postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

two grid method for the two dimensional time dependent schrodinger equation by the finite element method abstract in this paper we construct a backward euler full discrete two grid finite element scheme for the two dimensional time dependent schrodinger equation with this method the solution of the original problem on the fine grid is reduced to the solution of same problem on a much coarser grid together with the solution of two poisson equations on the same fine grid we analyze the error estimate of the standard finite element solution and the two grid solution in the h 1 norm it is shown that the two grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h k k 1 finally a numerical experiment indicates that our two grid algorithm is more efficient than the standard finite element method

a new a posteriori error estimate for convection reaction diffusion problems a new a posteriori error estimate is derived for the stationary convection reaction diffusion equation in order to estimate the approximation error in the usual energy norm the underlying bilinear form is decomposed into a computable integral and two other terms which can be estimated from above using elementary tools of functional analysis two auxiliary parameter functions are introduced to construct such a splitting and tune the resulting bound if these functions are chosen in an optimal way the exact energy norm of the error is recovered which proves that the estimate is sharp the presented methodology is completely independent of the numerical technique used to compute the approximate solution in particular it is applicable to approximations which fail to satisfy the galerkin orthogonality e g due to an inconsistent stabilization flux limiting low order quadrature rules round off and iteration errors etc moreover the only constant that appears in the proposed error estimate is global and stems from the friedrichs poincare inequality numerical experiments illustrate the potential of the proposed error estimation technique

a multi level discontinuous galerkin method for solving the stationary navier stokes equations abstract in this paper we consider the multi level discontinuous finite element method for solving the stationary incompressible navier stokes equations on the coarsest mesh the discrete nonlinear navier stokes equations are solved by using piecewise polynomial functions which are totally discontinuous across inter element boundaries and are pointwise divergence free on each element for the velocity and are continuous functions for the pressure subsequent approximations are generated on a succession of refined grids by solving the newton linearized navier stokes equations using piecewise polynomial functions which are similar to that on the coarsest mesh finally the well posedness and the optimal error estimate for the multi level discontinuous galerkin method are provided the error analysis shows that when the mesh scales k j 1 o k j 2 h j 1 o h j 2 with j 0 1 j 1 are chosen the multi level finite element method can save a large amount of computational time compared with the one level finite element method

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

two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

n widths and singularly perturbed boundary value problems using the theory of n widths the approximability of solutions of singularly perturbed linear reaction diffusion problems in two dimensions is quantified

a posteriori error estimates for combined finite volume finite element discretizations of reactive transport equations on nonmatching grids abstract we derive in this paper guaranteed and fully computable a posteriori error estimates for vertex centered finite volume type discretizations of transient linear convection diffusion reaction equations our estimates enable actual control of the error measured either in the energy norm or in the energy norm augmented by a dual norm of the skew symmetric part of the differential operator lower bounds global in space but local in time are also derived these lower bounds are fully robust with respect to convection or reaction dominance and the final simulation time in the augmented norm setting on the basis of the derived estimates we propose an adaptive algorithm which enables to automatically achieve a user given relative precision this algorithm also leads to efficient calculations as it balances the time and space error contributions as an example we apply our estimates to the combined finite volume finite element scheme including such features as use of mass lumping for the time evolution or reaction terms of upwind weighting for the convection term and discretization on nonmatching meshes possibly containing nonconvex and non star shaped elements a collection of numerical experiments illustrates the efficiency of our estimates and the use of the space time adaptive algorithm

convergence of a shock capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions convergence of a shock capturing streamline diffusion nite element method for a conservation law in two space dimensions

adaptive lagrange galerkin methods for unsteady convection diffusion problems in this paper we derive an a posteriori error bound for the lagrange galerkin discretisation of an unsteady linear convection diffusion problem assuming only that the underlying space time mesh is nondegenerate the proof of this error bound is based on strong stability estimates of an associated dual problem together with the galerkin orthogonality of the finite element method based on this a posteriori bound we design and implement the corresponding adaptive algorithm to ensure global control of the error with respect to a user defined tolerance

an adaptive support vector regression based on a new sequence of unified orthogonal polynomials in practical engineering small scale data sets are usually sparse and contaminated by noise in this paper we propose a new sequence of orthogonal polynomials varying with their coefficient unified chebyshev polynomials ucp which has two important properties namely orthogonality and adaptivity based on these new polynomials a new kernel function the unified chebyshev kernel uck is constructed which has been proven to be a valid svm kernel to find the optimal polynomial coefficient and the optimal kernel we propose an adaptive algorithm based on the evaluation criterion for adaptive ability of uck to evaluate the performance of the new method we applied it to learning some benchmark data sets for regression and compared it with other three algorithms the experiment results show that the proposed adaptive algorithm has excellent generalization performance and prediction accuracy and does not cost more time compared with other svms therefore this method is suitable for practical engineering application

