multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

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

finite calculus formulation for incompressible solids using linear triangles and tetrahedra summary many finite elements exhibit the so called volumetric locking in the analysis of incompressible or quasi incompressible problems in this paper a new approach is taken to overcome this undesirable effect the starting point is a new setting of the governing differential equations using a finite calculus fic formulation the basis of the fic method is the satisfaction of the standard equations for balance of momentum equilibrium of forces and mass conservation in a domain of finite size and retaining higher order terms in the taylor expansions used to express the different terms of the differential equations over the balance domain the modified differential equations contain additional terms which introduce the necessary stability in the equations to overcome the volumetric locking problem the fic approach has been successfully used for deriving stabilized finite element and meshless methods for a wide range of advective diffusive and fluid flow problems the same ideas are applied in this paper to derive a stabilized formulation for static and dynamic finite element analysis of incompressible solids using linear triangles and tetrahedra examples of application of the new stabilized formulation to linear static problems as well as to the semi implicit and explicit 2d and 3d non linear transient dynamic analysis of an impact problem and a bulk forming process are presented copyright 2004 john wiley sons ltd

stability of runge kutta nystrom methods in this paper a general and detailed study of linear stability of runge kutta nystrom rkn methods is given in the case that arbitrarily stiff problems are integrated we establish a condition that rkn methods must satisfy so that a uniform bound for stability can be achieved this condition is not satisfied by any method in the literature therefore a stable method is constructed and some numerical comparisons are made

stabilized formulation for the advection diffusion absorption equation using finite calculus and linear finite elements abstract a stabilized finite element method fem for the steady state advection diffusion absorption equation is presented the stabilized formulation is based on the modified governing differential equations derived via the finite calculus fic method the basis of the method is detailed for the 1d problem it is shown that the stabilization terms act as a non linear additional diffusion governed by a single stabilization parameter a critical constant value of this parameter ensuring a stabilized solution using two node linear elements is given more accurate numerical results can be obtained by using a simple expression of the non linear stabilization parameter depending on the signs of the numerical solution and of its derivatives a straightforward two steps algorithm yielding a stable and accurate solution for all the range of physical parameters and boundary conditions is described the extension to the multi dimensional case is briefly described numerical results for 1d and 2d problems are presented showing the efficiency and accuracy of the new stabilized formulation

difference approximations for the second order wave equation difference approximations are derived for the second order wave equation in one and two space dimensions without first writing it as a first order system both the dirichlet and the neumann problems are treated for the one dimensional case relations between the boundary error and the interior phase error are derived for a fully second order accurate discretization as well as a scheme that is fourth order accurate in the interior and second order accurate at the boundary general two dimensional domains are considered for the dirichlet problem where the domain is embedded in a cartesian grid and the boundary conditions are approximated by interpolation a stable conservative scheme is derived where the time step is determined only by the interior discretization formula discretization cells cut by the boundary are treated implicitly but the resulting scheme becomes explicit because the implicit dependence only is pointwise numerical examples are provided to verify the stability and accuracy of the proposed method

a posteriori dual mixed adaptive finite element error control for lame and stokes equations a unified and robust mathematical model for compressible and incompressible linear elasticity can be obtained by rephrasing the herrmann formulation within the hellinger reissner principle this quasi optimally converging extension of peers plane elasticity element with reduced symmetry is called dual mixed hybrid formulation dmh explicit residual based a posteriori error estimates for dmh are introduced and are mathematically shown to be locking free reliable and efficient the estimator serves as a refinement indicator in an adaptive algorithm for effective automatic mesh generation numerical evidence supports that the adaptive scheme leads to optimal convergence for lame and stokes benchmark problems with singularities

computer experiments on quadratic programming algorithms abstract this paper compares the computational performance of five quadratic programming algorithms these include wolfe s simplex method lemke s complementary pivot method convex simplex method and quadratic differential algorithm execution time and iteration count are used as the major criteria for comparison since lemke s algorithm out performed all other methods in the study a detailed statistical analysis was performed to determine the relative importance of problem parameters on the efficiency of lemke s algorithm an analysis of variance showed that the number of variables the percent of positive linear terms in the objective the number of constraints and their interactions were the significant factors for both iteration count and execution time finally regression equations for iteration count and execution time are derived as a function of fifteen problem parameters

rosenbrock methods for partial differential equations and fractional orders of convergence the authors apply rosenbrock methods for the solution of initial value problems for linear partial differential equations u t x t mathcal l x partial u x t f x t under weak assumptions on the operator mathcal l and the source function f a sharp lower bound for the order of convergence of these numerical methods is given it is further shown that the order of convergence is in general fractional it depends on the l r norm used to estimate the global error this analysis also applies to systems arising from spatial discretization of partial differential equations by finite differences or finite element techniques numerical examples illustrate the results

