random attractors for stochastic porous media equations perturbed by space time linear multiplicative noise unique existence of solutions to porous media equations driven by continuous linear multiplicative space time rough signals is proven for initial data in l 1 o on bounded domains o the generation of a continuous order preserving random dynamical system rds on l 1 o and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained the random attractor is shown to be compact and attracting in l o norm uniform l bounds and uniform space time continuity of the solutions is shown general noise including fractional brownian motion for all hurst parameters is treated a pathwise wong zakai result for driving noise given by a continuous semimartingale is obtained for fast diffusion equations driven by continuous linear multiplicative space time rough signals existence of solutions is proven for initial data in l m 1 o

on the stochastic porous medium equation abstract in this paper we discuss an initial boundary value problem and a cauchy problem for the stochastic porous medium equation our basic estimates are based on the known results due to math sbornik 67 1965 609 642 arch rational mech anal 25 1967 64 80 by the procedure developed in trans amer math soc 354 2001 1117 1135 we obtain solutions over the given probability space rather than martingale solutions we will also establish the existence of invariant measures when the space domain is bounded

mesh refinement processes based on the generalized bisection of simplices mesh refinement processes based on the generalized bisection of simplices are discussed and characterized in the context of the general theory of rheinboldt siam j numer anal 17 1980 pp 766 778 then it is proved that such processes allow the construction of sequences of naturally smooth conforming and nested nonuniform meshes in fact it is shown that there exists a generalized mesh refinement operator that for any conforming triangulation delta and for any v subset delta produces a nested smooth conforming triangulation delta containing successors of all elements of v and such that the minimum cell size of delta is bounded from below by half the minimum cell size of delta two conforming mesh refinement algorithms that allow the selective refinement of computational triangulations are explicitly given

analysis of the dynamics of a delayed hiv pathogenesis model in this paper considering full logistic proliferation of cd4 t cells we study an hiv pathogenesis model with antiretroviral therapy and hiv replication time we first analyze the existence and stability of the equilibrium and then investigate the effect of the time delay on the stability of the infected steady state sufficient conditions are given to ensure that the infected steady state is asymptotically stable for all delay furthermore we apply the nyquist criterion to estimate the length of delay for which stability continues to hold and investigate the existence of hopf bifurcation by using a delay t as a bifurcation parameter finally numerical simulations are presented to illustrate the main results

pointwise error estimates for differences in piecewise linear finite element approximations we consider piecewise linear finite element approximations uh to u the solution of an elliptic boundary value problem new estimates for the differences e x1 e x2 where e x u x u h x is the error and x1 and x2 are any two points in the domain are obtained in terms of weighted l infty norms as a consequence so called asymptotic expansion inequalities are derived that have been applied to obtain asymptotically exact a posteriori estimators for the gradient on each element

renormalised solutions of nonlinear parabolic problems with l1 data existence and uniqueness in this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problem where the data f and u 0 belong to l 1 0 t and l 1 and where the function a 0 t n n is monotone but not necessarily strictly monotone and defines a bounded coercive continuous operator from the space into its dual space the renormalised solution is an element of c 0 0 t l 1 such that its truncates t k u belong to with this solution satisfies the equation formally obtained by using in the equation the test function s u where belongs to and where s belongs to c with

a differential equation model of hiv infection of cd4 t cells with cure rate abstract a differential equation model of hiv infection of cd4 t cells with cure rate is studied we prove that if the basic reproduction number r 0 1 the hiv infection is cleared from the t cell population and the disease dies out if r 0 1 the hiv infection persists in the host we find that the chronic disease steady state is globally asymptotically stable if r 0 1 furthermore we also obtain the conditions for which the system exists an orbitally asymptotically stable periodic solution numerical simulations are presented to illustrate the results

adaptive finite element methods for elliptic equations with non smooth coefficients nous considerons une equation elliptique du second ordre a coefficients discontinus ou anisotropes dans un domaine borne en dimension 2 ou 3 et sa discretisation par elements finis le but de cet article est de demontrer des estimations d erreur a priori et a posteriori dans une norme appropriee qui soient independantes de la variation des coefficients