the relations of go and stop wave to car accidents in a cellular automaton with velocity dependent randomization abstract in this paper we numerically study the probability p ac of the occurrence of traffic accidents in the nagel schreckenberg ns model with velocity dependent randomization vdr numerical results show that there is a critical density over which car accidents occur but below which no car accidents happen different from the accident probability in the ns model the accident probability in the vdr model monotonously decreases with increase of car density above the critical density the value of the accident probability is only determined by the stochastic noise and the number of cars on road in the stochastic vdr model with the speed limit v max 1 no critical density exists and car accidents happen in the whole density region the braking probabilities of standing cars and moving cars have different influences on the accident probability a mean field theory reveals that the accident probability is proportional to the mean density of go and stop wave per time step theoretical analyses give excellent agreement with numerical results in the vdr model

a velocity pressure streamline diffusion finite element method for navier stokes equations a streamline diffusion finite element method is introduced for the time dependent incompressible navier stokes equations in a bounded domain in r squared and r cubed in the case of a flow with a high reynolds number an error estimate is proved and numerical results are given the method is based on a mixed velocity pressure formulation using the same finite element discretization of space time for the velocity and the pressure spaces which consist of piecewise linear functions together with certain least squares modifications of the galerkin variational formulation giving added stability without sacrificing accuracy

a two grid block centered finite difference method for darcy forchheimer flow in porous media a two grid block centered finite difference method is proposed for solving the two dimensional darcy forchheimer model describing non darcy flow in porous media to construct the two grid method we modify the original nonlinear elliptic operator of darcy forchheimer flow to a twice continuously differentiable one by introducing a small and positive parameter varepsilon by using the two grid method solving a nonlinear equation on a fine grid is reduced to solving a nonlinear equation on a coarse grid together with solving a linear equation on a fine grid optimal order error estimates for pressure and velocity in discrete l 2 norms are obtained some numerical examples are given to show the accuracy and efficiency of the presented method

large scale computation of incompressible viscous flow by least squares finite element method abstract the least square finite element method lsfem based on the velocity pressure vorticity formulation is applied to large scale three dimensional steady incompressible navier stokes problems this method can accommodate equal order interpolations and results in a symmetric positive definite algebraic system which can be solved effectively by simple iterative methods the first order velocity bernoulli pressure vorticity formulation for incompressible viscous flows is also tested the first order velocity pressure stress formulation is not elliptic in the ordinary sense so we do not recommend its use for newtonian flows for three dimensional flows a compatibility equation i e zero divergence of vorticity vector is included to make the first order system elliptic as a by product of proving the ellipticity of first order systems a rigorous mathematical technique has been developed to justify the number of permissible boundary conditions for the navier stokes equations the simple substitution or newton s method is employed to linearize the partial differential equations the lsfem is used to obtain discretized equations and the system of algebraic equations is solved using the jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices matrix free to achieve high efficiency to show the validity of this method for large scale computation we give numerical results for the 2d driven cavity problem at re 10000 with 408 400 bilinear elements the flow in a 3d cavity is calculated at re 100 400 and 1000 with 50 52 50 trilinear elements the taylor gortler like vortices are observed for re 1000

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

nonconforming streamline diffusion finite element methods for convection diffusion problems we analyze nonconforming finite element approximations of streamline diffusion type for solving convection diffusion problems both the theoretical and numerical investigations show that additional jump terms have to be added in the nonconforming case in order to get the same o h k 1 2 order of convergence in l 2 as in the conforming case for convection dominated problems a rigorous error analysis supported by numerical experiments is given

a dual petrovgalerkin finite element method for the convectiondiffusion equation we present a minimum residual finite element method based on a dual petrovgalerkin formulation for convectiondiffusion problems in a higher order adaptive continuous galerkin setting the method borrows concepts from both the discontinuous petrovgalerkin dpg method by demkowicz and gopalakrishnan 2011 and the method of variational stabilization by cohen dahmen and welper 2012 and it can also be interpreted as a variational multiscale method in which the fine scales are defined through a dual orthogonality condition a key ingredient in the method is the proper choice of dual norm used to measure the residual and we present two choices which are observed to be robust in both convection and diffusion dominated regimes as well as a proof of stability for quasi uniform meshes and a method for the weak imposition of boundary conditions numerically obtained convergence rates in 2d are reported and benchmark numerical examples are given to illustrate the behavior of the method