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

a stabilized finite element method for generalized stationary incompressible flows abstract in this paper we describe a finite element formulation for the numerical solution of the stationary incompressible navier stokes equations including coriolis forces and the permeability of the medium the stabilized method is based on the algebraic version of the sub grid scale approach we first describe this technique for general systems of convection diffusion reaction equations and then we apply it to the linearized flow equations the important point is the design of the matrix of stabilization parameters that the method has this design is based on the identification of the stability problems of the galerkin method and a scaling of variables argument to determine which coefficients must be included in the stabilization matrix this together with the convergence analysis of the linearized problem leads to a simple expression for the stabilization parameters in the general situation considered in the paper the numerical analysis of the linearized problem also shows that the method has optimal convergence properties

numerical methods for nonlinear second order hyperbolic partial differential equations ii rothe s techniques for 1 d problems abstract two families of rothe s methods that are based on the discretization of the time variable and keeping the spatial one continuous for the solution of second order hyperbolic equations with damping and nonlinear source terms are presented the first family is based on time integration and results in a volterra integro differential equation which upon approximating certain time integrals can be written as an ordinary differential equation in space upon discretizing the spatial derivatives by means of finite difference formula this family results in nonlinear algebraic equations which contain exponential terms that depend on the time step and its accuracy is shown to be lower than that of an explicit second order accurate discretization in both space and time of second order hyperbolic equations the second family of methods is based on the discretization of the time derivatives and time linearization and results in linear ordinary differential equations in space upon freezing the coefficients of these equations one can integrate analytically the resulting linear ordinary differential equations to obtain piecewise exponential solutions which are continuous in space and three point finite difference equations which depend exponentially on the time step spatial grid size and the diffusion relaxation damping and reaction times the finite difference equations are shown to result in non diagonally dominant matrices unless the time step is smaller than the characteristic relaxation diffusion damping and reaction times to avoid this problem two rothe s techniques that do not account for the jacobian of the reaction terms in the ordinary differential operator are developed and it is shown that the rothe s techniques belonging to the second family are as accurate as the linearly implicit methods presented in part i provided that the relaxation time is smaller than the critical time and the solution does not oscillate in space when the relaxation time is greater than the critical one rothe s methods have been found to be less accurate than the finite difference techniques of part i the rothe s techniques belonging to the second family are generalized to systems of nonlinear second order hyperbolic equations and mixed systems of parabolic and second order hyperbolic partial differential equations

a relationship between stabilized finite element methods and the galerkin method with bubble functions abstract a relation between stabilized finite element methods and the galerkin method employing interpolations with bubble functions is established for the advective diffusive model and for the linearized compressible navier stokes equations the bubble functions are shown to help in stabilizing the advective operator without recourse to upwinding or any other numerical strategy in particular for the advective diffusive model the galerkin method employing piecewise linears with bubble functions is shown to be equivalent to the streamline upwind petrov galerkin supg method in the diffusive limit

stabilized mixed finite element formulations for materially nonlinear partially saturated two phase media standard and enhanced bbar or eas low order finite elements applied to the problem of consolidation of two phase nonlinear continua do not satisfy the lbb condition when the same interpolation functions are used for both displacement and pore pressure fields strong spatial pressure oscillations are the main consequence of the lbb condition violation the class of direct stabilized methods widely used in the field of fluid mechanics is a powerful tool to circumvent violation of the aforementioned condition three stabilized formulations designed for the problem of consolidation of fully or partially saturated media are presented galerkin least squares gls with the least squares term construction based on the residuum of the local fluid mass conservation equation the pressure stabilized formulation fpl in which the rate of the pore pressure laplacian residual free in the incompressibility limit is added to the fluid mass conservation equation and finally a stabilized formulation in which the stabilization term is constructed based on the rate of the residuum of the local momentum equation an h convergence study and test problems for the three stabilized schemes are discussed

on the mathematical modeling for elastoplastic contact problem and its solution by quadratic programming in this paper a variational inequality model is created for the elastoplastic contact problem in which the constraint of the constitutional relation for elastoplastic material and the contact condition are relaxed totally it gives an effective and strict mathematical modeling for the problem the quadratic programming is used here for the numerical solution this algorithm has great advantages of convergence and computational effectiveness over the conventional methods avoiding the tedious procedure of iterations

an ale pfem method for the numerical simulation of two phase mixture flow abstract a new finite element method is proposed to simulate two phase mixture problems with free surfaces and or large moving boundaries the two phases are treated as interpenetrating continua within a two fluid model in contrast to the conventional two fluid model formulations in which the governing equations are written in the eulerian eulerian description the modified two fluid model has the governing equations of one fluid phase written in the lagrangian description while the respective equations of the other phase is formulated in the arbitrary lagrangian eulerian ale description with relative velocities defined with respect to the moving mesh associated with the first fluid phase the finite element method fem together with stabilization techniques the characteristic based split cbs and the algebraic flux correction afc method is employed to solve the equations numerically in space and time the nodes of the computational mesh are updated together with the flow of the phase described in the lagrangian description the computational mesh is updated by applying techniques used in the so called particle finite element method pfem characterized by regenerating the mesh in each computational step according to the updated nodal positions while the current boundary of the domain is identified using shape method the performance of the method is demonstrated by means of three 2d and 3d examples including a mixing process of two fluids in a stirred tank a free surface flow a 3d two phase channel flow a phase separation problem and the flow of two immiscible fluids

reliability and efficiency of an anisotropic zienkiewicz zhu error estimator abstract in this paper we study the efficiency and the reliability of an anisotropic a posteriori error estimator in the case of the poisson problem supplied with mixed boundary conditions the error estimator may be classified as a residual based one but its novelty is twofold firstly it employs anisotropic estimates of the interpolation error for linear triangular finite elements and secondly it makes use of the zienkiewicz zhu recovery procedure to approximate the gradient of the exact solution finally we describe the adaptive procedure used to obtain a numerical solution satisfying a given accuracy and we include some numerical test cases to assess the robustness of the proposed numerical algorithm

