error estimation and adaptation for functional outputs in time dependent flow problems the paper presents an adjoint based approach for determining global error in the time domain that is relevant to functional outputs from unsteady flow simulations the algorithm is derived for the unsteady euler equations that are discretized for second order accuracy in both space and time and takes into account the effect of dynamic meshes in addition to error due to temporal resolution the formulation also takes into account algebraic error arising from partial convergence of the governing equations at each implicit time step the resulting error distributions are then used to drive adaptation of the temporal resolution and the convergence tolerances for the governing equations at each time step the method is demonstrated in the context of both time integrated and instantaneous functionals and the results are compared against traditional adaptation methods

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

adaptive weak approximation of diffusions with jumps this work develops adaptive time stepping algorithms for the approximation of a functional of a diffusion with jumps based on a jump augmented monte carlo euler maruyama method which achieve a prescribed precision the main result is the derivation of new expansions for the time discretization error with computable leading order term in a posteriori form which are based on stochastic flows and discrete dual backward functions combined with proper estimation of the statistical error they lead to efficient and accurate computation of global error estimates extending the results by a szepessy r tempone and g e zouraris comm pure appl math 54 2001 pp 1169 1214 adaptive algorithms for either deterministic or trajectory dependent time stepping are proposed numerical examples show the performance of the proposed error approximations and the adaptive schemes

asymptotically exact a posteriori ldg error estimates for one dimensional transient convection diffusion problems in this paper new a posteriori error estimates for the local discontinuous galerkin ldg formulation applied to transient convection diffusion problems in one space dimension are presented and analyzed these error estimates are computationally simple and are computed by solving a local steady problem with no boundary conditions on each element we first show that the leading error term on each element for the solution is proportional to a p 1 degree right radau polynomial while the leading error term for the solution s derivative is proportional to a p 1 degree left radau polynomial when polynomials of degree at most p are used these results are used to prove that for smooth solutions these error estimates at a fixed time converge to the true spatial errors in the l 2 norm under mesh refinement more precisely we prove that our ldg error estimates converge to the true spatial errors at o h p 5 4 rate finally we prove that the global effectivity indices in the l 2 norm converge to unity at o h 1 2 rate our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates

blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions we describe and test an adaptive algorithm for evolution problems that employs a sequence of blocks consisting of fixed though nonuniform space meshes this approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing remeshing matrix reassembly and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes a major issue with a block adaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy we describe several strategies for achieving this goal using adjoint based a posteriori error estimates and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples

an adaptive discontinuous galerkin method for very stiff systems of ordinary differential equations abstract we present a discontinuous galerkin dg method for the numerical solution of stiff systems of ordinary differential equations odes we use a standard dg variational formulation with polynomials of degree k in each time interval we show that the method is a stable for every k we then introduce a hierarchical legendre finite element basis and we show that a whole family of approximations can be obtained simply by truncating the last p degrees of freedom from the computed solution we show that these approximations converge to order k 1 p in l2 norm and to order k 1 2 p in supremum norm we then show how this can be used to control the error and the time step length we present numerical examples of solutions on very stiff problems and on stiff problems with very long time integration where the time step length can vary on many orders of magnitude

a posteriori analysis of a multirate numerical method for ordinary differential equations in this paper we analyze a multirate time integration method for systems of ordinary differential equations that present significantly different scales within the components of the model the main purpose of this paper is to present a hybrid a priori a posteriori error analysis that accounts for the effects of projections between the coarse and fine scale discretizations the use of only a finite number of iterations in the iterative solution of the discrete equations the numerical error arising in the solution of each component and the effects on stability arising from the multirate solution the hybrid estimate has the form of a computable a posteriori leading order expression and a provably higher order a priori expression we support this estimate by an a priori convergence analysis we present several examples illustrating the accuracy of multirate integration schemes and the accuracy of the a posteriori estimate

a posteriori error bounds and global error control for approximation of ordinary differential equations the author analyzes a finite element method for the integration of initial value problems in ordinary differential equations general and contractive problems are treated and quasi optimal a priori and a posteriori error bounds obtained in each case in particular good results are obtained for a class of stiff dissipative problems these results are used to construct a rigorous and robust theory of global error control the author also derives an asymptotic error estimate that is used in a discussion of the behavior of the error in conclusion the properties of the error control are exhibited in a series of numerical experiments