an a posteriori error estimate for finite element approximations of a singularly perturbed advection diffusion problem abstract in this paper the author presents an a posteriori error estimator for approximations of the solution to an advection diffusion equation with a non constant vector valued diffusion coefficient in a conforming finite element space based on the complementary variational principle we show that the error of an approximate solution in an associated energy norm is bounded by the sum of the weighted l 2 norms of solutions to a set of independent complementary variational problems each defined on only one element of the partition this error bound guarantees the over estimation of the true error and does not depend unfavourably on as goes to zero although the original equation is a non self adjoint problem the strong form of each local variational problem is always a poisson equation with neumann boundary conditions the approximation of these local problems is then discussed and it is shown that omitting a higher order term the finite element solutions of these local complementary variational problems provide a computable upper error bound for the original finite element approximation in the energy norm numerical results presented to validate the theoretical results show that the computed error bounds are tight for a wide range of values of and always over estimate the true errors

stabilized finite elements on anisotropic meshes a priori error estimates for the advection diffusion and the stokes problems stabilized finite elements on strongly anisotropic meshes are considered the design of the stability coefficients is addressed for both the advection diffusion and the stokes problems when using continuous piecewise linear finite elements on triangles using the polar decomposition of the jacobian of the affine mapping from the reference triangle to the current one k and from a priori error estimates a new definition of the stability coefficients is proposed our analysis shows that these coefficients do not depend on the element diameter hk but on a characteristic length associated with k via the polar decomposition a numerical assessment of the theoretical analysis is carried out

a posteriori error estimation for two level discretizations of flows of electrically conducting incompressible fluids abstract we develop a posteriori error estimates for two finite element discretizations of a model due to peterson 1 describing the flow of an electrically conducting viscous incompressible fluid the model involves five equations for the fluid velocity u the pressure p and the electric potential estimates are derived for both the usual finite element method as well as a two level newton finite element discretization

iterative two grid methods for semilinear elliptic equations abstract we design and analyze a new finite element discretization technique based on iterative two grid methods for semilinear elliptic partial differential equations pdes we first present an iterative two grid method that just applies the classical two grid for semilinear elliptic pdes in a successive fashion namely to solve a semilinear problem on the coarse space and then to solve a symmetric positive definite problem on the fine space thus we only need to deal with semilinear problems in the coarse space in the first algorithm secondly we replace the semilinear term by using the corresponding first order taylor expansion in the second iterative two grid we provide the error estimates for the second iterative algorithm and present numerical experiments to show the efficiency of our methods

variational multiscale a posteriori error estimation for multi dimensional transport problems this paper presents an explicit a posteriori error estimator for the multi dimensional transport equation based on an approximation to the variational multiscale theory the technique is adequate for methods with a local error distribution such as stabilized methods yielding efficiencies close to one for both the hyperbolic and diffusive limit regarding the hyperbolic limit previous work is extended to multi dimensions adequate norms are proposed for the computation of the error and proper error intrinsic time scales are calculated for the bilinear quadrilateral and the linear triangle furthermore the model considers the element interface error along the element edges enabling the error prediction in the diffusive limit the success of the method can be explained by the fact that in stabilized methods the element local problem captures most of the error and the proposed error intrinsic time scales are an approximation to the solution of the dual problem the proposed technology which can be implemented straightforwardly in existing codes is extremely economical because it is a simple explicit postprocessing

a posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection diffusion equations this paper is devoted to the study of a posteriori and a priori error estimates for the scalar nonlinear convection diffusion equation c t nabla cdot u f c varepsilon delta c 0 the estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the l 1 norm in the situation where the diffusion parameter varepsilon is smaller or comparable to the mesh size numerical experiments underline the theoretical results

trust region methods for nonlinear elliptic equations with radial basis functions we consider the numerical solution of nonlinear elliptic boundary value problems with kansas method we derive analytic formulas for the jacobian and hessian of the resulting nonlinear collocation system and exploit them within the framework of the trust region algorithm this ansatz is tested on semilinear quasilinear and fully nonlinear elliptic pdes including plateaus problem heleshaw flow and the mongeampre equation with excellent results the new approach distinctly outperforms previous ones based on linearization or finite difference jacobians

investigations on two kinds of two level stabilized finite element methods for the stationary navier stokes equations abstract this article considers two kinds of two level stabilized finite element methods based on local gauss integral technique for the two dimensional stationary navier stokes equations approximated by the lowest equal order elements which do not satisfy the inf sup condition the two level methods consist of solving a small non linear system on the coarse mesh and then solving a linear system on the fine mesh the error analysis shows that the two level stabilized finite element methods provide an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the navier stokes equations on a fine mesh for a related choice of mesh widths therefore the two level methods are of practical importance in scientific computation finally the performance of two kinds of two level stabilized methods are compared in efficiency and precision aspects by a series of numerical experiments the conclusion is that simple two level stabilized method is a viable choice for the lowest equal order approximations of the stationary navier stokes problem