stability and convergence of staggered runge kutta schemes for semilinear wave equations a staggered runge kutta staggered rk scheme is a runge kutta type scheme using a time staggered grid as proposed by ghrist et al in 2000 6 afterwards verwer in two papers investigated the efficiency of a scheme proposed by ghrist et al 6 for linear wave equations we study stability and convergence properties of this scheme for semilinear wave equations in particular we prove convergence of a fully discrete scheme obtained by applying the staggered rk scheme to the mol approximation of the equation

derivation of stabilized equations for numerical solution of advective diffusive transport and fluid flow problems abstract the concept of the so called artificial or balancing diffusion used to stabilize the numerical solution of advective diffusive transport and fluid flow problems is revised in this paper it is shown that the standard forms of the balancing diffusion terms usually chosen in a heuristic manner can be naturally found by introducing higher order approximations in the derivation of the governing differential equations via standard conservation or equilibrium principles this allows us to reinterpret many stabilization algorithms and concepts used in every day practice by numerical analysts and also provides an expression for computing the stabilization parameter

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

adaptive coupled arbitrary lagrangian eulerian finite element and meshfree method for injection molding process this paper presents an adaptive method that couples the finite element fe method and the meshfree mf method in the arbitrary lagrangian eulerian ale description for numerical simulation of injection molding processes the ale feature is used to accurately capture moving free surfaces of flow problems and meanwhile to alleviate mesh distortion and its influence on the accuracy and robustness of numerical solutions based on the continuous blending method originally presented for properly imposing the essential boundary conditions in mf methods the coupled ale fe and mf method is developed to exploit the respective advantages of both fe and mf methods but to avoid their respective weak points the method features self adaptivity in view of the fact that moving free surfaces in an injection molding process are continuously advanced the pressure stabilized fractional step algorithm in which the modified version of the finite increment calculus process is extended to the coupled ale fe and mf method is used to solve incompressible navier stokes equations governing the injected polymer flow so that equal low order u p interpolations that violate the lbb condition can be effectively used numerical experiments demonstrate the significance and ability of the proposed method for numerical simulation of injection molding processes copyright 2007 john wiley sons ltd

error estimation and mesh adaptivity in incompressible viscous flows using a residual power approach abstract a methodology for error estimation and mesh adaptation for finite element fe analysis of incompressible viscous flow is presented the error estimation method is based on the evaluation of the energy rate the power of the fe residuals of the momentum and incompressibility equations the residuals are computed using recovered values of the derivatives of the velocity and pressure variables obtained via a nodal derivative recovery technique two mesh adaptation procedures based on a the equidistribution of the residual power among the elements in the mesh and b the equidistribution of the density of the total residual power are presented the stabilized form of the navier stokes equations using a finite calculus fic formulation is solved with a fractional step fe scheme this allows the use of linear triangles and tetrahedra with an equal order interpolation for the velocity and pressure variables a nodal based approach is used for computing the residual power integrals the examples show the ability of the error estimation and the mesh adaptation process to capture the high gradients of the solution in the vicinity of boundary layers and in zones of high vorticity of the fluid for both steady state and transient flow situations

runge kutta methods without order reduction for linear initial boundary value problems it is well known the loss of accuracy when a runge kutta method is used together with the method of lines for the full discretization of an initial boundary value problem we show that this phenomenon called order reduction is caused by wrong boundary values in intermediate stages with a right choice the order reduction can be avoided and the optimal order of convergence in time is achieved we prove this fact for time discretizations of abstract initial boundary value problems based on implicit runge kutta methods moreover we apply these results to the full discretization of parabolic problems by means of galerkin finite element techniques we present some numerical examples in order to confirm that the optimal order is actually achieved

a posteriori error estimation for standard finite element analysis a new residual type estimator based on projections of the error on subspaces of locally supported functions is presented the estimator is defined by a standard element by element refinement first an approximation of the energy norm of the error is obtained solving local problems with homogeneous dirichlet boundary conditions a later enrichment of the estimation is performed by adding the contributions of projections on a new family of subspaces this estimate is a lower bound of the measure of the actual error the estimator does not need to approximate local boundary conditions for the error equation therefore computation of flux jumps is not necessary moreover the estimator can be applied in mixed meshes containing elements of different shapes and its implementation in a standard finite element code is straightforward the presented results show the effectiveness of the estimator approximating both the distribution and the global measure of the error as well as its usefulness in adaptive procedures

the superconvergent patch recovery spr and adaptive finite element refinement abstract the error estimation processes which have been developed by the authors depend on the accuracy with which stresses gradients can be recovered the recovery methods available currently such as the l 2 projection and averaging are inaccurate and an improved process is needed this paper presents a very much improved recovery process yielding superconvergent values throughout the domain details of this method named the superconvergent patch recovery are presented with test results showing its efficiency examples of adaptive h based refinement are given showing high local and overall efficiency indices for error estimation

the correct formulation of intermediate boundary conditions for runge kutta time integration of initial boundary value problems a disturbing feature of applying explicit high order runge kutta rk time integrators to initial boundary value problems solved by the method of lines is the loss of accuracy that results from wrong specifications of intermediate stage boundary conditions the solution proposed here is to prescribe analytically those values that would result from applying the rk solver at the boundaries and hence maintain the order of the rk method without a reduction in the step size limit the procedure is described in detail for nonlinear hyperbolic equations and both scalar and vector examples are presented