continuous and discontinuous galerkin time stepping methods for nonlinear initial value problems with application to finite time blow up we consider continuous and discontinuous galerkin time stepping methods of arbitrary order as applied to first order initial value ordinary differential equation problems in real hilbert spaces our only assumption is that the nonlinearities are continuous in particular we include the case of unbounded nonlinear operators specifically we develop new techniques to prove general peano type existence results for discrete solutions in particular our results show that the existence of solutions is independent of the local approximation order and only requires the local time steps to be sufficiently small independent of the polynomial degree the uniqueness of local solutions is addressed as well in addition our theory is applied to finite time blow up problems with nonlinearities of algebraic growth for such problems we develop a time step selection algorithm for the purpose of numerically computing the blow up time and provide a convergence result

approximation of weak adjoints by reverse automatic differentiation of bdf methods we shed light on the relation between the discrete adjoints of multistep backward differentiation formula bdf methods and the solution of the adjoint differential equation to this end we develop a functional analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions of ordinary differential equations we devise a petrov galerkin finite element fe interpretation of the bdf method and its discrete adjoint scheme obtained by reverse internal numerical differentiation we show how the fe approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its convergence in the space of normalized functions of bounded variation we also show convergence of the discrete adjoints to the classical adjoints on the inner time interval finally we give numerical results for non adaptive and fully adaptive bdf schemes the presented framework opens the way to carry over techniques on global error estimation from fe methods to bdf methods

reconsideration on the elastic damage degradation theory for the modeling of microcrack closure reopening mcr effects abstract despite the substantial and noteworthy contributions the modeling of damage induced anisotropy remains an unsolved issue especially when the microcrack closure reopening mcr effects are accounted for theoretically speaking the most challenging problem might be the lack of energy conservation for all the existing models employing the spectral decomposition of the stress or strain tensor and the associated positive negative projection operators carol and willam 1996 in this paper this crucial problem is solved by introducing a new definition of thermodynamically consistent projection operators into the classical elastic damage degradation model the orthogonality between the rates of the new projection operators and the stress or strain automatically guarantees the fulfillment of energy conservation under any arbitrary proportional or non proportional load history moreover with this extra orthogonal property the thermodynamically consistent projection operators can be exclusively determined in a unique form the above advantages lend to their promising use in the modeling of damage induced anisotropy and mcr effects simultaneously with the aid of the thermodynamically consistent projection operators the existing elastic damage degradation models can be reformulated so that the energy conservation is restored only with minor modifications as a prototype example a stress based elastic damage degradation model with multiple loading surfaces is developed analogously to the classical multisurface plasticity model finally a simple closed loop load path accompanied with rotation of principal stress directions is constructed to verify its thermodynamically consistency

an embedded formula of the chebyshev collocation method for stiff problems abstract in this study we have developed an embedded formula of the chebyshev collocation method for stiff problems based on the zeros of the generalized chebyshev polynomials a new strategy for the embedded formula using a pair of methods to estimate the local truncation error as performed in traditional embedded runge kutta schemes is proposed the method is performed in such a way that not only the stability region of the embedded formula can be widened but by allowing the usage of larger time step sizes the total computational costs can also be reduced in terms of concrete convergence and stability analysis the constructed algorithm turns out to have an 8th order convergence and it exhibits a stability through several numerical experimental results we have demonstrated that the proposed method is numerically more efficient compared to several existing implicit methods

dynamic time step estimates for two dimensional transient field problems using square elements this study presents an experimental approach to estimate time steps that integrate the two dimensional field equation over a square element within 5 per cent accuracy from the exact solution the time step estimates were determined for three finite element and three finite difference schemes comparisons between finite element and finite difference methods and the various time stepping schemes were conducted the dynamic time step estimates are functions of grid size and the smallest eigenvalue of the system of odes 1 the results indicate that the finite element and finite difference methods generate similar time step estimates and are at similar accuracy levels the central difference scheme is superior to the other two schemes as far as the flexibility in allowing larger time step while maintaining the accuracy the backward difference and the forward difference schemes were very close in their level of accuracy 1998 john wiley sons ltd

