on monotonicity preserving stabilized finite element approximations of transport problems the aim of this work is to design monotonicity preserving stabilized finite element techniques for transport problems as a blend of linear and nonlinear shock capturing stabilization as linear stabilization we consider and analyze a novel symmetric projection stabilization technique based on a local scott zhang projector next we design a weighting of the aforementioned linear stabilization such that when combined with a finite element discretization enjoying a discrete maximum principle usually attained via nonlinear stabilization it does not spoil these monotonicity properties then we propose novel nonlinear stabilization schemes in the form of an artificial viscosity method where the amount of viscosity is proportional to gradient jumps at either finite element boundaries or nodes for the nodal scheme we prove a discrete maximum principle for time dependent multidimensional transport problems numerical experiments support the numerical analysis and we show that the resulting methods prov

on the convergence of a finite element method for a nonlinear hyperbolic conservation law we consider a space time finite element discretization of a time dependent nonlinear hyperbolic conservation law in one space dimension burgers equation the finite element method is higher order accurate and is a petrov galerkin method based on the so called streamline diffusion modification of the test functions giving added stability we first prove that if a sequence of finite element solutions converges boundedly almost everywhere as the mesh size tends to zero to a function u then u is an entropy solution of the conservation law this result may be extended to systems of conservation laws with convex entropy in several dimensions we then prove using a compensated compactness result of murat tartar that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of burgers equation we also consider a further modification of the test functions giving a method with improved shock capturing finally we present the results of some numerical experiments

a time space flux corrected transport finite element formulation for solving multi dimensional advection diffusion reaction equations abstract we present a time space flux corrected transport fct finite element formulation for the multi dimensional time dependent advection diffusion reaction equation monotonic solutions can be achieved with the presented method while large time steps with courant number c r 1 are used numerical verification as well as a grid convergence analysis are carried out for 1d and 2d benchmark problems a 1d burgers equation with various reynolds numbers is solved with the time space fct method in addition a 2d transport problem with and without a nonlinear reaction inside of a cavity is considered finally the newly developed time space fct formulation is applied for modeling biofilm growth problems based on a continuum mathematical model it turns out that the time space fct method helps to reduce numerical dissipation and guarantees comparable numerical dispersion of the solution at the same time comparing to numerical solutions obtained by a time space finite incremental calculus fic method

differentiable monotonicity preserving schemes for discontinuous galerkin methods on arbitrary meshes this work is devoted to the design of interior penalty discontinuous galerkin dg schemes that preserve maximum principles at the discrete level for the steady transport and convection diffusion problems and the respective transient problems with implicit time integration monotonic schemes that combine explicit time stepping with dg space discretization are very common but the design of such schemes for implicit time stepping is rare and it had only been attained so far for 1d problems the proposed scheme is based on a piecewise linear dg discretization supplemented with an artificial diffusion that linearly depends on a shock detector that identifies the troublesome areas in order to define the new shock detector we have introduced the concept of discrete local extrema the diffusion operator is a graph laplacian instead of the more common finite element discretization of the laplacian operator which is essential to keep monotonicity on general meshes and in multi dimension the resulting nonlinear stabilization is non smooth and nonlinear solvers can fail to converge as a result we propose a smoothed twice differentiable version of the nonlinear stabilization which allows us to use newton with line search nonlinear solvers and dramatically improve nonlinear convergence a theoretical numerical analysis of the proposed schemes shows that they satisfy the desired monotonicity properties further the resulting operator is lipschitz continuous and there exists at least one solution of the discrete problem even in the non smooth version we provide a set of numerical results to support our findings

computing rank dependent utility in graphical models for sequential decision problems this paper is devoted to automated sequential decision in ai more precisely we focus here on the rank dependent utility rdu model this model is able to encompass rational decision behaviors that the expected utility model cannot accommodate however the non linearity of rdu makes it difficult to compute an rdu optimal strategy in sequential decision problems this has considerably slowed the use of rdu in operational contexts in this paper we are interested in providing new algorithmic solutions to compute an rdu optimal strategy in graphical models specifically we present algorithms for solving decision tree models and influence diagram models of sequential decision problems for decision tree models we propose a mixed integer programming formulation that is valid for a subclass of rdu models corresponding to risk seeking behaviors this formulation reduces to a linear program when mixed strategies are considered in the general case i e when there is no particular assumption on the parameters of rdu we propose a branch and bound procedure to compute an rdu optimal strategy among the pure ones after highlighting the difficulties induced by the use of rdu in influence diagram models we show how this latter procedure can be extended to optimize rdu in an influence diagram finally we provide empirical evaluations of all the presented algorithms

