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

sharp l 1 a posteriori error analysis for nonlinear convection diffusion problems we derive sharp l l 1 a posteriori error estimates for initial boundary value problems of nonlinear convection diffusion equations of the form u t div f u a u g under the nondegeneracy assumption a s 0 for any s r the problem displays both parabolic and hyperbolic behavior in a way that depends on the solution itself it is discretized implicitly in time via the method of characteristic and in space via continuous piecewise linear finite elements the analysis is based on the kruzkov doubling of variables device and the recently introduced boundary layer sequence technique to derive the entropy error inequality on bounded domains the derived a posteriori error estimators have the correct convergence order in the region where the solution is smooth and recover the standard a posteriori error estimators known for parabolic equations with strong diffusions

quantitative error control for finite element methods for one dimensional compressible flow we prove a posteriori l 2 l 2 and l infty h 1 residual based error estimates the estimates contain certain strong stability factors or weights related to the solution of a linearized dual problem associated with the euler equations we compute the stability factors and weights by solving the dual problem numerically and show that quantitative error control is possible for one dimensional 1 d compressible flow

a posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions in this paper we shall derive a posteriori error estimates in the l 1 norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions this result is mainly based on some fundamental a priori error estimates published in a recent paper by c chainais hillairet the theoretical results are confirmed by numerical experiments

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

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

stability of perturbed nonlinear parabolic equations with sturmian property abstract we study stability of an equilibrium f of autonomous dynamical systems under asymptotically small perturbations of the equation we show that such stability takes place if the domain of attraction of the equilibrium f contains a one parametric ordered family f in the stability analysis we need a special s relation a kind of restricted partial ordering to be preserved relative to the family f this s relation is inherited from the sturmian zero set properties for linear parabolic equations as main applications we prove stability of the self similar blow up behaviour for the porous medium equation the p laplacian equation and the dual porous medium equation in r with nonlinear lower order perturbations for such one dimensional parabolic equations the s relation is sturm s theorem on the nonincrease of the number of intersections between the solutions and particular solutions with initial data in f this sturmian property plays a key role and is true for the unperturbed pme but is not true for perturbed equations

an adaptive finite element method for unsteady convection dominated flows with stiff source terms we consider nonstationary convection dominated flows with stiff source terms as a unified approach to such problems a combined finite element method in space and time with streamline diffusion is examined numerically it has good shock capturing features and is implicit and l stable moreover being a galerkin method it admits residual based weighted a posteriori error estimates of optimal order we avoid the use of global stability constants by actually solving the dual problem this in turn leads to efficient and mathematically rigorous mesh refinement strategies where the streamline diffusion parameter is used for optimizing the resulting adaptive scheme all theoretical results are substantiated by numerical test examples for a physical application we turn to detonation waves at moderate relaxation times in one space dimension the finite element method well reproduces the chapman jouguet speed of detonations and their wave structure our error estimator proves accurate and sensitive to parameters at least on time intervals of moderate length the adaptive scheme based on this estimator produces fairly accurate results which appear to be the best available with the present strategy all ingredients of the numerical scheme can be extended in a natural way to higher space dimensions

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

hamilton jacobi equations on a manifold and applications to grid generation or refinement in this paper we consider hamilton jacobi equations on a manifold typically on the graph of some previously computed function z x y and we show how the corresponding level set method allows us to generate and or to refine a mesh in regions where this function z has large derivatives such as it is the method needs to be strongly improved and accelerated but the principle is awfully natural and in principle the method is fully automatic similar ideas are also useful in image processing in particular for the active contours method

continuation of blowup solutions of nonlinear heat equations in several space dimensions the possible continuation of solutions of the nonlinear heat equation in r n r ut u m u p with m 0 p 1 after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p thus for m p 2 we find a phenomenon of nontrivial continuation where the regionfx u x t 1g is bounded and propagates with finite speed this we call incomplete blowup for n 3 and p m n 2 n 2 we find solutions that blow up at finite t t and then become bounded again for t t otherwise we find that blowup is complete for a wide class of initial data in the analysis of the behavior for large p a list of critical exponents appears whose role is described we also discuss a number of related problems and equations we apply the same technique of analysis to the problem of continuation after the onset of extinction for example for the equation ut u m u p m 0