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

efficient goal oriented global error estimators for bdf methods using discrete adjoints in this paper we develop new goal oriented global error estimators for variable multistep backward differentiation formulae bdf methods these estimators use for the first time in the context of multistep methods discrete adjoints computed by adjoint differentiation of the nominal integration scheme the derivation is based on the recently developed petrov galerkin finite element formulation of bdf methods and their discrete adjoint schemes and makes use of the dual weighted residual methodology defect integrals or local truncation errors are used as local error quantities we prove the asymptotic correctness and optimal convergence of the novel estimators for the one step bdf method for multistep bdf methods using a selfstarting procedure we show that the estimator with defect integrals converges but suboptimal and that its effectivity index converges to an offset value on the other hand the estimator with local truncation errors is again asymptotically correct and of optimal order we confirm these results numerically with a real world example from chemical engineering we give promising numerical results for the estimation accuracy in variable bdf type methods with changing orders and stepsizes finally we give a first use of the novel estimators for goal oriented global error control of a challenging stiff test problem

a subdomain residual method for a posteriori error estimates of finite element solutions of viscoelastic fluid flows abstract in this paper we present a subdomain residual method which gives a posteriori estimates of the errors in finite element solutions of viscoelastic fluid flows we employ a coupled approach in which the estimated errors in both velocity and extra stress are computed simultaneously by solving a series of local problems the subdomains on which the local problems are posed consist of patches of elements a given parameter which is independent of the element size is employed to determine the size of the subdomains to ensure that the error estimate improves as the element size is reduced the error estimator is applied to finite element solutions of newtonian and non newtonian fluid flows past a cylindrical tube in a duct the contour plots of the components of the estimated errors in velocity and extra stress are presented these errors are compared with reference errors

an error embedded method based on generalized chebyshev polynomials in this paper we develop an error embedded method based on generalized chebyshev polynomials for solving stiff initial value problems the solution and the error at each integration step are calculated by generalized chebyshev polynomials of two consecutive degrees having overlapping zeros which enables us to minimize overall computational costs further the errors at each integration step are embedded in the algorithm itself in terms of concrete convergence and stability analysis the constructed algorithm turns out to have the 6th order convergence and an almost l stability we assess the proposed method with several numerical results showing that it uses larger time step sizes and is numerically more efficient

time discretization of parabolic problems by the hp version of the discontinuous galerkin finite element method the discontinuous galerkin finite element method dgfem for the time discretization of parabolic problems is analyzed in the context of the hp version of the galerkin method error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp dgfem gives spectral convergence in problems with smooth time dependence in conjunction with geometric time partitions it is proved that the hp dgfem results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms for the h version dgfem algebraically graded time partitions are determined that give the optimal algebraic convergence rates a fully discrete hp scheme is discussed exemplarily for the heat equation the use of certain mesh design principles for the spatial discretizations yields exponential rates of convergence in time and space numerical examples confirm the theoretical results

discontinuous galerkin finite element method for parabolic problems in this paper we develop a time and its corresponding spatial discretization scheme based upon the assumption of a certain weak singularity of u t t l 2 w email protected u email protected 2 for the discontinuous galerkin finite element method for one dimensional parabolic problems optimal convergence rates in both time and spatial variables are obtained a discussion of automatic time step control method is also included

adaptive weak approximation of stochastic differential equations adaptive time stepping methods based on the monte carlo euler method for weak approximation of ito stochastic differential equations are developed the main result is new expansions of the computational error with computable leading order term in a posteriori form based on stochastic flows and discrete dual backward problems the expansions lead to efficient and accurate computation of error estimates adaptive algorithms for either stochastic time steps or deterministic time steps are described numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps stochastic time steps use brownian bridges and require more work for a given number of time steps deterministic time steps may yield more time steps but require less work for example in the limit of vanishing error tolerance the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps c 2001 john wiley sons inc

elliptic reconstruction and a posteriori error estimates for parabolic problems it is known that the energy technique for a posteriori error analysis of finite element discretizations of parabolic problems yields suboptimal rates in the norm l infty 0 t l 2 omega in this paper we combine energy techniques with an appropriate pointwise representation of the error based on an elliptic reconstruction operator which restores the optimal order and regularity for piecewise polynomials of degree higher than one this technique may be regarded as the dual a posteriori counterpart of wheeler s elliptic projection method in the a priori error analysis

