optimal designs of two phase studies the two phase design is a cost effective sampling strategy to evaluate the effects of covariates on an outcome when certain covariates are too expensive to be measured on all study subjects under

an exponential fitting finite element method for convection diffusion problems abstract in this paper we consider an exponential fitting finite element method effem for the discretization of convection dominated convection diffusion equation hemker s basis functions 5 are very robust for solving the one dimensional problem however they cannot resolve the two dimensional problem well the effem provides an exponential fitting for the solution on each grid to stabilize the solution the error bound for the effem in the energy norm is in the order of o h e e ch e for uniform mesh size h on the bilinear elements we compare the performance of the effem with various streamline upwind schemes and it shows that the exponential terms of the basis functions are tailored the solution accurately in the flow direction our numerical results demonstrate the efficiency and accuracy of the proposed method

positive solutions of difference equations with two point right focal boundary conditions abstract we consider the following boundary value problem formula formula where 1 p n 1 is fixed using a fixed point theorem for operators on a cone we develop criteria for the existence of two positive solutions of the boundary value problem in addition upper and lower bounds for these positive solutions are established for special cases we also include several examples to dwell upon the importance of the results obtained

nonresonant singular boundary value problems with sign changing nonlinearities existence results are established for nonresonant singular dirichlet boundary value problems our nonlinearity may be singular in its dependent variable and is allowed to change sign

existence criteria for singular boundary value problems with sign changing nonlinearities abstract a general existence theory is presented for boundary value problems where our nonlinearity may be singular in its dependent variable and may also change sign in addition from our theory we give easily verifiable conditions to check existence in practice

power calculation for case cohort studies with nonrare events summary case cohort design has been advocated in many epidemiologic studies when studying rare diseases or events in this design with a rare event all the events are selected for risk factor assessment when the event is not rare it is desirable to consider a generalized case cohort design where only a fraction of events are sampled we provide a valid test statistic to compare hazards functions between two samples for this generalized design and give a method for calculating power our result generalizes the result in cai and zeng 2004 biometrics60 1015 1024 and it shows numerically that efficiency loss due to sampling only part of the events is very low under nonrare events situation

more efficient estimators for case cohort studies the case cohort study design used to reduce costs in large cohort studies is a random sample of the entire cohort named the subcohort augmented with subjects having the disease of interest but not in the subcohort sample when several diseases are of interest several case cohort studies may be conducted using the same subcohort with each disease analyzed separately ignoring the additional exposure measurements collected on subjects with the other diseases this is not an efficient use of the data and in this paper we propose more efficient estimators we consider both joint and separate analyses for the multiple diseases we propose an estimating equation approach with a new weight function and we establish the consistency and asymptotic normality of the resulting estimator simulation studies show that the proposed methods using all available information gain efficiency we apply our proposed method to the data from the busselton health study

risk ratio and rate ratio estimation in case cohort designs hypertension and cardiovascular mortality multivariate analysis in case base designs depends on approximate methods in the present study new pseudo likelihood methods are developed for this design with these methods the case cohort risk ratio and rate ratio as well as their standard errors are easily estimated using logistic regression and poisson regression respectively this is illustrated by the association between hypertension and cardiovascular mortality in a cohort estimated by case cohort analysis using samples of several sizes the estimates are compared with those obtaining in full cohort and nested case control designs the results indicate that these methods which require nothing but widely available computer software are valid the case cohort design therefore is a good sometimes even advantageous alternative to the nested case control design in studying a disease that is not very rare application of the risk ratio method to the full cohort using a sample of 100 per cent follows logically whenever the true risk ratio is desired instead of the odds ratio a multivariate model for its estimation is therefore available

singular nonlinear boundary value problems for second order ordinary differential equations abstract the boundary value problem y f x y 0 y 0 y 0 0 y 1 y 1 0 is studied under conditions which are modeled on the problem y a x y p 0y 0 y 1 0 or y 0 y 1 0 which occurs in a number of applications the technical arguments involve properties of solutions i e concavity and the use of iterative techniques a new fixed point theorem for cones for decreasing mappings is proved and used it is of interest in its own right since most such theorems are for monotone increasing mappings