remarks on the stability of shock profiles for conservation laws with dissipation two remarks are made the first is to establish the stability of monotone shock profiles of the kdv burgers equation based on an energy method of goodman the second remark illustrates specifically in burgers equation that uniform rates of decay are not to be expected for perturbations of shock profiles in typical norms introduction we will prove two results the first is to establish the orbital stability of monotone shock profile solutions 4 x st of the kdv burgers equation 1 1 ut uux uxx uxxx 0 0 the second is to show that for shock profile solutions of viscous conservation laws and specifically for burgers equation 1 2 ut uux uxx one cannot expect perturbations to decay at any uniform rate algebraic or otherwise in ordinary norms such as lp the first result is related to goodman s conditional stability result in 5 for small amplitude shock profiles x st in systems of genuinely nonlinear conservation laws with viscosity u f u x buxx u e rm the second is intended to be relevant to the question of full stability for shock profiles of such systems the nonlinear stability to small perturbations of constant state solutions u0 x c of the system u f u x b u ux x where b c is a strictly stable viscosity matrix see 9 can be demonstrated based on a uniform rate of decay of solutions to the linearized equation the method used is similar to that developed in 8 and 10 for systems in n 1 space dimensions faster decay rates in higher space dimensions make the method easier to apply when n is large but it does just work for conservation laws with stable viscosities in one space dimension see 13 thus for example assuming c 0 if iiuo iih6 ll is sufficiently small then iiu t llh4 ct 1 4iiuoii h4 n la in this note we want to point out that such uniform decay rates cannot be valid for perturbations of shock profiles in particular for the shock profile solution of burgers equations we construct a family of perturbations which decay to zero but do so at no uniform rate in most ordinary norms including all lp norms thus for received by the editors october 15 1984 and in revised form december 19 1984 1980 mathematics subject classification primary 35q20 35l65 1985 american mathematical society 0002 9947 85 1 00 25 per page

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

stability and convergence of a class of finite element schemes for hyperbolic systems of conservation laws we propose a class of finite element schemes for systems of hyperbolic conservation laws that are based on finite element discretizations of appropriate relaxation models we consider both semidiscrete and fully discrete finite element schemes and show that the schemes are stable and when the compensated compactness theory is applicable do converge to a weak solution of the hyperbolic system the schemes use piecewise polynomials of arbitrary degree and their consistency error is of high order we also prove that the rate of convergence of the relaxation system to a smooth solution of the conservation laws is of order o eps

asymptotic behaviour near finite time extinction for the fast diffusion equation we study the cauchy problem for the fast diffusion equation u t delta u m u gee 0 qquad hbox in bf r n times bf r when n gee 3 and 0 m n 2 over n for a class of nonnegative radially symmetric finite mass solutions which vanish identically at a given time t 0 we show that their asymptotic behaviour as t nearrow t is described by a uniquely determined self similar solution of the second kind u r t t t alpha f eta where eta r t t beta and alpha 1 2 beta over 1 m and r x here beta is determined from a nonlinear eigenvalue problem involving an ordinary differential equation for the function f special attention is paid to the case when m n 2 over n 2 then beta 0 and the function f can be found explicity the proof is based on a geometric lyapunov type argument and comparison arguments based on the intersection properties of the solution graphs

goal oriented adaptive finite element methods for elliptic problems revisited a goal oriented a posteriori error estimation of an output functional for elliptic problems is presented continuous finite element approximations are used in quadrilateral and triangular meshes the algorithm is similar to the classical dual weighted error estimation however the dual weight contains solutions of the proposed patch problems the patch problems are introduced to apply clement and scott zhang type interpolation operators to estimate point values with the finite element polynomials the algorithm is shown to be reliable efficient and convergent

error control for a class of runge kutta discontinuous galerkin methods for nonlinear conservation laws we propose an a posteriori error estimate for the runge kutta discontinuous galerkin method rk dg of arbitrary order in arbitrary space dimensions for stabilization of the scheme a general framework of projections is introduced finally it is demonstrated numerically how the a posteriori error estimate is used to design both an efficient grid adaption and gradient limiting strategy numerical experiments show the stability of the scheme and the gain in efficiency in comparison with computations on uniform grids

numerical assessment of error estimators for euler equations grid convergence studies are conducted to assess four error estimators for their asymptotic behavior explicit residual solution reconstruction richardson extrapolation and solution of the error equations their accuracy reliability and efficitivity to predict the true error are verified on the quasi one dimensional euler equations solved by a second order accurate finite volume method

convergence rate of approximate solutions to conservation laws with initial rarefactions the authors address the question of local convergence rate of conservative textit lip stable approximations u varepsilon x t to the entropy solution u x t of a genuinely nonlinear conservation law this question has been answered in the case of rarefaction free i e textit lip bounded initial data it has been shown that by post processing u varepsilon pointwise values of u and its derivatives may be recovered with an error as close to o varepsilon as desired where varepsilon measures in w 1 1 the truncation error of the approximate solution u varepsilon this paper extends the previous results by including textit lip unbounded initial data specifically it is shown that for arbitrary l infty cap bv initial data u and its derivatives may be recovered with an almost optimal modulo a spurious log factor error of o varepsilon ln varepsilon this analysis relies on obtaining new lip stability estimates for the speed a

