kinematics the numerical model is extended to incorporate surface tension effects a novel partitioned solution procedure is developed based on the newton raphson methodology which incorporates full linearisation of the overall incremental problem several numerical examples are provided to demonstrate the efficiency of the methodology

general linear methods connection to one step methods and invariant curves we generalize a result of kirchgraber 1986 on multistep methods we show that every strictly stable general linear method is essentially conjugate to a one step method of the same order this result may be used to show that general properties of one step methods carry over to general linear methods as examples we treat the existence of invariant curves and the construction of attracting sets

the mol solution of time dependent partial differential equations abstract a modified bdf scheme is proposed for the numerical integration of the ordinary differential equations that arise in the method of lines solution of time dependent partial differential equations it is to be expected that this new approach will be superior to the use of bdf for at least three important classes of problems namely for advection diffusion problems where advection dominates and here we expect very large gains in efficiency for problems where high accuracy is required and for problems where function evaluations are very expensive some analytic and numerical results are given to confirm these expectations

discretization of free surface flows and other moving boundary problems a system of elliptic partial differential equations and boundary conditions has been developed for generating boundary fitted element discretizations of two dimensional free and moving boundary problems terms in the differential equations are scaled for dimensional homogeneity and adjustable weighting of orthogonality smoothness and concentration of the coordinate mesh they govern grid points become finite element nodes mapped isoparametrically or subparametrically from a simple or patched computational domain concentration terms contain control functions and parameters that influence node spacing along each coordinate independently overall control is by patchwise parameters and functions successful selection of these to follow deforming flow regions is straightforward and is illustrated by analysis of steady and transient slide coating flows

a comparison of time discretization linearization approaches for the incompressible navier stokes equations abstract this paper presents a numerical study of two ways for discretizing and linearizing the time dependent incompressible navier stokes equations one approach consists in first applying a semi discretization in time by a fully implicit scheme then in each discrete time the equations are linearized by a fixed point iteration the number of iterations to reach a given stopping criterion is a priori unknown in this approach in the second approach rosenbrock schemes with s stages are used as temporal discretization the non linearity of the navier stokes equations is treated internally in the rosenbrock methods in each discrete time exactly s linear systems of equations have to be solved the numerical study considers five two dimensional problems with distinct features four implicit time stepping schemes and five rosenbrock methods are involved

a parallel block multi level preconditioner for the 3d incompressible navier stokes equations the development of robust and efficient algorithms for both steady state simulations and fully implicit time integration of the navier stokes equations is an active research topic to be effective the linear subproblems generated by these methods require solution techniques that exhibit robust and rapid convergence in particular they should be insensitive to parameters in the problem such as mesh size time step and reynolds number in this context we explore a parallel preconditioner based on a block factorization of the coefficient matrix generated in an oseen nonlinear iteration for the primitive variable formulation of the system the key to this preconditioner is the approximation of a certain schur complement operator by a technique first proposed by kay loghin and wathen siam j sci comput 2002 and silvester elman kay and wathen j comput appl math 128 2001 261 the resulting operator entails subsidiary computations solutions of pressure poisson and convection diffusion subproblems that are similar to those required for decoupled solution methods however in this case these solutions are applied as preconditioners to the coupled oseen system one important aspect of this approach is that the convection diffusion and poisson subproblems are significantly easier to solve than the entire coupled system and a solver can be built using tools developed for the subproblems in this paper we apply smoothed aggregation algebraic multigrid to both subproblems previous work has focused on demonstrating the optimality of these preconditioners with respect to mesh size on serial two dimensional steady state computations employing geometric multi grid methods we focus on extending these methods to large scale parallel three dimensional transient and steady state simulations employing algebraic multigrid amg methods our results display nearly optimal convergence rates for steady state solutions as well as for transient solutions over a wide range of cfl numbers on the two dimensional and three dimensional lid driven cavity problem