analysis and application of finite volume element methods to a class of partial differential equations abstract the finite volume element fve methods for a class of partial differential equations are discussed and analyzed in this paper the new initial values are introduced in the finite volume element schemes and we obtain optimal error estimates in l p and w 1 p 2 p as well as some superconvergence estimates in w 1 p 2 p the main results in this paper perfect the theory of the finite volume element methods

crosswind smear and pointwise errors in streamline diffusion finite element methods for a model convection dominated singularly perturbed convection diffusion prob lem it is shown that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion the ensuing method with piecewise linear elements converges with a pointwise accuracy of almost h 5 4 under local smoothness assumptions 1 introduction the streamline diffusion method is a finite element method for convection dominated convection diffusion problems which combines formal high accuracy with decent stability properties the method was introduced in the case of stationary problems by hughes and brooks 7 cf raithby and torrance 14 and wahlbin 15 for earlier thoughts in this direction the mathematical analysis of the method was started in johnson and navert 8 and continued with extensions to e g time dependent problems in navert 12 johnson navert and pitkaranta 9 and johnson and saranen 10 in these papers local error estimates in l2 of order o h k 1 2 in regions of smoothness with piecewise polynomial finite elements of degree k were derived together with estimates stating as a typical example that in the zero diffusion limit a sharp discontinuity in the exact solution across a streamline will be captured in a numerical crosswind layer of width 0 h1 2 essentially the purpose of the present paper is first to improve the result just mentioned on numerical crosswind smear to 0 h3 4 the improvement from 0 h1 2 to 0 h3 4 is obtained by adding a small amount 0 h3 2 of artificial crosswind diffusion to the method in the piecewise linear case k 1 this does not destroy the known o h3 2 accuracy in l2 in smooth regions using our first result we then obtain our second main result localized pointwise error estimates of order 0 h5 4 in regions of smoothness the previously known best pointwise error estimate in the piecewise linear situation is 0 h1 2 another consequence is a global l estimate of order 0 h 2 in the presence of typical crosswind and downwind singularities we shall consider the model problem of finding u u x y such that

focal boundary value problems for singular difference equations abstract solutions are shown to exist for the boundary value problem n u m f m u u n 2 u 0 m 0 t satisfying focal boundary conditions where f m y 1 y n 1 is singular at y i 0 1 i n the techniques involve concavity properties iterations and a fixed point theorem for mappings which are decreasing with respect to a cone in a banach space

an explicit finite element method for convection diffusion equations using rational basis functions abstract an explicit galerkin method is formulated by using rational basis functions the characteristics of the rational difference scheme are investigated with regard to consistency stability and numerical convergence of the method numerical results are also presented

an integral equation approach to the unsteady convection diffusion equations an integral equation approach is proposed to solve the unsteady convection diffusion equations in this approach the second order adams moulton method is firstly utilized for the time discretization then by using the green s function of the laplace equation in the series form the convection diffusion equation is transformed into an integral equation that is further converted into an algebraic equation system the accuracy convergence and the stability of this integral equation approach are examined by four examples in comparison with the characteristic variational multiscale method and the finite volume element method the integral equation approach shows a higher accuracy compared with the finite volume element method the integral equation approach has a better convergence in solving the convection dominated convection diffusion problems the integral equation approach demonstrates a good stability

the finite volume element method for diffusion equations on general triangulations this paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in mathcal r 2 using a special type of control volume the theory applies to diffusion equations of the form begin gathered nabla a nabla u f quad text in omega hfill u 0 quad text on partial omega hfill end gathered under fairly general conditions the theory establishes o h estimates of the error in a discrete mathcal h 1 seminorm under an additional assumption concerning local uniformity of the triangulation the estimate is improved to o h 2