a robust dpg method for convection dominated diffusion problems ii adjoint boundary conditions and mesh dependent test norms we introduce a dpg method for convection dominated diffusion problems the choice of a test norm is shown to be crucial to achieving robust behavior with respect to the diffusion parameter demkowicz and heuer 2011 18 we propose a new inflow boundary condition which regularizes the adjoint problem allowing the use of a stronger test norm the robustness of the method is proven and numerical experiments demonstrate the method s robust behavior

adaptive streamline diffusion finite element methods for stationary convection diffusion problems adaptive finite element methods for stationary convectiondiffusion problems are designed and analyzed the underlying discretization scheme is the shock capturing streamline diffusion method the adaptive algorithms proposed are based on a posteriori error estimates for this method leading to reliable methods in the sense that the desired error control is guaranteed a priori error estimates are used to show that the algorithms are efficient in a certain sense

density induced margin support vector machines this paper proposes a new classifier called density induced margin support vector machines dmsvms dmsvms belong to a family of svm like classifiers thus dmsvms inherit good properties from support vector machines svms e g unique and global solution and sparse representation for the decision function for a given data set dmsvms require to extract relative density degrees for all training data points these density degrees can be taken as relative margins of corresponding training data points moreover we propose a method for estimating relative density degrees by using the k nearest neighbor method we also show the upper bound on the leave out one error of dmsvms for a binary classification problem and prove it promising results are obtained on toy as well as real world data sets

a two grid method for incompressible miscible displacement problems by mixed finite element and eulerian lagrangian localized adjoint methods abstract in this paper we present a scheme for solving two dimensional miscible displacement problems using eulerian lagrangian localized adjoint methods and mixed finite element methods since only the velocity and not the pressure appears explicitly in the concentration equation an eulerian lagrangian localized adjoint method is used to solve the concentration equation and a mixed finite element method is used for the pressure equation to linearize and decouple the mixed method equations we use a two grid algorithm based on the newton iteration method for this fully discrete problems first we solve the original nonlinear equations on the coarse grid then we solve the linearized problem on the fine grid using newton iteration once it is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h 1 2 in this paper

a two grid eulerian lagrangian localized adjoint method to miscible displacement problems with dispersion term abstract in this paper we propose a novel two grid method of eulerian lagrangian localized adjoint method ellam mixed finite element method mfem solutions for nonlinear and coupled miscible displacement problems which describe complex fluid flow processes in porous media an ellam is used to solve the nonlinear convection diffusion equation for concentration and we present a mfem to compute the pressure equation for the pressure and darcy velocity a two grid scheme is investigated to linearize and decouple the mixed method equations it is illustrated that the coarse space can be selected as coarse as h o h 1 2 and the asymptotically optimal approximation as the nonlinear schemes can be still achieved as a result a large class of such nonlinear and coupled problems can be solved efficiently with large time steps

a two grid discretization method for decoupling systems of partial differential equations in this paper we propose a two grid finite element method for solving coupled partial differential equations e g the schrodinger type equation with this method the solution of the coupled equations on a fine grid is reduced to the solution of coupled equations on a much coarser grid together with the solution of decoupled equations on the fine grid it is shown both theoretically and numerically that the resulting solution still achieves asymptotically optimal accuracy

an effective explicit pressure gradient scheme implemented in the two level non staggered grids for incompressible navier stokes equations in this paper an improved two level method is presented for effectively solving the incompressible navier stokes equations this proposed method solves a smaller system of nonlinear navier stokes equations on the coarse mesh and needs to solve the oseen type linearized equations of motion only once on the fine mesh level within the proposed two level framework a prolongation operator which is required to linearize the convective terms at the fine mesh level using the convergent navier stokes solutions computed at the coarse mesh level is rigorously derived to increase the prediction accuracy this indispensable prolongation operator can properly communicate the flow velocities between the two mesh levels because it is locally analytic solution convergence can therefore be accelerated for the sake of numerical accuracy momentum equations are discretized by employing the general solution for the two dimensional convection diffusion reaction model equation the convective instability problem can be simultaneously eliminated thanks to the proper treatment of convective terms the converged solution is thus very high in accuracy as well as in yielding a quadratic spatial rate of convergence for the sake of programming simplicity and computational efficiency pressure gradient terms are rigorously discretized within the explicit framework in the non staggered grid system the proposed analytical prolongation operator for the mapping of solutions from the coarse to fine meshes and the explicit pressure gradient discretization scheme which accommodates the dispersion relation preserving property have been both rigorously justified from the predicted navier stokes solutions

minimum residual adaptive multilevel finite element procedure for the solution of nonlinear stationary problems a multilevel minimum residual type procedure for an adaptive finite element fe solution of a general nonlinear differential problem is analyzed with respect to linearization and adaptive fe approximation errors simple sufficient conditions for the linear and quadratic convergence to the exact solution of the original problem with increasing number of levels are established a nontrivial numerical example illustrates the efficiency of the method