maximum norm estimates for an immunology model using reaction diffusion equations with stochastic source terms this paper describes numerical methods and the corresponding maximum norm error estimates for a chemotaxis model proposed by kepler upon detecting pathogens immune cells secrete soluble factors that attract other immune cells to the site of the infection the motion of the model cells is stochastic but biased toward the gradient of one or more of the soluble factors the soluble factors are modeled by a system of reaction diffusion equations with sources that are centered on the cells previously i presented a first order splitting in time for solving the reaction diffusion stochastic system numerically the diffusion reaction and stochastic differential equations can be approximated separately to first order in the supremum norm the three dimensional domain is discretized using finite elements and the diffusion is solved using a backward euler scheme combined with multigrid the reaction is solved using a simple semi implicit first order scheme the stochastic differential equations are given by a langevin process which can be simulated exactly the paper concludes by demonstrating first order convergence of the entire simulation and providing a sample simulation of the reaction diffusion stochastic system

an adaptive strategy for elliptic problems including a posteriori controlled boundary approximation we derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary the numerical solution is assumed to be defined on a finite element mesh whose boundary vertices are located on the boundary of the continuous problem no assumption is made on a geometrically fitting shape a posteriori error estimates are given in the energy norm and the l 2 norm and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation furthermore a strategy is presented to compute the effect of the non discretized part of the domain on the error starting from a coarse mesh this especially implies that parts of the domain where the measured error is small stay non discretized the presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary the reliable computation of the a posteriori estimates and a mesh adaptation based on delaunay techniques numerical examples illustrate that errors outside the initial discretization will be detected

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 estimates for resolvents of elliptic finite element operators let be a convex domain with smooth boundary in rd it has been shown recently that the semigroup generated by the discrete laplacian for quasi uniform families of piecewise linear finite element spaces on is analytic with respect to the maximum norm uniformly in the mesh width this implies a resolvent estimate of standard form in the maximum norm outside some sector in the right halfplane and conversely here we show directly that such a resolvent estimate holds outside any sector around the positive real axis with arbitrarily small angle this is useful in the study of fully discrete approximations based on a stable rational functions with small

pointwise a posteriori error estimates for elliptic problems on highly graded meshes pointwise a posteriori error estimates are derived for linear second order elliptic problems over general polygonal domains in 2d the analysis carries over regardless of convexity accounting even for slit domains and applies to highly graded unstructured meshes as well a key ingredient is a new asymptotic a priori estimate for regularized green s functions the estimators lead always to upper bounds for the error in the maximum norm along with lower bounds under very mild regularity and nondegeneracy assumptions the effect of both point and line singularities is examined three popular local estimators for the energy norm are shown to be equivalent when suitably interpreted to those introduced here

asymptotic error expansions for the finite element method for second order elliptic problems in r n n geq2 i local interior expansions our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h valid for h sufficiently small such expansions are obtained from the so called asymptotic expansion inequalities valid in mathbf r n for n geq2 studies by schatz in math comp 67 1998 pp 877 899 and siam j numer anal 38 2000 pp 1269 1293

maximum norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter

weak solutions to the stochastic porous media equation via kolmogorov equations the degenerate case a stochastic version of the porous medium equation with coloured noise is studied the corresponding kolmogorov equation is solved in the space l 2 h where is an infinitesimally excessive measure then a weak solution is constructed

sharply local pointwise a posteriori error estimates for parabolic problems we prove pointwise a posteriori error estimates for semi and fully discrete finite element methods for approximating the solution u to a parabolic model problem our estimates may be used to bound the finite element error u u h l d where d is an arbitrary subset of the space time domain of the definition of the given pde in contrast to standard global error estimates these estimators de emphasize spatial error contributions from space time regions removed from d our results are valid on arbitrary shape regular simplicial meshes which may change in time and also provide insight into the contribution of mesh change to local errors when implemented in an adaptive method these estimates require only enough spatial mesh refinement away from d in order to ensure that local solution quality is not polluted by global effects

on a random scaled porous media equation abstract it is shown that a random scaled porous media equation arising from a stochastic porous media equation with linear multiplicative noise through a random transformation is well posed in l in the fast diffusion case we show existence in l p

global properties for virus dynamics model with mitotic transmission and intracellular delay abstract in this paper we study the basic model of viral infections with mitotic transmission and intracellular delay discrete the delay corresponds to the time between infection of uninfected cells and the emission of virus on a cellular level by means of volterra type lyapunov functionals we provide the global stability for this model let be the number of virus produced per infected cell if crit the critical number satisfies crit then the virus free steady state is globally asymptotically stable on the contrary if crit then the infected steady state is globally asymptotically stable if a sufficient condition is satisfied