positive solutions to singular multi point dynamic eigenvalue problems with mixed derivatives abstract this paper considers a singular m point dynamic eigenvalue problem on time scales t p t u t f t u t t 0 1 t u 0 i 1 m 2 a i u i u 1 p 1 u 1 i 1 m 2 b i p i u i we allow f t w to be singular at w 0 and t 0 by constructing the green s function and studying its positivity eigenvalue intervals in which there exist positive solutions of the above problem are obtained by making use of the fixed point index theory

a case cohort design for epidemiologic cohort studies and disease prevention trials summary suppose that a cohort of individuals is to be followed in order to relate failure rates to preceding covariate histories a design is proposed which involves covariate data only for cases experiencing failure and for members of a randomly selected subcohort odds ratio and relative risk estimation procedures are presented for such a case cohort design a small simulation study compares case cohort relative risk estimation procedures to full cohort and synthetic case control analyses relevance to epidemiologic cohort studies and disease prevention trials is discussed

a system of convection diffusion equations with small diffusion coefficient arising in semiconductor physics abstract we consider a system of two convection diffusion equations with a small diffusion parameter in one space dimension subject to dirichlet boundary conditions the system governs the evolution of the flow of electrons and holes in semiconductor devices on the dielectrical relaxation time scale the equations are coupled by nonlinear nonlocal electric field driven convection terms we prove the convergence in suitable topologies of the solutions of the diffusion convection problem to the unique solution of the convective limit problem subject to inflow boundary conditions as the diffusion coefficient tends to zero

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

existence of multiple positive solutions of singular nonlinear boundary value problems for a given positive integer n we provide conditions on the nonlinear function f which guarantee that the boundary value problem y f t y 0 0

difference methods for solving convection diffusion equations abstract difference methods for solving the convection diffusion equation are discussed the superiority of allen s approximation over central or upwind differences for one dimensional problems is confirmed the superiority being greatest when the boundary layer is very thin higher order methods give improved accuracy with negligible increase in cost a new iterative scheme is proposed for the two dimensional problem which requires orders of magnitude fewer arithmetic operations than existing procedures currently available in the literature

singular dirichlet second order bvps with impulses abstract the paper deals with the existence of solutions to singular second order differential equations with impulse effects and with the dirichlet boundary conditions the right hand side of the differential equation can be singular in its phase variable

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

a high order scheme for unsteady heat conduction equations abstract in this work a high order scheme is proposed for solving the heat conduction problems in this scheme the governing equation of heat conduction is firstly written into a system of first order partial differential equations which are integrated over the control volumes around each node point and a finite time step the resulting time integrals are approximated by numerical quadrature with an undetermined weighting parameter the error analysis allows us to determine the parameter and the ratio of the time step to the square of grid spacing by making the error of the scheme as small as possible both theoretical and numerical results show that the proposed high order scheme can achieve at least sixth order accuracy at the expense of the same computing cost as the traditional finite volume method and the high order scheme demonstrates a better performance than the high order compact finite difference method and the runge kutta discontinuous galerkin method

methods for the analysis of sampled cohort data in the cox proportional hazards model methods are provided for regression parameter and cumulative baseline hazard estimation in the cox proportional hazards model when the cohort is sampled according to a predictable sampling probability law the key to the methodology is to define counting processes which count joint failure and sampled risk sets occurrences and to derive the appropriate intensities for these counting processes leading to estimation methods parallel to those for full cohort data these techniques are illustrated for a number of sampling designs including three novel designs counter matching with additional randomly sampled controls quota sampling of controls and nested case control sampling with number of controls dependent on the failure s exposure status general asymptotic theory is developed for the maximum partial likelihood estimator and cumulative baseline hazard estimator and is used to derive the asymptotic distributions for estimators from a large class of designs

the semidiscrete finite volume element method for nonlinear convection diffusion problem abstract in this work a semidiscrete finite volume element method for the nonlinear convection diffusion problem is considered under some regular assumptions we obtain the l2 and h1 norm error estimates of numerical solution furthermore we investigate two grid finite volume element method for the considered equations compared with the standard method the two grid method is of the same order as the standard method in the h1 norm as long as the mesh sizes satisfy h o h 1 3 however the two grid method involves much less work than the standard method finally some numerical results are provided to verify the established theoretical analysis

the existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the p laplacian operator abstract in this paper we study the existence of countable many positive solutions for a class of nonlinear singular boundary value systems with p laplacian operator p 1 u a 1 t f u v 0 0 t 1 p 2 v a 2 t g u v 0 0 t 1 1 p 1 u 0 1 p 1 u 0 0 1 p 1 u 1 1 p 1 u 1 0 2 p 2 v 0 2 p 2 v 0 0 2 p 2 v 1 2 p 2 v 1 0 where p i s s p i 2 s p i 1 f g are lower semi continuous functions a i t has countable many singularities on 0 1 2 i 1 2 by using the fixed point theorem of cone expansion and compression of norm type the existence of countable many positive solutions for nonlinear singular boundary value system with p laplacian operator are obtained

a generalization of concavity for finite differences abstract the concept of concavity is generalized to discrete functions u satisfying the n th order difference inequality 1 n k n u m 0 m 0 1 n and the homogeneous boundary conditions u 0 u k 1 0 u n k 1 u n n 0 for some k 1 n 1 a piecewise polynomial is constructed which bounds u below the piecewise polynomial is employed to obtain a positive lower bound on u m for m k n k where the lower bound is proportional to the supremum of u an analogous bound is obtained for a related green s function

analysis of stabilized higher order finite element approximation of nonstationary and nonlinear convection diffusion reaction equations abstract in this work the numerical approximation of unsteady and nonlinear convection diffusion reaction models with small diffusion by higher order finite element methods is studied streamline upwind petrov galerkin supg stabilization together with anisotropic shock capturing as an additional stabilization in crosswind direction are used error analyses and the parameter design given in the literature are generalized to the case of nonlinear reaction rates by numerical studies it is illustrated that the combination of these approximation techniques reduce spurious oscillations in crosswind direction and improve the accuracy of simulations

the existence of positive solutions for a nonlinear four point singular boundary value problem with a p laplacian operator in this paper we study the existence of positive solutions for the nonlinear four point singular boundary value problem with a p laplacian operator by using the fixed point index theory the existence of a positive solution and many positive solutions for a nonlinear four point singular boundary value problem with a p laplacian operator are obtained

a finite difference scheme for multidimensional convection diffusion reaction equations abstract in this paper a finite difference scheme is proposed for multidimensional convection diffusion reaction equations particularly designed to treat the most interesting case of small diffusion it is based closely on the work endur and nesliturk 2011 application of the method to multidimensional convection diffusion reaction equation is based on a simple splitting of the convection diffusion reaction equation and then joining their approximations obtained with endur and nesliturk 2011 the method adapts very well to all regimes with continuous transitions from one regime to another numerical tests show good performance of the method and superiority with respect to well known stabilized finite element methods

upper and lower solutions method and a superlinear singular boundary value problem abstract the singular boundary value problem u g t u u 0 for t 0 1 u 0 u 1 0 is studied in this paper the singularity may appear at u 0 t 0 and t 1 and the function g may be superlinear at u and change sign the existence of solutions is obtained via an upper and lower solutions method

regularity of an elliptic problem with a singular nonlinearity we study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular we prove that the positive solution is in general holder continuous up to the boundary and has even better regularity in some special cases

solvability of some nonlinear boundary value problems on etablit l existence des solutions positives de problemes a deux points limites singuliers pour des equations differentielles de la forme y f t y y o o t 1

a mixed hybrid finite element method for convection diffusion problems a mixed hybrid finite element method for scalar diffusion convection problems is presented in this method the lowest degree raviart thomas finite element is used for the diffusive term the convective term is treated by means of a lagrange multiplier which is introduced in the hybrid form the numerical results show that this method produces very accurate solutions for high peclet numbers