a robust a posteriori estimator for the residual free bubbles method applied to advection diffusion problems we develop the a posteriori error analysis for the rfb method applied to the linear advection diffusion problem the numerical error measured in suitable norms is estimated in terms of the numerical residual the robustness is investiged in the sense that we prove uniform equivalence between a norm of the numerical residual and a particular norm of the error

boundary effects on car accidents in a cellular automaton model in this paper we numerically study the probability pac of occurrence of car accidents in the nagel schreckenberg ns model with open boundary condition in the deterministic ns model numerical results show that there exists a critical value of extinction rate above which no car accidents occur and below which the probability pac is independent of the speed limit vmax and the injection rate but only determined by the extinction rate in the non deterministic ns model the probability pac is a non monotonic function of in the region of low value while it is independent of in the region of high value the stochastic braking not only reduces the occurrence of car accidents but splits degenerate effects of vmax on the probability pac theoretical analyses give an agreement with numerical results in the deterministic ns model and in the non deterministic ns model with vmax 1 in the case of low value region qualitative differences between open and periodic systems in the relations of pac to the bulk density imply that various correlations may exist between the two systems

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

convergence to steady state solutions for stabilized finite element simulations of compressible flows abstract the main objective of the paper is to compare the numerical performance of several pseudo time integration algorithms used for driving finite element solutions of compressible flow equations towards the steady state the algorithms are implementations of four strategies explicit linear implicit multistage linear implicit and nonlinear all except the explicit scheme employ a particular implementation of gmres to solve systems of linear equations the nonlinear algorithm uses at each time step two iterations of the newton method with an approximate jacobian the finite element formulation is a version of the stabilized fem with h adaptivity the comparison is done for three flow problems in three flow regimes subsonic transonic and supersonic final conclusions concerning relative performance of the methods are drawn

dangerous situations within the framework of the nagel schreckenberg model this paper investigates the occurrence of dangerous situations ds within the framework of the nagel schreckenberg model the conditions of the ds are modified it is shown that when vmax 1 there will be no ds in both deterministic and non deterministic cases the situation is different for vmax 1 we show that in the deterministic case the probability of ds covers a two dimensional region which depends on both the density and the initial configuration as for the non deterministic case our results are qualitatively the same as previous ones but are quantitatively different

a posteriori error estimates and adaptive finite elements for a nonlinear parabolic problem related to solidification a posteriori error estimates are derived for a nonlinear parabolic problem arising from the isothermal solidification of a binary alloy space discretization with continuous piecewise linear finite elements is considered the l 2 in time h 1 in space error is bounded above and below by an error estimator based on the equation residual numerical results show that the effectivity index is close to one an adaptive finite element algorithm is proposed and a solutal dendrite is computed c 2002 elsevier science b v all rights reserved

a new two grid nonconforming mixed finite element method for nonlinear benjamin bona mahoney equation abstract a new low order two grid mixed finite element method fem is developed for the nonlinear benjamin bona mahoney bbm equation in which the famous nonconforming rectangular c n q 1 r o t element and q0 q0 constant element are used to approximate the exact solution u and the variable p u t respectively then based on the special properties of these two elements and interpolation post processing technique the superconvergence results for u in broken h1 norm and p in l2 norm are obtained for the semi discrete and crank nicolson fully discrete schemes without the restriction between the time step and coarse mesh size h or the fine mesh size h which improve the results of the existing literature finally some numerical results are provided to confirm the theoretical analysis

a posteriori error estimation for interior penalty finite element approximations of the advection reaction equation in this note we consider residual based a posteriori error estimation for finite element approximations of the transport equation for the discretization we use piecewise affine continuous or discontinuous finite elements and symmetric stabilization of interior penalty type the lowest order discontinuous galerkin method using piecewise constant approximation is included as a special case the key elements in the analysis are a saturation assumption and an approximation result for interpolation between discrete spaces

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

genuinely nonlinear models for convection dominated problems this paper introduces a general nonlinear subgrid scale sgs model having boundedartificial viscosity for the numerical simulation of convection dominated problems we also present a numerical comparison error analysis and numerical experiments between this model and the most common sgs model of smagorinsky which uses a p laplacian regularization the numerical experiments for the 2 d convection dominated convection diffusion test problem show a clear improvement in solution quality for the new sgs model this improvement is consistent with the bounded amount of artificial viscosity introduced by the new sgs model in the sharp transition regions

rapid and brief communiction possibilistic support vector machines we propose new support vector machines svms that incorporate the geometric distribution of an input data set by associating each data point with a possibilistic membership which measures the relative strength of the self class membership by using a possibilistic distance measure based on the possibilistic membership we reformulate conventional svms in three ways the proposed methods are shown to have better classification performance than conventional svms in various tests