broken ergodicity and the geometry of rugged landscapes we study the nature and mechanisms of broken ergodicity be in specific random walk models corresponding to diffusion on random potential surfaces in both one dimension and high dimension using both rigorous results and nonrigorous methods we confirm several aspects of the standard be picture and show that others apply in one dimension but need to be modified in higher dimensions these latter aspects include the notions that a fixed temperature confining barriers increase logarithmically with time that components are necessarily bounded regions of state space which depend on the observational time scale and that the system continually revisits previously traversed regions of state space we examine our results in the context of several experiments and discuss some implications of our results for the dynamics of disordered and or complex systems

generation of orthogonal and nearly orthogonal coordinates with gridcontrol near boundaries a numerical procedure is presented for the generation of boundary fitted curvilinear coordinates with controllable mesh spacing and orthogonality the technique is based on a straightforward two step procedure involving the solution of poisson s equation the method is capable of generating orthogonal grids with partial control of the mesh spacing or nearly orthogonal grids with strict control of the mesh spacing the inhomogeneous terms in the poisson equations are automatically adjusted during the grid generation the technique is applicable to two dimensional simply connected regions typical of airfoils cascades diffusers and inlets

a low rank solver for the stochastic unsteady navier stokes problem abstract we study a low rank iterative solver for the unsteady navier stokes equations for incompressible flows with a stochastic viscosity the equations are discretized using the stochastic galerkin method and we consider an all at once formulation where the algebraic systems at all the time steps are collected and solved simultaneously the problem is linearized with picard s method to efficiently solve the linear systems at each step we use low rank tensor representations within the krylov subspace method which leads to significant reductions in storage requirements and computational costs combined with effective mean based preconditioners and the idea of inexact solve we show that only a small number of linear iterations are needed at each picard step the proposed algorithm is tested with a model of flow in a two dimensional symmetric step domain with different settings to demonstrate the computational efficiency

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

a computational framework for fluid structure interaction finite element formulation and applications abstract this work is concerned with the modelling of the interaction of fluid flow with flexibly supported rigid bodies the fluid flow considered is governed by the incompressible navier stokes equations and modelled by employing stabilised low order velocity pressure finite elements the motion of the fluid domain is accounted for by an arbitrary lagrangian eulerian ale strategy the rigid body motion excited by the fluid flow is restricted by the elastic and damping properties of the supports for the temporal discretisation the discrete implicit generalised method is employed the resulting strongly coupled set of nonlinear equations is solved by means of a novel partitioned solution procedure which is based on the newton raphson methodology and incorporates full linearisation of the overall incremental problem the strong coupling is resolved and optimal convergence of the residuals is achieved several numerical examples are presented to demonstrate the robustness and efficiency of the methodology the examples clearly capture the phenomena of vortex induced oscillations and galloping

a space time galerkin least squares finite element formulation of the navier stokes equations for moving domain problems a space time galerkin least squares finite element formulation of the navier stokes equations is presented for the analysis of free surface flows moving spatial configurations and deforming fluid structure interfaces the variational equation is based on the time discontinuous galerkin method employing the physical entropy variables the space time elements are oriented in time to accommodate the spatial deformations if the elements are oriented along the particle paths the formulation is lagrangian and if they are fixed in time it is eulerian consequently this formulation is analogous to the arbitrary lagrangian eulerian ale technique a novel mesh rezoning strategy is presented to orient the elements in time and adapt the fluid mesh to the changing spatial configuration numerical results are presented to show the performance of the method

a second order l stable time discretisation of the semiconductor device equations abstract the application of a second order and l stable predictor corrector method to the time discretisation of the semiconductor device equations is investigated the semidiscrete system is linearised at an equilibrium point and the resulting method is shown to be stable it is also shown that the method conserves the total current balance at each time level

outlets of 2d invasion percolation and multiple armed incipient infinite clusters we study invasion percolation in two dimensions focusing on properties of the outlets of the invasion and their relation to critical percolation and to incipient infinite clusters iic s first we compute the exact decay rate of the distribution of both the weight of the k th outlet and the volume of the k th pond next we prove bounds for all moments of the distribution of the number of outlets in an annulus this result leads to almost sure bounds for the number of outlets in a box b 2 n and for the decay rate of the weight of the k th outlet to pc we then prove existence of multiplearmed iic measures for any number of arms and for any color sequence which is alternating or monochromatic we use these measures to study the invaded region near outlets and near edges in the invasion backbone far from the origin

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