reliable a posteriori error control for nonconforming finite element approximation of stokes flow we derive computable a posteriori error estimates for the lowest order nonconforming crouzeix raviart element applied to the approximation of incompressible stokes flow the estimator provides an explicit upper bound that is free of any unknown constants provided that a reasonable lower bound for the inf sup constant of the underlying problem is available in addition it is shown that the estimator provides an equivalent lower bound on the error up to a generic constant

sharply localized pointwise and w 1 estimates for finite element methods for quasilinear problems we establish pointwise and w 1 estimates for finite element methods for a class of second order quasilinear elliptic problems defined on domains in r these estimates are localized in that they indicate that the pointwise dependence of the error on global norms of the solution is of higher order our pointwise estimates are similar to and rely on results and analysis techniques of schatz for linear problems we also extend estimates of schatz and wahlbin for pointwise differences e x 1 e x 2 in pointwise errors to quasilinear problems finally we establish estimates for the error in w 1 d where d is a subdomain these negative norm estimates are novel for linear as well as for nonlinear problems our analysis heavily exploits the fact that galerkin error relationships for quasilinear problems may be viewed as perturbed linear error relationships thus allowing easy application of properly formulated results for linear problems

maximum norm error estimators for three dimensional elliptic problems in this paper we define an a posteriori error estimator for finite element approximations of 3 d elliptic problems we prove that the estimator is equivalent up to logarithmic factors of the meshsize to the maximum norm of the error the results are valid for an arbitrary polyhedral domain and rather general meshes we also obtain analogous results for the nonconforming method of crouzeix raviart finally we present some numerical results comparing adaptive procedures based on controlling the error in different norms

error estimators for nonconforming finite element approximations of the stokes problem in this paper we define and analyze a posteriori error estimators for nonconforming approximations of the stokes equations we prove that these estimators are equivalent to an appropriate norm of the error for the case of piecewise linear elements we define two estimators both of them are easy to compute but the second is simpler because it can be computed using only the right hand side and the approximate velocity we show how the first estimator can be generalized to higher order elements finally we present several numerical examples in which one of our estimators is used for adaptive refinement

a posteriori pointwise error computation for 2 d transport equations based on the variational multiscale method abstract this article presents a general framework to estimate the pointwise error of linear partial differential equations the error estimator is based on the variational multiscale theory in which the error is decomposed in two components according to the nature of the residuals element interior residuals and inter element jumps the relationship between the residuals coarse scales and the error components fine scales is established yielding to a very simple model in particular the pointwise error is modeled as a linear combination of bubble functions and green s functions if residual free bubbles and the classical green s function are employed the technology leads to an exact explicit method for the pointwise error if bubble functions and free space green s functions are employed then a local projection problem must be solved within each element and a global boundary integral equation must be solved on the domain boundary as a consequence this gives a model for the so called fine scale green s functions the numerical error is studied for the standard galerkin and supg methods with application to the heat equation the reaction diffusion equation and the convection diffusion equation numerical results show that stabilized methods minimize the propagation of pollution errors which stay mostly locally

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

a posteriori error estimators for nonconforming approximation of some quasi newtonian flows this paper deals with a posteriori error estimators for nonconforming approximations of quasi newtonian flows we consider the crouzeix raviart piecewise linear approximations of scalar elliptic problems and define an error estimator when omega is a simply connected domain the error is dominated by the estimator this estimator can be generalized to higher order elements we define a posteriori error estimators for the fortin soulie piecewise quadratic approximations of quasi newtonian flows and prove that the error is dominated by the estimator this estimator can be computed locally in terms of the approximate solution and is therefore suitable for adaptive refinement

global stability and periodic solution of a model for hiv infection of cd4 t cells abstract it is well known that the mathematical models provide very important information for the research of human immunodeficiency virus type 1 and hepatitis c virus hcv however the infection rate of almost all mathematical models is linear the linearity shows the simple interaction between the t cells and the viral particles in this paper we consider the classical mathematical model with non linear infection rate the global dynamics of this model is rigorously established we prove that if the basic reproduction number r 0 1 the hiv infection is cleared from the t cells population if r 0 1 the hiv infection persists further the existence of a non trivial periodic solution is also studied by means of numerical simulation