convergence analysis of a multigrid method for a convection dominated model problem the paper presents a convergence analysis of a multigrid solver for a system of linear algebraic equations resulting from the discretization of a convection diffusion problem using a finite element method we consider piecewise linear finite elements in combination with a streamline diffusion stabilization we analyze a multigrid method that is based on canonical intergrid transfer operators a direct discretization approach for the coarse grid operators and a smoother of line jacobi type a robust diffusion and h independent bound for the contraction number of the two grid method and the multigrid w cycle are proved for a special class of convection diffusion problems namely with neumann conditions on the outflow boundary dirichlet conditions on the rest of the boundary and a flow direction that is constant and aligned with gridlines our convergence analysis is based on modified smoothing and approximation properties the arithmetic complexity of one multigrid iteration is optimal up to a logarithmic term

mean field theory for car accidents we study analytically the occurrence of car accidents in the nagel schreckenberg traffic model we obtain exact results for the occurrence of car accidents p ac as a function of the car density rho and the degree of stochastic braking p 1 in the case of speed limit v max 1 various quantities are calculated analytically the nontrivial limit p 1 0 is discussed

a two grid discretization scheme for eigenvalue problems a two grid discretization scheme is proposed for solving eigenvalue problems including both partial differential equations and integral equations with this new scheme the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy

effects of a type of quenched randomness on car accidents in a cellular automaton model this article reports on a numerical study of the probability pac of the occurrence of car accidents in the nagel schreckenberg ns model with a defect a bottleneck situation bottlenecks results in a distinct variety of traffic flow including saturated flow vehicle queuing changes of traffic flow and variation in the number of stopped cars the authors describe the model used and the conditions for the occurrence of accidents review the numerical studies of the car accidents and consider the effects of the stochastic braking and speed limiting on the probability of car accidents in the deterministic ns model numerical results show that there exists a critical value of car density below which no car accident happens the critical density is not related only to the maximum speed of cars but also to the braking probability at the defect the braking probability at a defect can enhance not suppress the occurrence of car accidents when its value is small only the braking probability at the defect is very large car accidents can be reduced by the bottleneck in the nondeterministic ns model the probability pac exhibits the same behaviors with that in the deterministic model except the case of vmax 1 under which the probability pac is only reduced by the defect the defect also induces the inhomogeneous distribution of car accidents over the whole road car accidents happen only in the high density region

error estimates of two level finite element method for smagorinsky model two level finite element method for simulating smagorinsky model in large eddy simulation is investigated in this two level algorithm a linearized discrete problem is solved by using the variational multiscale method on the coarse mesh corresponding to newton linearization method the linearized smagorinsky model is solved on the fine mesh the error estimates derived in this paper implies that by choosing appropriate mesh sizes and the radius of the spatial filter used in smagorinsky model the two level method provides the optimal convergence rates for the velocity in h1 norm and the pressure in l2 norm meanwhile the two different numerical experiments are given to support the optimal convergence rates and the high efficiency of two level algorithm

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

adaptive finite elements for a linear parabolic problem a posteriori error estimates for the heat equation in two space dimensions are presented a classical discretization is used euler backward in time and continuous piecewise linear triangular finite elements in space the error is bounded above and below by an explicit error estimator based on the residual numerical results are presented for uniform triangulations and constant time steps the quality of our error estimator is discussed an adaptive algorithm is then proposed successive delaunay triangulations are generated so that the estimated relative error is close to a preset tolerance again numerical results demonstrate the efficiency of our approach c 1998 elsevier science s a all rights reserved

on the convergence of shock capturing streamline diffusion finite element methods for hyperbolic conservation laws we extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the meh size with this term present we prove a maximum norm bound for finite element solutionsof burgers equation an thus complete an earlier convergence proof for this equation we further prove using entropy variables that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality asociated with the entropy variables results of some numerical experiments for the time dependent compressible euler equations in two dimensions are also reported

a two level method with backtracking for the navier stokes equations we consider a two level method for resolving the nonlinearity in finite element approximations of the equilibrium navier stokes equations the method yields l2 and h1 optimal velocity approximations and an l2 optimal pressure approximation the two level method involves solving one small nonlinear coarse mesh system one oseen problem hence linear with positive definite symmetric part on the fine mesh and one linear correction problem on the coarse mesh the algorithm we study produces an approximate solution with the optimal asymptotic in h accuracy for any fixed reynolds number we do not consider the behavior of the error for fixed h as re rightarrow infty i e for flows in transition to turbulence

variational multiscale a posteriori error estimation for systems the euler and navier stokes equations this paper extends explicit a posteriori error estimators based on the variational multiscale theory to systems of equations in particular the emphasis is placed on flow problems the euler and navier stokes equations three error estimators are proposed the standard the naive and the upper bound numerical results show that with a very economical algorithm the attained global and local efficiencies for the naive approach are reasonably close to unity whereas the standard and upper bound approaches give respectively approximate lower and higher error estimates