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

an hp version discontinuous galerkin method for integro differential equations of parabolic type we study the numerical solution of a class of parabolic integro differential equations with weakly singular kernels we use an hp version discontinuous galerkin dg method for the discretization in time we derive optimal hp version error estimates and show that exponential rates of convergence can be achieved for solutions with singular temporal behavior near t 0 caused by the weakly singular kernel moreover we prove that by using nonuniformly refined time steps optimal algebraic convergence rates can be achieved for the h version dg method we then combine the dg time stepping method with a standard finite element discretization in space and present an optimal error analysis of the resulting fully discrete scheme our theoretical results are numerically validated in a series of test problems

a universally acceptable smoothing factor for kernel density estimates we define a minimum distance estimate of the smoothing factor for kernel density estimates based on a methodology first developed by yatracos it is shown that if f nh denotes the kernel density estimate on r d for an i i d sample of size n drawn from an unknown density f where h is the smoothing factor and if f n is the kernel estimate with the same kernel and with the proposed new data based smoothing factor then under a regularity condition on the kernel k formula math this is the first published smoothing factor that can be proven to have this property

a posteriori error estimates for variable time step discretizations of nonlinear evolution equations we study the backward euler method with variable time steps for abstract evolution equations in hilbert spaces exploiting convexity of the underlying potential or the angle bounded condition thereby assuming no further regularity we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual these estimators depend solely on the discrete solution and data and impose no constraints between consecutive time steps we also prove that they converge to zero with an optimal rate with respect to the regularity of the solution we apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character 2000 john wiley sons inc

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

a posteriori error estimation for hp version time stepping methods for parabolic partial differential equations the aim of this paper is to develop an hp version a posteriori error analysis for the time discretization of parabolic problems by the continuous galerkin cg and the discontinuous galerkin dg time stepping methods respectively the resulting error estimators are fully explicit with respect to the local time steps and approximation orders their performance within an hp adaptive refinement procedure is illustrated with a series of numerical experiments

the concept of b convergence a useful concept of convergence for nonlinear stiff ode s will be developed which permits the derivation of uniform global error bounds independent of the stiffness of the considered problem this concept will be discussed for three simple methods for stiff systems the implicit euler scheme the implicit midpoint rule and the implicit trapezoidal rule

a posteriori error estimation of steady state finite element solutions of the navier stokes equations by a subdomain residual method abstract we present a subdomain residual method as well as its mathematical basis for estimating errors in steady state finite element solutions of the incompressible navier stokes equations the estimated errors are obtained by solving a series of local problems in which velocity boundary condition is used wherever the exact traction boundary condition is not available an iterative procedure similar to the newton method is employed to improve the error estimates the performance of the method is demonstrated in two numerical examples i e the channel flow over a backward facing step and past a circular cylinder at low reynolds numbers

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

multiadaptive galerkin methods for odes iii a priori error estimates the multiadaptive continuous discontinuous galerkin methods mcgq and mdgq for the numerical solution of initial value problems for ordinary differential equations are based on piecewise polynomial approximation of degree q on partitions in time with time steps which may vary for different components of the computed solution in this paper we prove general order a priori error estimates for the mcgq and mdgq methods to prove the error estimates we represent the error in terms of a discrete dual solution and the residual of an interpolant of the exact solution the estimates then follow from interpolation estimates together with stability estimates for the discrete dual solution

a high order discontinuous galerkin method for ito stochastic ordinary differential equations in this paper we develop a high order discontinuous galerkin dg method for strong solution of ito stochastic ordinary differential equations sdes driven by one dimensional wiener processes motivated by the dg method for deterministic ordinary differential equations odes we first construct an approximate deterministic ode with a random coefficient on each element using the well known wong zakai approximation theorem since the resulting ode converges to the solution of the corresponding stratonovich sde we apply a transformation to the drift term to obtain a deterministic ode which converges to the solution of the original sde the corrected equation is then discretized using the standard dg method for deterministic odes we prove that the proposed stochastic dg sdg method is equivalent to an implicit stochastic runge kutta method then we study the numerical stability of the sdg scheme applied to linear sdes with an additive noise term the method is shown to be numerically stable in the mean sense and also a stable as a result it is suitable for solving stiff sdes moreover the method is proved to be convergent in the mean square sense numerical evidence demonstrates that our proposed dg scheme for sdes with additive noise has a strong convergence order of 2 p 1 when p degree piecewise polynomials are used when applied to sdes with multiplicative noise the sdg method is strongly convergent with order p several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method in particular we demonstrate that our proposed scheme is suitable for stiff stochastic differential systems