pointwise a posteriori error control for elliptic obstacle problems we consider a finite element method for the elliptic obstacle problem over polyhedral domains in d which enforces the unilateral constraint solely at the nodes we derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle which is just assumed to be holder continuous they exhibit optimal order and localization to the non contact set we illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi optimal meshes

pointwise a posteriori error estimates for monotone semi linear equations we derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi linear equations the estimates hold for lagrange elements of any fixed order non smooth nonlinearities and take numerical integration into account the proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately and then applying the continuous maximum principle no geometric mesh constraints are thus required numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature

local pointwise a posteriori gradient error bounds for the stokes equations we consider the standard taylor hood finite element method for the stationary stokes system on polyhedral domains we prove local a posteriori error estimates for the maximum error in the gradient of the velocity field because the gradient of the velocity field blows up near reentrant corners and edges such local error control is necessary when pointwise control of the gradient error is desirable computational examples confirm the utility of our estimates in adaptive codes

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

stochastic generalized porous media and fast diffusion equations abstract we present a generalization of krylov rozovskii s result on the existence and uniqueness of solutions to monotone stochastic differential equations as an application the stochastic generalized porous media and fast diffusion equations are studied for finite reference measures where the drift term is given by a negative definite operator acting on a time dependent function which belongs to a large class of functions comparable with the so called n functions in the theory of orlicz spaces

asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes part 1 a smooth problem and globally quasi uniform meshes a class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems the meshes are unstructured and in particular it is not assumed that there are any known superconvergent points the estimators are based on averaging operators which are approximate gradients recovered gradients which are then compared to the actual gradient of the approximation on each element conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large in a controlled fashion to be detailed below compared to the size of the elements

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

finite element approximation of the p laplacian in this paper we consider the continuous piecewise linear finite el ement approximation of the following problem given p 1 oo and g find u such that v vu 2vu f iniicr2 u g on a the finite element approximation is defined over i2 a union of regular tri angles yielding a polygonal approximation to q for sufficiently regular so lutions u achievable for a subclass of data g and i2 we prove optimal error bounds for this approximation in the norm wlq q 1 q p for p 2 under the additional assumption that qh c 2 numerical results demonstrating these bounds are also presented

regularity estimates and optimal finite element methods in wp1 on polygonal domains abstract consider the poisson equation with the dirichlet boundary condition on a bounded convex polygonal domain r 2 we investigate the finite element approximation of singular solutions that are due to the non smoothness of the domain in the w p 1 norm 1 p in particular with analysis in weighted sobolev spaces and weighted holder spaces we provide regularity requirements on the given data and specific parameter selection criteria for graded meshes such that the resulting numerical approximation achieves the optimal convergence rate in w p 1 sample results from various numerical tests are provided to confirm the theory

quasi optimal and robust a posteriori error estimates in for the approximation of allen cahn equations past singularities quasi optimal a posteriori error estimates in l 0 t l 2 are derived for the finite element approximation of allen cahn equations the estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface the error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming nonconforming mixed and discontinuous galerkin methods numerical experiments illustrate the theoretical results

local a posteriori estimates on a nonconvex polygonal domain a class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems on a nonconvex polygonal domain the results are extensions of previous results for smooth domains w hoffmann et al math comp 70 2001 pp 897 909 a h schatz and l b wahlbin math comp 73 2004 pp 517 523

a posteriori error estimates of hp discontinuous galerkin method for strongly nonlinear elliptic problems abstract in this paper we study the residual based a posteriori error estimates of h p discontinuous galerkin finite element methods for strongly nonlinear elliptic boundary value problems computable upper and lower bounds on the error are derived in a natural mesh dependent energy norm the bounds are explicit in the local mesh size and the local degree of the approximating polynomial numerical experiments are also provided to illustrate the performance of the proposed estimators

strong solutions for stochastic partial differential equations of gradient type unique existence of analytically strong solutions to stochastic partial differential equations spde with drift given by the subdifferential of a quasi convex function and with general multiplicative noise is proven the proof applies a genuinely new method of weighted galerkin approximations based on the distance defined by the quasi convex function spatial regularization of the initial condition analogous to the deterministic case is obtained the results yield a unified framework which is applied to stochastic generalized porous media equations stochastic generalized reaction diffusion equations and stochastic generalized degenerated p laplace equations in particular higher regularity for solutions of such spde is obtained