exact results for car accidents in a traffic model within the framework of a recent model for car accidents on single lane highway traffic we study analytically the probability of the occurrence of car accidents exact results are obtained various scaling behaviours are observed the linear dependence of the occurrence of car accidents on density is understood as the dominance of a single velocity in the distribution

a multi level correction scheme for eigenvalue problems abstractin this paper a new type of multi level correction scheme is proposed forsolving eigenvalue problems by nite element method with this new scheme the accuracy of eigenpair approximations can be improved after each correc tion step which only needs to solve a source problem on ner nite elementspace and an eigenvalue problem on the coarsest nite element space thiscorrection scheme can improve the e ciency of solving eigenvalue problemsby nite element method keywords eigenvalue problem multi level correction scheme nite ele ment method multi space multi grid ams subject classi cations 65n30 65b99 65n25 65l15 1 introduction the purpose of this paper is to propose a new type of multi level correction schemebased on nite element discretization to solve eigenvalue problems the two gridmethod for solving eigenvalue problems has been proposed and analyzed by xu andzhou in 21 the idea of the two grid comes from 19 20 for nonsymmetric orinde nite problems and nonlinear elliptic equations since then there have existedmany numerical methods for solving eigenvalue problems based on the idea of two grid method 1 6 17

adaptive finite element methods for conservation laws based on a posteriori error estimates we prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension and design corresponding adaptive methods the proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem together with the galerkin orthogonality of the finite element method the strong stability estimate uses the entropy condition for the system in an essential way 1995 john wiley sons inc

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

adaptive finite element methods in computational mechanics we present a general approach to adaptivity for finite element methods and give applications to linear elasticity non linear elasto plasticity and nonlinear conservation laws including numerical results

the semi discrete streamline diffusion finite element method for time dependented convection diffusion problems abstract in this paper a semi discrete streamline diffusion finite element method sdfem for solving the time dependent convection dominated diffusion problem in one dimension is studied under the mild regularity assumptions the convergence stability results are derived finally some numerical experiments are given for a convection dominated problem with boundary layer at x 1 by using p1 element compared with the finite difference streamline diffusion method fdsdm the semi discrete sdfem is more stable and it can imitate the exact solution effectively

stability and convergence of a finite element method for a semilinear elliptical problem with small viscosity abstract in this paper we propose and analyze a new finite element method for a semilinear elliptical problem with small viscosity firstly we prove the existence and uniqueness of solution for its variational problem then we obtain the stability and convergence of the corresponding finite element galerkin approximation furthermore we derive the optimal error estimates in the l 2 and h 1 norms for the finite element approximations respectively finally several numerical experiments are provided to confirm the above theoretical results

long time unconditional stability of a two level hybrid method for nonstationary incompressible navier stokes equations abstract this paper presents a two level maccormack rapid solver method for solving a 2d time dependent navier stokes problem that models an incompressible fluid flow a decoupling approach based on the coupling term via spatial extrapolation is proposed for devising decoupled marching algorithms for the nonstationary incompressible model in this hybrid method an explicit maccormack scheme provides a h 1 optimal velocity approximation and a l 2 optimal pressure approximation by solving a nonlinear navier stokes problem on a coarse grid with mesh size h while an implicit crank nicolson algorithm consists in dealing with the fully discrete linear generalized stokes problem on a fine grid with mesh size h h the theoretical result suggests that our method is unconditionally stable over long time intervals while the numerical evidences both confirm the theoretic analysis and show that the algorithm is convergent the tests also suggest that our method is both cheaper and faster than the two level finite element galerkin scheme

a two grid method of a mixed stokes darcy model for coupling fluid flow with porous media flow we study numerical methods for solving a coupled stokes darcy problem in porous media flow applications a two grid method is proposed for decoupling the mixed model by a coarse grid approximation to the interface coupling conditions error estimates are derived for the proposed method both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two grid approach for solving multimodeling problems potential extensions and future directions are discussed

a streamline diffusion method for nonconforming finite element approximations applied to convection diffusion problems abstract we consider a nonconforming streamline diffusion finite element method for solving convection diffusion problems the theoretical and numerical investigation for triangular and tetrahedral meshes recently given by john maubach and tobiska has shown that the usual application of the sdfem gives not a sufficient stabilization additional parameter dependent jump terms have been proposed which preserve the same order of convergence as in the conforming case the error analysis has been essentially based on the existence of a conforming finite element subspace of the nonconforming space thus the analysis can be applied for example to the crouzeix raviart element but not to the nonconforming quadrilateral elements proposed by rannacher and turek in this paper parameter free new jump terms are developed which allow to handle both the triangular and the quadrilateral case numerical experiments support the theoretical predictions