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

on the stability of finite element methods for shock waves this paper we study the large time asymptotic stability of solutions for systems of nonlinear viscous conservation laws of the form 1 1 u t f u x u xx x 2 r i t 0 u 2 r i u delta 0 u 0 delta we treat systems which are strictly hyperbolic such systems possess a smooth travelling wave solution which is called a viscous p shock wave solution u x t oe x gamma oet x sigma1 oe x u sigma provided that the shock strength ffl j ju gamma u gamma j is small 19 the constant states u sigma and the wave speed oe are related by the rankine hugoniot condition 1 3a f u gamma gamma f u oe u gamma gamma u

viscous limits for piecewise smooth solutions to systems of conservation laws in this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity it is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles

discontinuous galerkin error estimation for linear symmetric hyperbolic systems in this manuscript we present an error analysis for the discontinuous galerkin discretization error of multi dimensional first order linear symmetric hyperbolic systems of partial differential equations we perform a local error analysis by writing the local error as a series and showing that its leading term can be expressed as a linear combination of legendre polynomials of degree p and p 1p 1 we apply these asymptotic results and solve relatively small local problems to compute efficient and asymptotically exact estimates of the finite element error we present computational results for several linear hyperbolic systems in one two and three dimensions

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

asymptotic stability of rarefaction waves for 2 2 viscous hyperbolic conservation laws this paper concerns the asymptotic behavior toward rarefaction waves of the solution of a genera 2 x 2 hyperbolic conservation laws with positive viscosity we prove that if the initial data is close to a constant state and its values at a3 lie on the kth rarefaction curve for the corresponding hyperbolic conservation laws then the solution tends as i co to the rarefaction wave determined by these states

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

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 variational principle for adaptive approximation of ordinary differential equations a variational principle inspired by optimal control yields a simple derivation of an error representation global error local error weight for general approximation of functions of solutions to ordinary differential equations this error representation is then approximated by a sum of computable error indicators to obtain a useful global error indicator for adaptive mesh refinements a uniqueness formulation is provided for desirable error representations of adaptive algorithms

a local discontinuous galerkin method for the second order wave equation abstract in this paper we present new superconvergence results for the local discontinuous galerkin ldg method applied to the second order scalar wave equation in one space dimension numerical experiments show o h p 1 l 2 convergence rate for the ldg solution and o hp 2 superconvergent solutions at radau points more precisely a local error analysis reveals that at a fixed time t the leading terms of the discretization errors for the solution and its derivative using p degree polynomial approximations are proportional to the p 1 degree right radau and p 1 degree left radau polynomials respectively thus the p degree ldg solution is o hp 2 superconvergent at the roots of the p 1 degree right radau polynomial and the derivative of the ldg solution is o hp 2 superconvergent at the roots of the p 1 degree left radau polynomial these results are used to construct simple efficient and asymptotically correct a posteriori error estimates in regions where solutions are smooth finally we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori errors estimates under mesh refinement

rarefactions and large time behavior for parabolic equations and monotone schemes we consider the large time behavior of monotone semigroups associated with degenerate parabolic equations and monotone difference schemes for an appropriate class of initial data the solution is shown to converge to rarefaction waves at a determined asymptotic rate

asymptotic behaviour of nonlinear parabolic equations with critical exponents a dynamical systems approach we investigate the large time behaviour of solutions to the nonlinear heat conduction equation with absorption ut u 1 u in q rn 0 e with n 1 0 and critical absorption exponent 1 2n the initial function u x 0 0 is assumed to be integrable nonnegative and compactly supported we prove that u converges as t to a unique self similar function which is a contracted version of one of the asymptotic profiles of the nonabsorptive problem ut u 1 the same for any initial data the cornerstone of the proof is a result about limits of infinite dimensional asymptotical dynamical systems combining this result with an asymptotic evaluation of the mass function as well as typical pde estimates gives the behaviour of e for large times similar unusual asymptotic behaviour is obtained for the equation ut div du du u with same conditions on and u x 0 and critical value for 1 2 n

some remarks on li sinai type solutions of the burgers equation for a class of complex valued initial conditions we construct solutions for the inviscid burgers equations of the same type as those recently introduced by li and sinai obtained by very simple considerations these solutions correspond to stable distributions with 1 2 we furthermore show that solutions of non gaussian type can be constructed for the viscous burgers equation

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

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