langevin equation for the density of a system of interacting langevin processes we present a simple derivation of the stochastic equation obeyed by the density function for a system of langevin processes interacting via a pairwise potential the resulting equation is considerably different from the phenomenological equations usually used to describe the dynamics of non conserved model a and conserved model b particle systems the major feature is that the spatial white noise for this system appears not additively but multiplicatively this simply expresses the fact that the density cannot fluctuate in regions devoid of particles the steady state for the density function may however still be recovered formally as a functional integral over the coursed grained free energy of the system as in models a and b

well posedness of nonlinear diffusion equations with nonlinear conservative noise we prove the pathwise well posedness of stochastic porous media and fast diffusion equations driven by nonlinear conservative noise as a consequence the generation of a random dynamical system is obtained this extends results of the second author and souganidis who considered analogous spatially homogeneous and first order equations and earlier works of lions perthame and souganidis

two grid finite element method and its a posteriori error estimates for a nonmonotone quasilinear elliptic problem under minimal regularity of data abstract in this paper we propose and analyze a two grid finite element method for a class of quasilinear elliptic problems under minimal regularity of data in a bounded convex polygonal r 2 which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space with this technique solving a quasilinear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasilinear elliptic problem on a coarse space convergence estimates in the h 1 norm are derived to justify the efficiency of the proposed two grid algorithm moreover we propose a natural and computationally efficient residual based a posteriori error estimator of the two grid finite element method for this nonmonotone quasilinear elliptic problem and derive the global upper and lower bounds on the error in the h 1 norm numerical experiments are provided to confirm our theoretical findings

localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second order quasilinear elliptic problems two types of pointwise a posteriori error estimates are presented for gradients of finite element approximations of second order quasilinear elliptic dirichlet boundary value problems over convex polyhedral domains omega in space dimension n ge 2 we first give a residual estimator which is equivalent to nabla u u h l infty omega up to higher order terms the second type of residual estimator is designed to control nabla u u h locally over any subdomain of omega it is a novel a posteriori counterpart to the localized or weighted a priori estimates of sch98 this estimator is shown to be equivalent up to higher order terms to the error measured in a weighted global norm which depends on the subdomain of interest all estimates are proved for general shape regular meshes which may be highly graded and unstructured the constants in the estimates depend on the unknown solution u in the nonlinear case but in a fashion which places minimal restrictions on the regularity of u

local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems we prove local a posteriori error estimates for pointwise gradient errors in finite element methods for a second order linear elliptic model problem first we split the local gradient error into a computable local residual term and a weaker global norm of the finite element error the pollution term using a mesh dependent weight the residual term is bounded in a sharply localized fashion in specific situations the pollution term may also be bounded by computable residual estimators on nonconvex polygonal and polyhedral domains in two and three space dimensions we may choose estimators for the pollution term which do not employ specific knowledge of corner singularities and which are valid on domains with cracks the finite element mesh is only required to be simpliciai and shape regular so that highly graded and unstructured meshes are allowed

existence of strong solutions for stochastic porous media equation under general monotonicity conditions this paper addresses existence and uniqueness of strong solutions to stochastic porous media equations dx x dt b x dw t in bounded domains of r d with dirichlet boundary conditions here is a maximal monotone graph in r r possibly multivalued with

asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes part ii the piecewise linear case we extend results from part i about estimating gradient errors elementwise a posteriori given there for quadratic and higher elements to the piecewise linear case the key to our new result is to consider certain technical estimates for differences in the error e x 1 e x 2 rather than for e x itself we also give a posteriori estimators for second derivatives on each element

adaptive finite element methods for an optimal control problem involving dirac measures the purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so called pointwise tracking optimal control problem this linear quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the state thus leading to an adjoint problem with dirac measures on the right hand side control constraints are also considered the proposed error estimator relies on a posteriori error estimates in the maximum norm for the state and in muckenhoupt weighted sobolev spaces for the adjoint state we present an analysis that is valid for two and three dimensional domains we conclude by presenting several numerical experiments which reveal the competitive performance of adaptive methods based on the devised error estimator

speed of propagation for hamilton jacobi equations with multiplicative rough time dependence and convex hamiltonians we show that the initial value problem for hamilton jacobi equations with multiplicative rough time dependence typically stochastic and convex hamiltonians satisfies finite speed of propagation we prove that in general the range of dependence is bounded by a multiple of the length of the skeleton of the path that is a piecewise linear path obtained by connecting the successive extrema of the original one when the driving path is a brownian motion we prove that its skeleton has almost surely finite length we also discuss the optimality of the estimate