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

pointwise error estimates for a streamline diffusion finite element scheme pointwise error estimates for a streamline diffusion scheme for solving a model convection dominated singularly perturbed convection diffusion problem are given these estimates improve pointwise error estimates obtained by johnson et al 5

anisotropic mesh refinement in stabilized galerkin methods the numerical solution of a convection diffusion reaction model problem is considered in two and three dimensions a stabilized finite element method of galerkin least square type accomodates diffusion dominated as well as convection and or reaction dominated situations the resolution of boundary layers occuring in the singularly perturbed case is achieved using anisotropic mesh refinement in boundary layer regions in this paper the standard analysis of the stabilized galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement simplicial lagrangian elements of arbitrary order are used

on the stabilization parameter in the subgrid scale approximation of scalar convection diffusion reaction equations on distorted meshes abstract in this paper we revisit the definition of the stabilization parameter in the finite element approximation of the convection diffusion reaction equation the starting point is the decomposition of the unknown into its finite element component and a subgrid scale that needs to be approximated in order to incorporate the distortion of the mesh into this approximation we transform the equation for the subgrid scale within each element to the shape regular reference domain the expression for the subgrid scale arises from an approximate fourier analysis and the identification of the wave number direction where instabilities are most likely to occur the final outcome is an expression for the stabilization parameter that accounts for anisotropy and the dominance of either convection or reaction terms in the equation

a positivity preserving variational method for multi dimensional convectiondiffusionreaction equation a new positivity preserving variational ppv procedure is proposed to solve the convectiondiffusionreaction cdr equation through the generalization of stabilized finite element methods the present variational procedure offers minimal phase and amplitude errors for different regimes associated with convection diffusion and reaction effects by means of fourier analysis we first review the shortcomings of the galerkin least squares gls and the subgrid scale sgs methods during the change in sign of the reaction coefficient that motivates us for the present linear stabilization as a combined gls sgs methodology discrete upwind operator with a solution dependent nonlinear term is then introduced in high gradient regions which enables the positivity preserving property in the variational formulation direct extension to multi dimensions is carried out by considering the principle streamline and crosswind directions the efficacy of the method is demonstrated by the systematic accuracy and stability analyses in one and two dimensions results show the reduction of oscillations in the solution in one and two dimensional cases and a remarkable reduction in the phase error is observed for the cases with negative reaction coefficient the proposed formulation provides a superior solution in the reaction dominated as well as the convection dominated regimes due to the minimization of spurious oscillations and accurate capturing of the high gradient regions

sub mach 1 sound due to an arbitrary time dependent source near an absorbing half plane we discuss the diffraction of a spherical acoustic wave emanating from an arbitrary time dependent source near an absorbing half plane in the presence of a moving fluid the method of solution consists of complex fourier series integral transforms and asymptotic methods

blending low order stabilised finite element methods a positivity preserving local projection method for the convection diffusion equation in this work we propose a nonlinear blending of two low order stabilisation mechanisms for the convection diffusion equation the motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions the approach is to blend a first order artificial diffusion method which will be active only in the vicinity of layers and extrema with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution we prove existence of discrete solutions as well as convergence under appropriate assumptions on the nonlinear terms and on the exact solution numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle while the layers are not smeared significantly

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

on mathematical modeling of fluid structure interactions with nonlinear effects in this paper the numerical simulation of aeroelastic interactions of flexibly supported two degrees of freedom 2 dof airfoil in two dimensional 2d incompressible viscous turbulent flow subjected to a gust sudden change of flow conditions is considered the flow is modeled by reynolds averaged navier stokes equations rans and by k turbulence model the considered flow problem is discretized in space using the fully stabilized finite element fe method implemented in the developed in house program which allows to solve interaction problems in order to treat the time dependent inlet boundary condition the standard stabilization procedure was modified further the under relaxation procedure was introduced in order to overcome the artificial instability of the coupling algorithm the aeroelastic response to a sudden gust is numerically analyzed with the aid of the developed fe code

variational multi scale stabilized formulations for the stationary three field incompressible viscoelastic flow problem abstract in this paper three field finite element stabilized formulations are proposed for the numerical solution of incompressible viscoelastic flows these methods allow one to use equal interpolation for the problem unknowns u p elastic deviatoric stress velocity pressure and to stabilize dominant convective terms starting from residual based stabilized formulations the proposed method introduces a term by term stabilization which is shown to have a superior behavior when there are stress singularities a general discontinuity capturing technique for the elastic stress component is also proposed which allows one to eliminate the local oscillations that can appear when the weissenberg number we is high and the fluid flow finds an abrupt change in the geometry the formulations are tested in the classical 4 1 planar contraction benchmark up to we 5 in the inertial case with reynolds number re 1 and up to we 6 5 in the quasi non inertial case with re 0 01 the standard oldroyd b constitutive model is used for the rheological behavior and linear and quadratic elements for the spatial approximation