a new stabilized finite element method for reaction diffusion problems the source stabilized petrov galerkin method this paper proposes a new stabilized finite element method to solve singular diffusion problems described by the modified helmholtz operator the galerkin method is known to produce spurious oscillations for low diffusion and various alternatives were proposed to improve the accuracy of the solution the mostly used methods are the well known galerkin least squares and galerkin gradient least squares ggls the ggls method yields the exact nodal solution in the one dimensional case and for a uniform mesh however the behavior of the method deteriorates slightly in the multi dimensional case and for non uniform meshes in this work we propose a new stabilized finite element method that leads to improved accuracy for multi dimensional problems for the one dimensional case the new method leads to the same results as the ggls method and hence provides exact nodal solutions to the problem on uniform meshes the proposed method is a galerkin discretization used to solve a modified equation that includes a term depending on the gradient of the original partial differential equation copyright 2008 john wiley sons ltd

oddls a new unstructured mesh finite element method for the analysis of free surface flow problems this paper introduces a new stabilized finite element method based on the finite calculus comput methods appl mech eng 1998 151 233 267 and arbitrary lagrangian eulerian techniques comput methods appl mech eng 1998 155 235 249 for the solution to free surface problems the main innovation of this method is the application of an overlapping domain decomposition concept in the statement of the problem the aim is to increase the accuracy in the capture of the free surface as well as in the resolution of the governing equations in the interface between the two fluids free surface capturing is based on the solution to a level set equation the navier stokes equations are solved using an iterative monolithic predictor corrector algorithm encyclopedia of computational mechanics wiley new york 2004 where the correction step is based on imposing the divergence free condition in the velocity field by means of the solution to a scalar equation for the pressure examples of application of the oddls formulation for overlapping domain decomposition level set to the analysis of different free surface flow problems are presented copyright 2008 john wiley sons ltd

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

a finite element method for fluid structure interaction with surface waves using a finite calculus formulation abstract a stabilized semi implicit fractional step finite element method fem for solving coupled fluid structure interaction problems involving free surface waves is presented the stabilized governing equations for the viscous incompressible fluid and the free surface are derived at a differential level via a finite calculus fic procedure a mesh updating technique based on solving a fictitious elastic problem on the moving mesh is described examples of the efficiency of the stabilized semi implicit algorithm for the analysis of fluid structure interaction problems in totally or partially submerged bodies is presented

modeling incompressible flows at low and high reynolds numbers via a finite calculus finite element approach we present a formulation for incompressible flows analysis using the finite element method fem the necessary stabilization for dealing with convective effects and the incompressibility condition is modeled via the finite calculus fic method the stabilization terms introduced by the fic formulation allow to solve a wide range of fluid flow problems for low and high reynolds numbers flows without the need for a turbulence model examples of application of the fic fem formulation to the analysis of 2d and 3d incompressible flows with moderate and large reynolds numbers are presented

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

modified nystrom methods for semi discrete hyperbolic differential equations first and second order nystrom type methods are derived for second order differential equations without first derivatives possessing the following properties i the stability interval equals 4m 2 m denoting the number of stages per integration step ii the method is internally stable irrespective of the value of m iii the storage requirements are limited and iv the costs per integration step are one right hand side evaluation one evaluation of the jacobian matrix and m 1 matrix vector multiplications these four properties are of interest in the integration of the usually very large systems of ordinary differential equations resulting from the semi discretization of partial differential equations which are of hyperbolic type and of second order in time

the treatment of nonhomogeneous dirichlet boundary conditions by thep version of the finite element method the paper addresses the problem of the implementation of nonhomogeneous essential dirichlet type boundary conditions in thep version of the finite element method

on stabilized finite element formulations for incompressible advective diffusive transport and fluid flow problems abstract a new approach is presented to obtain stabilized finite element formulations such as streamline upwind petrov galerkin supg and galerkin least squares gls the procedure consists in modifying the equations to be solved and then obtaining the variational equations by the standard galerkin method the new formulation generates additional terms involving boundary integrals to standard stabilization techniques these terms compensate for the lack of consistency of the traditional supg and gls methods for which stabilization terms are added only on the element interiors while jumps of the residual across element faces are neglected a physical interpretation is provided of how the modified equations are obtained it is shown how stabilized formulations such as streamline upwind su and supg are recovered as special cases stabilization terms defined on the element interiors are always accompanied by additional boundary integrals the presence of the boundary integrals is shown to improve the numerical prediction for various viscous and nearly inviscid flows

fluid structure interaction using the particle finite element method abstract in the present work a new approach to solve fluid structure interaction problems is described both the equations of motion for fluids and for solids have been approximated using a material lagrangian formulation to approximate the partial differential equations representing the fluid motion the shape functions introduced by the meshless finite element method mfem have been used thus the continuum is discretized into particles that move under body forces gravity and surface forces due to the interaction with neighboring particles all the physical properties such as density viscosity conductivity etc as well as the variables that define the temporal state such as velocity and position and also other variables like temperature are assigned to the particles and are transported with the particle motion the so called particle finite element method pfem provides a very advantageous and efficient way for solving contact and free surface problems highly simplifying the treatment of fluid structure interactions