the integral equation method for a steady kinematic dynamo problem with only a few exceptions the numerical simulation of cosmic and laboratory hydromagnetic dynamos has been carried out in the framework of the differential equation method however the integral equation method is known to provide robust and accurate tools for the numerical solution of many problems in other fields of physics the paper is intended to facilitate the use of integral equation solvers in dynamo theory in concrete the integral equation method is employed to solve the eigenvalue problem for a hydromagnetic dynamo model with an isotropic helical turbulence parameter for the case of spherical geometry three examples of the function r with steady and oscillatory solutions are considered a convergence rate proportional to the inverse squared of the number of grid points is achieved based on this method a convergence accelerating strategy is developed and the convergence rate is improved remarkably typically quite accurate results can be obtained with a few tens of grid points in order to demonstrate its suitability for the treatment of dynamos in other than spherical domains the method is also applied to x2 dynamos in rectangular boxes the magnetic fields and the electric potentials for the first eigenvalues are visualized

boundary value problems for singular second order functional differential equations abstract we consider general linear boundary value problems for equations of the form y f x y x 0 0 x x is continuous and f x y has a singularity at y 0 the results improve and extend earlier results for the case x x due to taliaferro 1979 and gatica 1989

marginal hazards model for case cohort studies with multiple disease outcomes case cohort study designs are widely used to reduce the cost of large cohort studies while achieving the same goals especially when the disease rate is low a key advantage of the case cohort study design is its capacity to use the same subcohort for several diseases or for several subtypes of disease in order to compare the effect of a risk factor on different types of diseases times to different events need to be modelled simultaneously valid statistical methods that take the correlations among the outcomes from the same subject into account need to be developed to this end we consider marginal proportional hazards regression models for case cohort studies with multiple disease outcomes we also consider generalized case cohort designs that do not require sampling all the cases which is more realistic for multiple disease outcomes we propose an estimating equation approach for parameter estimation with two different types of weights consistency and asymptotic normality of the proposed estimators are established large sample approximation works well in small samples in simulation studies the proposed methods are applied to the busselton health study

singular lidstone boundary value problem with given maximal values for solutions abstract the singular problem 1 nx 2n f t x x 2n 2 x 2j 0 x 2j t 0 0 j n 1 max x t 0 t t a depending on the parameter is considered here the positive caratheodory function f may be singular at the zero value of all its phase variables the paper presents conditions which guarantee that for any a 0 there exists a 0 such that the above problem with a has a solution x ac2n 1 0 t which is positive on 0 t the proofs are based on the regularization and sequential techniques and use the leray schauder degree and vitali s convergence theorem

selecting an efficient design for assessing exposure disease relationships in an assembled cohort summary we develop approximate methods to compare the efficiencies and to compute the power of alternative potential designs for sampling from a cohort before beginning to collect exposure data our methods require only that the cohort be assembled meaning that the numbers of individuals nkj at risk at pairs of event times tk and ti tk are available to compute nkj one needs to know the entry follow up censoring and event history but not the exposure for each individual our methods apply to any unbiased control sampling design in which cases are compared to a random sample of noncases at risk at the time of an event we apply our methods to approximate the efficiencies of the nested case control design the case cohort design and an augmented case cohort design compared to the full cohort design in an assembled cohort of 17 633 members of an insurance cooperative who were followed for mortality from prostatic cancer the assumptions underlying the approximation are that exposure is unrelated both to the hazard of an event and to the hazard for censoring the approximations performed well in simulations when both assumptions held and when the exposure was moderately related to censoring

a nonstandard finite difference scheme for convection diffusion equations having constant coefficients in this note we derive using the subequation method a new nonstandard finite difference scheme nsfd for a class of convection diffusion equations having constant coefficients despite the fact that this scheme has nonlinear denominator functions of the step sizes even for linear pdes it has a couple of favourable properties it is explicit and due to its construction it reproduces important properties of the solution of the parabolic pde this proposed method conserves by construction the positivity of the solution if one choses a right combination of spatial and temporal step sizes and hence it is perfectly suited for solving for example air pollution problems or the black scholes equation for the valuation of standard options since it avoids negative values for the calculated prices finally we illustrate the usefulness of this newly proposed method on a classical benchmark example from the literature