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

a nonlinear subgrid method for advection diffusion problems abstract this work presents a general framework for approximating advection diffusion equations based on principles of scale separation a two level decomposition of the discrete approximation space is performed and the local problem is modified to capture both local and nonlocal discontinuities the new feature is the local control resulting from decomposing the velocity field into the resolved and unresolved scales and requiring the satisfaction of the discrete model problem at the element level for a minimum kinetic energy associated to the unresolved scales this procedure leads to a nonlinear subgrid model that acts only on the unresolved scales but does not require any tuned up parameter it can be considered a self adaptive method such that the amount of the subgrid viscosity is automatically introduced according to the residual of the resolved scale at element level

stabilized reduced order models for the advection diffusion reaction equation using operator splitting reduced order modeling rom coupled with finite element methods has been used effectively in many disciplines to efficiently solve complex problems however for advection dominated flows numerical simulations often contain spurious nonphysical oscillations which will also be apparent in the rom simulations in this work we consider stabilization methods for rom for the advection diffusion reaction adr equation when it is solved both with and without operator splitting specifically we consider the streamline upwind petrov galerkin supg stabilization method and the spurious oscillations at layers diminishing sold stabilization method we build on these methods by constructing a coherent framework which successfully integrates these model reduction stabilization and operator splitting approaches and we provide numerical examples detailing the application of this framework in the adr setting the stabilized rom results are compared numerically with their corresponding full finite element simulations

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

an initial value approach to the production of discrete orthogonal coordinates abstract a numerical procedure is described for the orthogonalization of a discrete non orthogonal coordinate system in two dimensions one set of the non orthogonal coordinate lines is preserved by the orthogonalization procedure the other set is shifted by interpolation to form the new orthogonal system the procedure is based upon an initial value formulation with the non orthogonal coordinates as the independent variables a first order linear partial differential equation is derived which describes the orthogonal trajectories a discrete representation of the equation is suggested and an interpolation procedure is described examples of non orthogonal with orthogonalized meshes are given including closed and open coordinate lines the method may be used to generate boundary fitted coordinates for boundary value problems or eulerian time dependent codes it may also be used at each timestep in a waterbag or semi lagrangian code

on discrete maximum principles for discontinuous galerkin methods the aim of this work is to propose a monotonicity preserving method for discontinuous galerkin dg approximations of convection diffusion problems to do so a novel definition of discrete maximum principle dmp is proposed using the discrete variational setting of the problem and we show that the fulfilment of this dmp implies that the minimum maximum depending on the sign of the forcing term is on the boundary for multidimensional problems then an artificial viscosity av technique is designed for convection dominant problems that satisfies the above mentioned dmp the noncomplete stabilized interior penalty dg method is proved to fulfil the dmp property for the one dimensional linear case when adding such av with certain parameters the benchmarks for the constant values to satisfy the dmp are calculated and tested in the numerical experiments section finally the method is applied to different test problems in one and two dimensions to show its performance

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

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

a monotone streamline upwind finite element method for convection dominated flows a new streamline upwind approximation for treating advection dominatcd transport is presented the proposed method is quite simple to implement with existing finite element methods for this work the method is used in a two dimensional finite element computer program the new method is applied to several test cases that have been presented in previous articles to evaluate the stability and accuracy of proposed methods for treating advection dominatcd flows the results demonstrate that the new method is able to predict the test cases with a significant increase in accuracy compared to previous techniques in addition these results demonstrate conclusively that the method does not exhibit the non physical spatial oscillations that have plagued previous methods

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

stability analysis of the forward euler scheme for the convection diffusion equation using the supg formulation in space the stability and accuracy of the forward euler scheme for the semidiscrete problem arising from the space discretization of the convection diffusion equation using the supg formulation are analysed both linear and quadratic finite elements are considered the stability limits are derived for the one dimensional problem and a heuristic criterion is proposed for the multidimensional equation some numerical experiments are conducted in order to assess the performance of different finite elements

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