convergence and order reduction of runge kutta schemes applied to evolutionary problems in partial differential equations we address the question of convergence of fully discrete runge kutta approximations we prove that under certain conditions the order in time of the fully discrete scheme equals the conventional order of the runge kutta formula being used however these conditions which are necessary for the result to hold are not natural as a result in many problems the order in time will be strictly smaller than the conventional one a phenomenon called order reduction this phenomenon is extensively discussed both analytically and numerically as distinct from earlier contributions we here treat explicit runge kutta schemes although our results are valid for both parabolic and hyperbolic problems the examples we present are therefore taken from the hyperbolic field as it is in this area that explicit discretizations are most appealing

a petrov galerkin finite element method for convection dominated flows an accurate upwinding technique for satisfying the maximum principle abstract in this paper we present a new petrov galerkin method for convection dominated flow problems which is conservative and satisfies the discrete maximum principle this method also possesses no spurious crosswind diffusion and gives very accurate solutions the method is representative of a class of methods which may be described as fixed mesh adaptive

multiscale computational analysis in mechanics using finite calculus an introduction abstract the paper introduces a general procedure for computational analysis of a wide class of multiscale problems in mechanics using a finite calculus fic formulation the fic approach is based in expressing the governing equations in mechanics accepting that the domain where the standard balance laws are established has a finite size this introduces naturally additional terms into the classical equations of infinitesimal theory in mechanics which are useful for the numerical solution of problems involving different scales in the physical parameters the discrete nodal values obtained with the fic formulation and the finite element method fem can be effectively used as the starting point for obtaining a more refined solution in zones where high gradients of the relevant variables occur using hierarchical or enriched fem typical multiscale problems in mechanics which can be solved with the fic method include convection diffusion reaction problems with high localized gradients incompressible problems in solid and fluid mechanics localization problems such as prediction of shear bands in solids and shock waves in compressible fluids turbulence etc the paper presents an introduction of the treatment of multiscale problems using the fic approach in conjunction with the fem examples of application of the fic fem formulation to the solution of simple multiscale convection diffusion problems are given

multi fluid flows with the particle finite element method abstract particle methods are those in which the problem is represented by a discrete number of particles each particle moves accordingly with its own mass and the external internal forces applied on it in this paper the particle finite element method based on finite element shape functions is used to solve the continuous fluid mechanics equations in the case of heterogeneous density to evaluate the external applied forces to each particle the incompressible navier stokes equations are solved at each time step using a lagrangian formulation all the information in the fluid is transmitted via the particles all kinds of density heterogeneous fluids and multiphase flows with internal interfaces including or not free surfaces breaking waves and fluid separations may be easily solved with this methodology

a modified gautschi s method without order reduction when integrating boundary value nonlinear wave problems abstract in this paper we analyse the order reduction which turns up when integrating nonlinear wave problems with non homogeneous and time dependent boundary conditions with the well known gautschi s method moreover a technique is suggested to avoid that order reduction so that the classical local order 4 and global order 2 are recovered on the other hand the usual approximation for the derivative which is used together with this method is also analysed and a substantial improvement is suggested some numerical results are shown which corroborate the performed analysis

formulation and comparison of algorithms for frictional contact problems this paper presents two algorithms for solving the discrete quasi static small displacement linear elastic contact problem with coulomb friction the algorithms are adoptions of a newton method for solving b differentiable equations and an interior point method for solving smooth constrained equations for the application of the former method the contact problem is formulated as a system of b differentiable equations involving the projection operator onto sets with simple structure for the application of the latter method the contact problem is formulated as a system of smooth equations involving complementarity conditions and with the non negativity of variables treated as constraints the two algorithms are numerically tested for two dimensional problems containing up to 100 contact nodes and up to 100 time increments results show that at the present stage of development the newton method is superior both in robustness and speed additional comparison is made with a commercial finite element code

an interior point algorithm for elastoplasticity the problem of small deformation rate independent elastoplasticity is treated using convex programming theory and algorithms a finite step variational formulation is first derived after which the relevant potential is discretized in space and subsequently viewed as the lagrangian associated with a convex mathematical program next an algorithm based on the classical primal dual interior point method is developed several key modifications to the conventional implementation of this algorithm are made to fully exploit the nature of the common elastoplastic boundary value problem the resulting method is compared to state of the art elastoplastic procedures for which both similarities and differences are found finally a number of examples are solved demonstrating the capabilities of the algorithm when applied to standard perfect plasticity hardening multisurface plasticity and problems involving softening copyright 2006 john wiley sons ltd

a discontinuity capturing crosswind dissipation for the finite element solution of the convection diffusion equation abstract to avoid the local oscillations that still remain using the streamline upwind petrov galerkin formulation for the scalar convection diffusion equation the introduction of a nonlinear crosswind dissipation is proposed it is shown that the method is less overdiffusive than other discontinuity capturing techniques and has better numerical behavior the design of the crosswind diffusion is based on the study of the discrete maximum principle for some simple cases

enhanced linear triangle for plasticity problems in j 2 solids due to their economy and capability to reproduce high gradient zones low order elements are of great interest in the calculations involving localized failure however the performance of the simplest element the three nodes triangle under incompressible and bending dominated situations is very poor and the computed results unreliable this paper presents an inexpensive three fields u p formulation solving these issues as shown in the examples of application

finite element formulation for convective diffusive problems with sharp gradients using finite calculus abstract a finite element method fem for steady state convective diffusive problems presenting sharp gradients of the solution both in the interior of the domain and in boundary layers is presented the necessary stabilization of the numerical solution is provided by the finite calculus fic approach the fic method is based in the solution by the galerkin fem of a modified set of governing equations which include characteristic length parameters it is shown that the fic balance equation for the multidimensional convection diffusion problem written in the principal curvature axes of the solution introduces an orthotropic diffusion which stabilizes the numerical solution both in smooth regions as well in the vicinity of sharp gradients the dependence of the stabilization terms with the principal curvature directions of the solution makes the method non linear details of the iterative scheme to obtain stabilized results are presented together with examples of application which show the efficiency and accuracy of the approach

a fic based stabilized finite element formulation for turbulent flows abstract we present a new stabilized finite element fem formulation for incompressible flows based on the finite increment calculus fic framework onate 1998 in comparison to existing fic approaches for fluids this formulation involves a new term in the momentum equation which introduces non isotropic dissipation in the direction of velocity gradients we also follow a new approach to the derivation of the stabilized mass equation inspired by recent developments for quasi incompressible flows onate et al 2014 the presented fic fem formulation is used to simulate turbulent flows using the dissipation introduced by the method to account for turbulent dissipation in the style of implicit large eddy simulation

on the effect of the bulk tangent matrix in partitioned solution schemes for nearly incompressible fluids the purpose of this paper is to study the effect of the bulk modulus on the iterative solution of free surface quasi incompressible fluids using a mixed partitioned scheme a practical rule to set up the value of a pseudo bulk modulus a priori in the tangent bulk stiffness matrix for improving the conditioning of the linear system of algebraic equations is also given the efficiency of the proposed strategy is tested in several problems analyzing the advantage of the modified bulk tangent matrix with regard to the stability of the pressure field the convergence rate and the computational speed of the analyses the technique has been tested on a finite calculus particle finite element method lagrangian formulation but it can be easily extended to other quasi incompressible stabilized finite element formulations copyright c 2014 john wiley sons ltd

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

order reduction and how to avoid it when explicit runge kutta nystrom methods are used to solve linear partial differential equations in this paper we study the order reduction which turns up when explicit runge kutta nystrom methods are used to discretize linear second order hyperbolic equations by means of the method of lines the order observed in practice including its fractional part is obtained it is also proved that the order reduction can be completely avoided taking the boundary values of the intermediate stages of the time semidiscretization the numerical experiments confirm that the optimal order can be reached

numerical modeling of viscoelastic flows using equal low order finite elements a mixed finite element scheme abbreviated as i ps devss cnbs scheme for modeling viscoelastic flow problems is presented the finite incremental calculus fic pressure stabilization process and the discrete elastic viscous stress splitting method devss are introduced into the general framework of the iterative version of the fractional step algorithm with the use of the crank nicolson based splitting inconsistent streamline upwinding method su is employed to spatially discretize the constitutive equation of viscoelastic fluids equal low order finite elements which violate the lbb compatibility conditions are successfully used in the proposed scheme in addition the oldroyd b and the ptt models have been integrated into the proposed scheme to solve the 4 1 sudden contraction flow problem the numerical results demonstrate prominent stability and accuracy of both pressure and stress distributions over the flow domains provided by the proposed scheme within the weissenberg number range studied in the present work as compared with the reference solutions reported in the literatures

finite calculus formulations for finite element analysis of incompressible flows eulerian ale and lagrangian approaches we present a general formulation for incompressible fluid flow analysis using the finite element method fem the standard eulerian formulation is described first the necessary stabilization for dealing with convective effects and the incompressibility condition are introduced via the finite calculus fic method a simple extension of the fluid flow equations to an arbitrary lagrangian eulerian ale frame adequate for treating fluid structure interaction problems is briefly presented a fully lagrangian formulation called the particle finite element method pfem is also described the pfem is particularly attractive for fluid structure interaction problems involving large motions of the free surface and breaking waves examples of application of the eulerian the ale and the fully lagrangian pfem formulations are presented

a new approach to the development of automatic quadrilateral mesh generation a methodology for the automatic mesh generation of quadrilateral elements is proposed the methodology is based on the facts that i a region can always be subdivided entirely into quadrilaterals if the polygon which forms the boundary of the region has an even number of sides ii a quadrilateral can be formed by two triangles which share a common side by taking advantage of the techniques developed in an existing automatic triangular mesh generator quadrilateral meshes can be generated according to the distribution of user specified element size over the domain of interest in particular an automatic quadrilateral mesh generator based on the advancing front technique for triangular mesh generation1 is described techniques to improve the quality of the generated quadrilateral meshes are introduced the robustness and versatility of the mesh generator are demonstrated by a variety of examples

an ale based iterative cbs algorithm for non isothermal non newtonian flow with adaptive coupled finite element and meshfree method this paper presents a generalized scheme of the fractional step algorithm using characteristic based split cbs in ale framework for the numerical simulation of incompressible non isothermal non newtonian fluid flows and applies it combined with the adaptive coupled finite element and meshfree method for spatial discretizations to the polymer injection molding process based on the idea to discretize the material time derivative instead of the spatial time derivative the characteristic galerkin method is derived in a more general way and is used to discretize the non newtonian momentum and the energy conservation equations an iterative procedure which makes the conservation equations satisfied in an implicit sense is introduced into the scheme to enhance its stability in the time domain numerical experiments with the power law fluid model demonstrate and validate the performance and the capability of the proposed scheme

mesh refinement strategies without mapping of nonlinear solutions for the generalized and standard fem analysis of 3 d cohesive fractures summary a robust and efficient strategy is proposed to simulate mechanical problems involving cohesive fractures this class of problems is characterized by a global structural behavior that is strongly affected by localized nonlinearities at relatively small sized critical regions the proposed approach is based on the division of a simulation into a suitable number of sub simulations where adaptive mesh refinement is performed only once based on refinement window s around crack front process zone s the initialization of newton raphson nonlinear iterations at the start of each sub simulation is accomplished by solving a linear problem based on a secant stiffness rather than a volume mapping of nonlinear solutions between meshes the secant stiffness is evaluated using material state information stored read on crack surface facets which are employed to explicitly represent the geometry of the discontinuity surface independently of the volume mesh within the generalized finite element method framework moreover a simplified version of the algorithm is proposed for its straightforward implementation into existing commercial software data transfer between sub simulations is not required in the simplified strategy the computational efficiency accuracy and robustness of the proposed strategies are demonstrated by an application to cohesive fracture simulations in 3 d copyright 2016 john wiley sons ltd

a perspective on adaptive modeling and meshing am m the purpose of this study is to develop an integrated approach to adaptive modeling in conjunction with adaptive meshing this involves consideration of the modeling error from various sources and the discretization error in the numerical scheme adapting the model to control error due to geometry boundary data constitutive relations and model parameters can be integrated with adaptive mesh control to improve numerical reliability other factors such as stabilization by artificial dissipation and regularization are considered a class of hybrid algorithms are developed to combine these approaches

the particle finite element method a powerful tool to solve incompressible flows with free surfaces and breaking waves summary particle methods are those in which the problem is represented by a discrete number of particles each particle moves accordingly with its own mass and the external internal forces applied to it particle methods may be used for both discrete and continuous problems in this paper a particle method is used to solve the continuous fluid mechanics equations to evaluate the external applied forces on each particle the incompressible navier stokes equations using a lagrangian formulation are solved at each time step the interpolation functions are those used in the meshless finite element method and the time integration is introduced by an implicit fractional step method in this manner classical stabilization terms used in the momentum equations are unnecessary due to lack of convective terms in the lagrangian formulation once the forces are evaluated the particles move independently of the mesh all the information is transmitted by the particles fluid structure interaction problems including free fluid surfaces breaking waves and fluid particle separation may be easily solved with this methodology copyright 2004 john wiley sons ltd

meshfree iterative stabilized taylor galerkin and characteristic based split cbs algorithms for incompressible n s equations abstract this paper presents two stabilized schemes of the fractional step algorithm using characteristic based split and taylor galerkin like based split respectively in the meshfree context an iterative procedure which makes the convective term satisfy the momentum conservation equation in an implicit sense is introduced into the two schemes to allow much larger time step sizes a modified version of finite increment calculus process for incompressibility stabilization is developed to avoid the calculation of high order spatial derivatives numerical experiments demonstrate the capability and stability of the proposed schemes of the meshfree based fractional step method in the simulation of incompressible flows

possibilities of finite calculus in computational mechanics abstract the expression finite calculus refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes forces etc in a domain of finite size the governing equations resulting from this approach are different from those of infinitessimal calculus theory and they incorporate new terms depending on the dimensions of the balance domain the new modified equations allow to derive naturally stabilized numerical schemes using finite element finite difference finite volume or meshless methods the paper briefly discusses the possibilities of the modified governing equations derived via the finite calculus technique for the numerical solution of convection diffusion problems incompressible flow and incompressible solid mechanic problems and strain localization problems

superconvergent derivatives a taylor series analysis an analysis based on the use of taylor series expansions is developed to determine accuracy estimates for derivatives in one and two dimensions computed by differentiation of a finite element interpolant or approximation the analysis clarifies some issues concerning special points at which the derivatives are believed to be exceptionally accurate with higher convergence rates superconvergence moreover it leads directly to a class of post processing strategies for the derivatives and offers a more direct constructive approach to the subject

the patch test a condition for assessing fem convergence this paper discusses various forms of the irons patch test and shows again that its satisfaction is a necessary convergence condition being equivalent to consistency further it is shown that the test can be applied for verification of stability requirements thus providing also a sufficient condition of convergence the patch test is useful in testing the order of asymptotic convergence and the satisfaction of the test gives a guide for development of certain incompatible element forms

polynomial preserving recovery on boundary in this paper we propose two systematic strategies to recover the gradient on the boundary of a domain the recovered gradient has comparable superconvergent property on the boundary as that in the interior of the domain this superconvergence property has been validated by several numerical experiments

a spectral deferred correction method for incompressible flow with variable viscosity abstract this paper presents a semi implicit spectral deferred correction sdc method for incompressible navier stokes problems with variable viscosity and time dependent boundary conditions the proposed method integrates elements of velocity and pressure correction schemes which yields a simpler pressure handling and a smaller splitting error than the sdpc method of minion and saye j comput phys 375 797 822 2018 combined with the discontinuous galerkin spectral element method for spatial discretization it can in theory reach arbitrary order of accuracy in time and space numerical experiments in three space dimensions demonstrate up to order 12 in time and 17 in space for constant as well as varying solution dependent viscosity compared to sdpc the present method yields a substantial improvement of accuracy and robustness against order reduction caused by time dependent boundary conditions

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

avoiding the order reduction when solving second order in time pdes with fractional step runge kutta nystrom methods we study some of the main features of fractional step runge kutta nystrom methods when they are used to integrate initial boundary value problems of second order in time in combination with a suitable spatial discretization we focus our attention on the order reduction phenomenon which appears if classical boundary conditions are taken at the internal stages this drawback is specially hard when time dependent boundary conditions are considered in this paper we present an efficient technique very simple and computationally cheap which allows us to avoid the order reduction such technique consists in modifying the boundary conditions for the internal stages of the method

unified lagrangian formulation for elastic solids and incompressible fluids application to fluid structure interaction problems via the pfem abstract we present a general lagrangian formulation for treating elastic solids and quasi fully incompressible fluids in a unified form the formulation allows to treat solid and fluid subdomains in a unified manner in fluid structure interaction fsi situations in our work the fsi problem is solved via the particle finite element method pfem the pfem is an effective technique for modeling complex interactions between floating and submerged bodies and free surface flows accounting for splashing of waves large motions of the bodies and frictional contact conditions applications of the unified lagrangian formulation to a number of fsi problems are given

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

triangles and tetrahedra in explicit dynamic codes for solids explicit dynamic codes which are used currently for the study of plastic deformations in impact or with some modification for metal forming suffer two serious limitations first only quadrilateral or hexahedral linear elements can be used thus limiting the possibilities of adaptive refinement and adaptive meshing second even with the use of such elements special devices such as reduced integration must be introduced to avoid locking and reduce costs these necessitate complex hour glass control mending type procedures the main difficulties are those due to the need of treatment of almost incompressible deformation modes recently similar difficulties have been overcome in the context of fluid dynamics soil dynamics and we show here how the processes introduced there can be adopted effectively to the present problem thus allowing an almost unrestricted choice of element interpolations 1998 john wiley sons ltd

p1 p0 elements for incompressible flows with discontinuous material properties abstract we present a 3 noded triangle and a 4 noded tetrahedra with a continuous linear velocity and a discontinuous linear pressure field formed by the sum of an unknown constant pressure field and a prescribed linear field that satisfies the steady state momentum equations for a constant body force the elements are termed p1 p0 as the effective pressure field is linear although the unknown pressure field is piecewise constant within each element the elements have an excellent behavior for incompressible viscous flow problems with discontinuous material properties formulated in either eulerian or lagrangian descriptions the necessary numerical stabilization for dealing with the inf sup condition imposed by the incompressibility constraint and high convective effects in eulerian flows is introduced via the finite calculus fic approach for the sake of clarity the element derivation is presented first for the simpler stokes equations written in the standard eulerian frame the extension of the formulation to the navier stokes equations written in the eulerian and lagrangian frameworks is straightforward and is presented in the second part of the paper the efficiency and accuracy of the new p1 p0 triangle is verified by solving a set of incompressible multifluid flow problems using a lagrangian approach and a classical eulerian description the excellent performance of the new triangular element in terms of mass conservation and general accuracy for analysis of fluids with discontinuous material properties is highlighted

to mesh or not to mesh that is the question in the last decade a family of methods called meshless methods has been developed both for structural and fluid mechanics problems after these ideas a possible classification for numerical formulations may be to separate the methods that make use of a standard finite element mesh such as those made of tetrahedra or hexahedra from those that do not need a standard mesh namely the meshless methods for solving a partial different equation by a numerical method a possible alternative may be either to use a mesh method or a meshless method this paper discusses this issue to show that this choice is not in the large majorities of the cases the right question

a model adaptive hanging node concept based on a new non linear solid shell formulation in this contribution we propose an innovative approach to automatically refine shell finite element meshes in thickness direction the procedure employs a new eight node solid shell element formulation based on the concept of reduced integration with hourglass stabilization apart from the important advantages of computational efficiency and robustness this element formulation offers the possibility to work with a variable mesh density in thickness direction in the present paper we suggest an adaptive approach using the hanging node concept to avoid additional mesh distortion among different stress and strain based error criteria the warping of the cross section and the residual error in the solution of the strong form of the balance of linear momentum have been proven to be suitable error measures the former represents the deviation of the realistically non linear dependence of the characteristic strain coefficient from its corresponding linear distribution applications show that the mesh refines as expected in loading domains in the neighbourhood of bearings or regions with strongly accumulating inelastic deformation on the other hand the discretization remains rather coarse in uncritical parts of the structure which decreases the computational effort noticeably

a lagrangian finite element approach for the analysis of fluid structure interaction problems a lagrangian finite element method for the analysis of incompressible newtonian fluid flows based on a continuous re triangulation of the domain in the spirit of the so called particle finite element method is here revisited and applied to the analysis of the fluid phase in fluid structure interaction problems a new approach for the tracking of the interfaces between fluids and structures is proposed special attention is devoted to the mass conservation problem it is shown that despite its lagrangian nature the proposed combined finite element particle method is well suited for large deformation fluid structure interaction problems with evolving free surfaces and breaking waves the method is validated against the available analytical and numerical benchmarks copyright 2010 john wiley sons ltd

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

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