on solvability and unique solvability of two point singular boundary value problems abstract for the differential equation u f t u u where the function f a b r 2 r has nonintegrable singularities with respect to the time variable at t a and t b new unimprovable sufficient conditions of solvability and unique solvability of the boundary value problems u a 0 u b 0 and u a 0 u b 0 are established

positive solutions of boundary value problems for disfocal ordinary differential equations we study the existence of positive solutions of a nonlinear eigenvalue problem for a two point boundary value problem for a family of second order disfocal ordinary differential equations we apply a cone theoretic fixed point theorem and obtain both sufficient conditions for the existence of positive solutions and sufficient conditions for the existence of multiple positive solutions

singular boundary value problems for the one dimension p laplacian abstract the singular boundary value problem u g t u u 0 for t 0 1 u 0 u 1 0 where s s p 2s p 1 is studied in this paper the singularity may appear at u 0 t 0 and t 1 and the function g may change sign the existence of solutions is obtained via an upper and lower solution method

estimation in the semiparametric accelerated failure time model with missing covariates improving efficiency through augmentation abstractthis article considers linear regression with missing covariates and a right censored outcome we first consider a general two phase outcome sampling design where full covariate information is only ascertained for subjects in phase two and sampling occurs under an independent bernoulli sampling scheme with known subject specific sampling probabilities that depend on phase one information e g survival time failure status and covariates the semiparametric information bound is derived for estimating the regression parameter in this setting we also introduce a more practical class of augmented estimators that is shown to improve asymptotic efficiency over simple but inefficient inverse probability of sampling weighted estimators estimation for known sampling weights and extensions to the case of estimated sampling weights are both considered the allowance for estimated sampling weights permits covariates to be missing at random according to a monotone but unknown mechanism the asymptotic pr

existence theory for single and multiple solutions to singular positone boundary value problems abstract in this paper we present some old and new existence results for singular boundary value problems our nonlinearity may be singular in its dependent variable

a symmetric characteristic fve method with second order accuracy for nonlinear convection diffusion problems the finite volume element fve methods used currently are essentially low order and unsymmetric in this paper by biquadratic elements and multistep methods we construct a second order fve scheme for nonlinear convection diffusion problem on nonuniform rectangular meshes to overcome the numerical oscillation we discretize the problem along its characteristic direction the choice of alternating direction strategy is critical in this paper which guarantees the high efficiency and symmetry of the discrete scheme optimal order error estimates in h 1 norm are derived and a numerical example is given at the end to confirm the usefulness of the method

positive solutions of singular dirichlet and periodic boundary value problems let a and t be positive numbers the singular differential equation r x x q t f t x is considered here r 0 on 0 a may be singular at x 0 and f t x 0 may be singular at x 0 and x a effective sufficient conditions imposed on r q and f are given for the existence of a solution x to the above equation satisfying either the dirichlet conditions x 0 x t 0 or the periodic conditions x 0 x t x 0 x t and in addition 0 x a on 0 t

cox regression with incomplete covariate measurements abstract this article provides a general solution to the problem of missing covariate data under the cox regression model the estimating function for the vector of regression parameters is an approximation to the partial likelihood score function with full covariate measurements and reduces to the pseudolikelihood score function of self and prentice in the special setting of case cohort designs the resulting parameter estimator is consistent and asymptotically normal with a covariance matrix for which a simple and consistent estimator is provided extensive simulation studies show that the large sample approximations are adequate for practical use the proposed approach tends to be more efficient than the complete case analysis especially for large cohorts with infrequent failures for case cohort designs the new methodology offers a variance covariance estimator that is much easier to calculate than the existing ones and allows multiple subcohort augmentations to improve efficiency real data taken f

the mmoc and mmocaa schemes for the finite volume element method of convection diffusion problems we present and analyze the modified method of characteristics mmoc and the modified method of characteristics with adjusted advection mmocaa for the finite volume element fve method of convection diffusion problems these two schemes maintain the advantages of both the mmoc and the fve method and the mmocaa scheme discussed herein conserves the conservation law globally at a minor additional computational cost optimal order error estimates in the h 1 norm are proved for these schemes a numerical example is presented to confirm the estimates

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