adaptive streamline diffusion methods for compressible flow using conservation variables we consider the streamline diffusion finite element method applied to compressible flow using conservation variables we propose some adaptive algorithms and present related numerical results

adaptive defect correction methods for convection dominated convection diffusion problems abstract we present a posteriori error estimates for a defect correction method for approximating solutions of convection diffusion problems the algorithms and estimators include the possibility of using in the discretization a nonlinear selection mechanism which we find improves solution quality in and near layers energy norm and l2 a posteriori error estimates are proven for the full algorithm two examples of fully adaptive finite element defect correction calculations are presented these examples illustrate the scheme and demonstrate the effectiveness of the method

traffic accidents in a cellular automaton model with a speed limit zone in this paper we numerically study the probability pac of the occurrence of car accidents in the nagel schreckenberg ns model with a speed limit zone numerical results show that the probability for car accidents to occur pac is determined by the maximum speed v max of the speed limit zone but is independent of the length lv of the speed limit zone in the deterministic ns model however in the nondeterministic ns model the probability of the occurrence of car accidents pac is determined not only by the maximum speed v max but also the length lv the probability pac increases accordingly with the increase of the maximum speed of the speed limit zone but decreases with the increase of the length of the speed limit zone in the low density region however in the case of v max 1 the probability pac increases with the increase of the length in the low density region but decreases in the interval between the low density and high density regions the speed limit zone also causes an inhomogeneous distribution of car accidents over the whole road theoretical analyses give an agreement with numerical results in the nondeterministic ns model with v max 1 and vmax 5

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

an adaptive multigrid scheme for bose einstein condensates in a periodic potential we present a novel multigrid continuation method for treating parameter dependent problems the proposed algorithm which can be flexibly implemented is a generalization of the two grid discretization schemes c s chien b w jeng a two grid discretization scheme for semilinear elliptic eigenvalue problems siam j sci comput 27 2006 1287 1304 that is approximating points on a solution curve do not necessarily lie on the same fine grid we apply the algorithm to compute energy levels and superfluid densities of bose einstein condensates bec in a periodic potential both positive and negative scattering lengths are considered in our numerical experiments for positive scattering length if the chemical potential is large enough and the domain is properly chosen the results show that the number of peaks of the first few energy states of the 2d bec in a periodic potential depends on the wave number of the periodic potential moreover for bright solitons the number of peaks of the ground state solutions is 1d 1 2 and 1d 2 where the periodic potential is expressed in terms of the sine or the cosine functions respectively however these formulae do not hold if the scattering length is negative the numerical study is extended to the two component 1d and 2d bec in a periodic potential

long time behavior of the two grid finite element method for fully discrete semilinear evolution equations with positive memory abstract based on two grid discretizations two fully discrete finite element algorithms for semilinear parabolic integro differential equations with positive memory are proposed with the backward euler scheme for the temporal discretization the basic idea of the space two grid finite element algorithms is to approximate the semilinear equations on a coarse space grid and to solve the linearized equations on a finer space grid at each time step to further decreases the amount of computational work a space time two grid algorithm based on a coarse space grid with large time stepsize t and a finer space grid with small time stepsize t for the evolutional equations is proposed in this paper the sharp long time stability and error estimates for the standard finite element method the space two grid finite element method and the space time two grid finite element method are derived it is showed that the two grid algorithms long time stability and error estimates are similar to those of the direct resolution of the semilinear problem on a fine grid

adaptive finite element methods for systems of reaction diffusion equations we consider an adaptive finite element method with quantitative error control for systems of reaction diffusion equations the adaptive algorithm is based on an a posteriori error estimate involving residuals of the computed solution and certain strong stability factors which are estimated by solving an associated linearized dual problem numerically to take into account that the stability properties in space and time are different separate stability factors are used for the space discretization and the time discretization

an adaptive finite element method for linear elliptic problems on propose une methode adaptative d elements finis basee sur une estimation d erreur optimale de norme maxima pour les problemes elliptiques lineaires

automatically stable discontinuous petrov galerkin methods for stationary transport problems quasi optimal test space norm we investigate the application of the discontinuous petrov galerkin dpg finite element framework to stationary convection diffusion problems in particular we demonstrate how the quasi optimal test space norm improves the robustness of the dpg method with respect to vanishing diffusion we numerically compare coarse mesh accuracy of the approximation when using the quasi optimal norm the standard norm and the weighted norm our results show that the quasi optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions we address the problems associated to the resolution of the optimal test functions with respect to the quasi optimal norm by studying their convergence numerically in order to facilitate understanding of the method we also include a detailed explanation of the methodology from the algorithmic point of view

goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations