adaptive finite element methods for parabolic problems vi analytic semigroups we continue our work on adaptive finite element methods with a study of time discretization of analytic semigroups we prove optimal a priori and a posteriori error estimates for the discontinuous galerkin method showing in particular that analytic semigroups allow long time integration without error accumulation

discontinuous galerkin finite element methods for second order hyperbolic problems abstract we prove a priori and a posteriori error estimates for a finite element method for linear second order hyperbolic equations based on a space time finite element discretization with the basis functions being continuous in space and discontinuous in time

on the a posteriori error analysis for equations of prescribed mean curvature we present two approaches to the a posteriori error analysis for prescribed mean curvature equations the main difference between them concerns the estimation of the residual without or with computable weights in the second case the weights are related to the eigenvalues of the underlying operator and thus provide local and computable information about the conditioning we analyze the two approaches from a theoretical viewpoint moreover we investigate and compare the performance of the derived indicators in an adaptive procedure our theoretical and practical results show that it is advantageous to estimate the residual in a weighted way

maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions a semilinear second order parabolic equation is considered in a regular and a singularly perturbed regime for this equation we give computable a posteriori error estimates in the maximum norm semidiscrete and fully discrete versions of the backward euler crank nicolson and discontinuous galerkin dg r methods are addressed for their full discretizations we employ elliptic reconstructions that are respectively piecewise constant piecewise linear and piecewise quadratic for r 1 in time we also use certain bounds for the green s function of the parabolic operator

a posteriori error analysis for the mean curvature flow of graphs we study the equation describing the motion of a nonparametric surface according to its mean curvature flow this is a nonlinear nonuniformly parabolic pde that can be discretized in space via a finite element method we conduct an aposteriori error analysis of the spatial discretization and derive upper bounds on the error in terms of computable estimators based on local residual indicators the reliability of the estimators is illustrated with two numerical simulations one of which treats the case of a singular solution

optimal order a posteriori error estimates for a class of runge kutta and galerkin methods we derive a posteriori error estimates which exhibit optimal global order for a class of time stepping methods of any order that include runge kutta collocation rk c methods and the continuous galerkin cg method for linear and nonlinear stiff odes and parabolic pdes the key ingredients in deriving these bounds are appropriate one degree higher continuous reconstructions of the approximate solutions and pointwise error representations the reconstructions are based on rather general orthogonality properties and lead to upper and lower bounds for the error regardless of the time step they do not hinge on asymptotics

elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems we derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations the space discretization uses finite element spaces that are allowed to change in time our main tool is an appropriate adaptation of the elliptic reconstruction technique introduced by makridakis and nochetto we derive novel a posteriori estimates for the norms of l 0 t l2 and the higher order spaces l 0 t h1 and h1 0 t l2 with optimal orders of convergence

an adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems an efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator an adaptive algorithm with variable time step sizes and space meshes is proposed and studied which at each time step delays the mesh coarsening until the final iteration of the adaptive procedure allowing only mesh and time step size refinements before it is proved that at each time step the adaptive algorithm is able to reduce the error indicators and thus the error below any given tolerance within a finite number of iteration steps the key ingredient in the analysis is a new coarsening strategy numerical results are presented to show the competitive behavior of the proposed adaptive algorithm

a space time finite element method for the nonlinear schro dinger equation the discontinuous galerkin method the convergence of the discontinuous galerkin method for the nonlinear cubic schrodinger equation is analyzed in this paper we show the existence of the resulting approximations and prove optimal order error estimates in l l 2 these estimates are valid under weak restrictions on the space time mesh

a posteriori error estimates for the crank nicolson method for parabolic equations we derive optimal order a posteriori error estimates for time discretizations by both the crank nicolson and the crank nicolson galerkin methods for linear and nonlinear parabolic equations we examine both smooth and rough initial data our basic tool for deriving a posteriori estimates are second order crank nicolson reconstructions of the piecewise linear approximate solutions these functions satisfy two fundamental properties i they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori and ii they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation the resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem as a consequence we provide alternative proofs for known a priori rates of convergence for the crank nicolson method

a posteriori error analysis for higher order dissipative methods for evolution problems we prove a posteriori error estimates for time discretizations by the discontinuous galerkin method dg q and the corresponding implicit runge kutta radau method irk r q of arbitrary order q 0 for both linear and nonlinear evolution problems of the form u prime mathfrak f u f with 2 angle bounded operator mathfrak f the key ingredient is a novel higher order reconstruction widehat u of the discrete solution u which restores continuity and leads to the differential equation widehat u prime pi mathfrak f u f for a suitable interpolation operator and piecewise polynomial approximation f of f we discuss applications to linear pde such as the convection diffusion equation 1 2 and the wave equation formally and nonlinear pde corresponding to subgradient operators 1 such as the p laplacian as well as lipschitz operators 1 2 we also derive conditional a posteriori error estimates for the time dependent minimal surface problem

an a posteriori error estimate and adaptive timestep control for a backward euler discretization of a parabolic problem a simple algorithm for adaptive timestep control is presented for a backward euler discretization of a linear parabolic problem the algorithm is based on an a posteriori error estimate involving the computed approximate solution it is proved that with only very rough a priori information on the exact solution the algorithm will choose a sequence of timesteps for which the error will be controlled up to a constant uniformly in time on a given tolerance level

asymptotically periodic solutions of abstract differential equations abstract in this paper we introduce the concept of s asymptotically periodicity in the stepanov sense we study some properties of the space consisting of s asymptotically periodic functions and we use this notion to establish the existence of s asymptotically periodic solutions to linear and semi linear first order abstract differential equations in particular we characterize the strongly continuous semigroups of bounded linear operators defined on reflexive spaces which are s asymptotically periodic as those semigroups which are the direct sum of an periodic and a strongly stable semigroup

optimized schwarz waveform relaxation and discontinuous galerkin time stepping for heterogeneous problems we design and analyze a schwarz waveform relaxation algorithm for domain decomposition of advection diffusion reaction problems with strong heterogeneities the interfaces are curved and we use optimized ventcell transmission conditions we analyze the semidiscretization in time with discontinuous galerkin as well we also show two dimensional numerical results using generalized mortar finite elements in space

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

a posteriori error estimates for discontinuous galerkin methods for the generalized korteweg de vries equation we construct analyze and numerically validate a posteriori error estimates for conservative discontinuous galerkin dg schemes for the generalized korteweg de vries gkdv equation we develop the concept of dispersive reconstruction i e a piecewise polynomial function which satisfies the gkdv equation in the strong sense but with a computable forcing term enabling the use of a priori error estimation techniques to obtain computable upper bounds for the error both semidiscrete and fully discrete approximations are treated

the dynamical behavior of the discontinuous galerkin method and related difference schemes we study the dynamical behavior of the discontinuous galerkin finite element method for initial value problems in ordinary differential equations we make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system we show that under certain conditions the discontinuous galerkin approximation also defines a dissipative dynamical system and we study the approximation properties of the associated discrete dynamical system we also study the behavior of difference schemes obtained by applying a quadrature formula to the integrals defining the discontinuous galerkin approximation and construct two kinds of discrete finite element approximations that share the dissipativity properties of the original method

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

analysis for time discrete approximations of blow up solutions of semilinear parabolic equations we prove a posteriori error estimates for time discrete approximations for semilinear parabolic equations with solutions that might blow up in finite time in particular we consider the backward euler and the crank nicolson methods the main tools that are used in the analysis are the reconstruction technique and energy methods combined with appropriate fixed point arguments the final estimates we derive are conditional and lead to error control near the blow up time

error analysis for implicit approximations to solutions to cauchy problems exact error analyses for two implicit approximation schemes to the cauchy problem du t dt mathcal a u t mathrel backepsilon 0 u 0 u 0 in d mathcal a are presented the operator mathcal a is required to be maximal monotone on a hilbert space mathcal h the approximations considered are u h kh i h mathcal a k u 0 and solutions u lambda to the approximating problems du lambda t dt mathcal a lambda u lambda 0 u lambda 0 u 0 where a lambda is the yosida approximation of mathcal a the special case of a being a subgradient is considered and tight estimates of the convergence of both u h and mathcal a h u h or u h and a h u h are given the new results in this case are the linear rates of convergence of u h in l infty 0 infty mathcal h and in the energy norm and the sublinear rate of convergence of a h u h in l 2 0 infty mathcal h applications to both linear and

bounds of functional outputs for parabolic problems part i exact bounds of the discontinuous galerkin time discretization classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest consequently the bounds obtained are only guaranteed asymptotically that is with respect to the reference solution computed with the fine mesh exact bounds that is bounds guaranteed with respect to the exact solution are needed to properly certify the accuracy of the results especially if the meshes are coarse the paper introduces a procedure to compute strict upper and lower bounds of the error in linear functional outputs of parabolic problems in this first part the bounds account for the error associated with the spatial discretization the error coming from the time marching scheme is therefore assumed to be negligible in front of the spatial error the time discretization is performed using the discontinuous galerkin method both for the primal and adjoint problems in the error estimation procedure equilibrated fluxes at interelement edges are calculated using hybridization techniques

space time domain decomposition methods for diffusion problems in mixed formulations this paper is concerned with global in time nonoverlapping domain decomposition methods for the mixed formulation of the diffusion problem two approaches are considered one uses the time dependent steklov poincare operator and the other uses optimized schwarz waveform relaxation oswr based on robin transmission conditions for each method a mixed formulation of an interface problem on the space time interfaces between subdomains is derived and different time grids are employed to adapt to different time scales in the subdomains demonstrations of the well posedness of the robin subdomain problems involved in the oswr method and a convergence proof of the oswr algorithm are given for the mixed formulation numerical results for two dimensional problems with strong heterogeneities are presented to illustrate the performance of the two methods

hp adaptive discontinuous galerkin finite element methods for first order hyperbolic problems we consider the a posteriori error analysis of hp discontinuous galerkin finite element approximations to first order hyperbolic problems in particular we discuss the question of error estimation for linear functionals such as the outflow flux and the local average of the solution based on our a posteriori error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance this involves exploiting both local polynomial degree variation and local mesh subdivision the theoretical results are illustrated by a series of numerical experiments

a posteriori analysis of the finite element discretization of some parabolic equations nous considerons la discretisation d equations paraboliques soit lineaires soit semi lineaires par un schema d euler implicite en temps et par elements finis en espace l idee de cet article est de construire des indicateurs d erreur lies a l approximation en temps et en espace et de prouver leur equivalence avec l erreur dans le but de travailler avec des pas de temps adaptatifs et des maillages d elements finis adaptes a la solution

adaptivity and blow up detection for nonlinear evolution problems this work is concerned with the development of a space time adaptive numerical method based on a rigorous a posteriori error bound for a semilinear convection diffusion problem which may exhibit blow up in finite time more specifically a posteriori error bounds are derived in the l infty l 2 l 2 h 1 type norm for a first order in time implicit explicit interior penalty discontinuous galerkin in space discretization of the problem although the theory presented is directly applicable to the case of conforming finite element approximations in space the choice of the discretization in time is made based on a careful analysis of adaptive time stepping methods for odes that exhibit finite time blow up the new adaptive algorithm is shown to accurately estimate the blow up time of a number of problems including one which exhibits regional blow up

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

adaptive finite element methods for parabolic problems iv nonlinear problems we extend our program on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems we prove a posteriori error estimates design corresponding adaptive algorithms and present some numerical results

the essential stability of local error control for dynamical systems although most adaptive software for initial value problems is designed with an accuracy requirement control of the local error it is frequently observed that stability is imparted by the adaptation this relationship between local error control and numerical stability is given a firm theoretical underpinning the dynamics of numerical methods with local error control are studied for three classes of ordinary differential equations dissipative contractive and gradient systems dissipative dynamical systems are characterised by having a bounded absorbing set mathcal b which all trajectories eventually enter and remain inside the exponentially contractive problems studied have a unique globally exponentially attracting equilibrium point and thus they are also dissipative since the absorbing set mathcal b may be chosen to be a ball of arbitrarily small radius around the equilibrium point the gradient systems studied are those for which the set of equilibria comprises isolated points and all traje

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 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 posteriori analysis of discontinuous galerkin schemes for systems of hyperbolic conservation laws in this work we construct reliable a posteriori estimates for some semi spatially discrete discontinuous galerkin schemes applied to nonlinear systems of hyperbolic conservation laws we make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework which leads to error control in the case of smooth solutions the methodology we use is quite general and allows for a posteriori control of discontinuous galerkin schemes with standard flux choices which appear in the approximation of conservation laws in addition to the analysis we conduct some numerical benchmarking to test the robustness of the resultant estimator

stiffly accurate runge kutta methods for nonlinear evolution problems governed by a monotone operator stiy accurate implicit runge kutta methods are studied for the time discretisation of nonlinear rst order evolution equations the equation is supposed to be governed by a time dependent hemicontinuous operator that is up to a shift monotone and coercive and fullls a certain growth condi tion it is proven that the piecewise constant as well as the piecewise linear interpolant of the time discrete solution converge towards the exact weak solu tion provided the runge kutta method is consistent and satises a stability criterion that implies algebraic stability examples are the radau iia and lo batto iiic methods the convergence analysis is also extended to problems involving a strongly continuous perturbation of the monotone main part

a posteriori error estimates and maximal regularity for approximations of fully nonlinear parabolic problems in banach spaces a posteriori error estimates are provided for discretizations in time of abstract nonlinear parabolic problems u f u by the backward euler method in the maximal regularity framework of banach spaces the estimates are of conditional type i e are valid under assumptions on the approximate solution and the proofs are based on appropriate fixed point arguments

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

a posteriori error estimates for leap frog and cosine methods for second order evolution problems we consider second order explicit and implicit two step time discrete schemes for wave type equations we derive optimal order a posteriori estimates controlling the time discretization error our analysis has been motivated by the need to provide a posteriori estimates for the popular leap frog method also known as verlet s method in the molecular dynamics literature it is extended however to general cosine type second order methods the estimators are based on a novel reconstruction of the time dependent component of the approximation numerical experiments confirm similarity of the convergence rates of the proposed estimators and the theoretical convergence rate of the true error

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

a posteriori error estimates for nonlinear problems l r o t p ω error estimates for finite element discretizations of parabolic equations using the abstract framework of 9 we analyze a residual a posteriori error estimator for space time finite element discretizations of quasilinear parabolic pdes the estimator gives global upper and local lower bounds on the error of the numerical solution the finite element discretizations in particular cover the so called 0 scheme which includes the implicit and explicit euler methods and the crank nicholson scheme

silting reduction and calabi yau reduction of triangulated categories it is shown that the silting reduction t thickp of a triangulated category t with respect to a presilting subcategory p can be realised as a certain subfactor category of t and that there is a one to one correspondence between the set of pre silting subcategories of t containing p and the set of pre silting subcategories of t thickp this is analogous to a result for calabi yau reduction this result is applied to show that the amiot guo keller construction of d calabi yau triangulated categories with d cluster tilting objects takes silting reduction to calabi yau reduction and conversely calabi yau reduction lifts to silting reduction

thirteen ways to estimate global error various techniques that have been proposed for estimating the accumulated discretization error in the numerical solution of differential equations particularly ordinary differential equations are classified described and compared for most of the schemes either an outline of an error analysis is given which explains why the scheme works or a weakness of the scheme is illustrated

galerkin and runge kutta methods unified formulation a posteriori error estimates and nodal superconvergence we unify the formulation and analysis of galerkin and runge kutta methods for the time discretization of parabolic equations this together with the concept of reconstruction of the approximate solutions allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods

a posteriori error estimation and adaptivity for degenerate parabolic problems two explicit error representation formulas are derived for degenerate parabolic pdes which are based on evaluating a parabolic residual in negative norms the resulting upper bounds are valid for any numerical method and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient they are applied to a practical space time discretization consisting of c 0 piecewise linear finite elements over highly graded unstructured meshes and backward finite differences with varying time steps two rigorous a posteriori error estimates are derived for this scheme and used in designing an efficient adaptive algorithm which equidistributes space and time discretization errors via refinement coarsening a simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach and hints at the potentials and reliability of the proposed method