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

global stability and periodic solution of the viral dynamics it is well known that the mathematical models provide very important information for the research of human immunodeficiency virus type 1 and hepatitis c virus hcv however the infection rate of almost all mathematical models is linear the linearity shows the simple interaction between the t cells and the viral particles in this paper we consider the classical mathematical model with saturation response of the infection rate by stability analysis we obtain sufficient conditions on the parameters for the global stability of the infected steady state and the infection free steady state we also obtain the conditions for the existence of an orbitally asymptotically stable periodic solution numerical simulations are presented to illustrate the results

fully localized a posteriori error estimators and barrier sets for contact problems we derive novel pointwise a posteriori error estimators for elliptic obstacle problems which except for obstacle resolution completely vanish within the full contact set localization we then construct a posteriori barrier sets for free boundaries under a natural stability or nondegeneracy condition we illustrate localization properties as well as reliability and efficiency for both solutions and free boundaries via several simulations in 2 and 3 dimensions

an a posteriori error analysis of an elliptic optimal control problem in measure space abstract we propose an a posteriori error estimator for an optimal control problem with sparsity promoting cost the control variable lies in the space of regular borel measures we consider a solution technique that relies on the discretization of the control variable as a linear combination of dirac measures the proposed a posteriori error estimator can be decomposed into the sum of two contributions an error estimator in the maximum norm for the discretization of the adjoint equation and an estimator in the l 2 norm that accounts for the approximation of the state equation we prove that the designed error estimator is locally efficient and we explore its reliability properties the analysis is valid for two and three dimensional domains we illustrate the theory with numerical examples

non monotone stochastic generalized porous media equations abstract by using the nash inequality and a monotonicity approximation argument existence and uniqueness of strong solutions are proved for a class of non monotone stochastic generalized porous media equations moreover we prove for a large class of stochastic pde that the solutions stay in the smaller l 2 space provided the initial value does so that some recent results in the literature are considerably strengthened

regularity in sobolev spaces for the fast diffusion and the porous medium equation abstract the degenerate parabolic equation u t u m 1 u m 0 is considered in a cylinder 0 t under homogeneous dirichlet boundary values the regularity of weak solutions for the fast diffusion equation m 1 and the porous medium equation m 1 are investigated regularity of u and u m 1 u in weighted sobolev spaces and in fractional order nikolskii spaces are proved

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

holder estimates for green s functions on convex polyhedral domains and their applications to finite element methods a model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered the aim of this paper is twofold first sharp holder estimates for the corresponding green s function are obtained as an applications of these estimates to finite element methods we show the best approximation property of the error in w 1 infty in contrast to previously known results w p 2 regularity for p 3 which does not hold for general convex polyhedral domains is not required furthermore the new green s function estimates allow us to obtain localized error estimates at a point

global superconvergence and a posteriori error estimates of the finite element method for second order quasilinear elliptic problems in this paper we are concerned with the linear finite element approximations to the second order quasi linear elliptic problems by means of an interpolation postprocessing technique we develop the global superconvergence estimates in the h 1 and w 1 norms provided the weak solutions are sufficiently smooth based on the global superconvergent approximations we introduce and analyze the efficient postprocessing based a posteriori error estimators measured by the h 1 and w 1 norms respectively these can be used to assess the accuracy of the finite element solutions in applications numerical experiments are given to illustrate the global superconvergence estimates and the performance of the proposed estimators

analysis of stability and hopf bifurcation for an hiv infection model with time delay a class of more general hiv infection models with time delay is proposed based on some important biological meanings the effect of time delay on stability of the equilibria of the infection model has been studied and the sufficient criteria for stability switch of the infected equilibrium and the local and global asymptotic stability of the viral free equilibrium are given using the normal form theory and center manifold argument the explicit formulaes which determine the stability the direction and the periodic of bifurcating period solutions are derived numerical simulations are carried out to explain the mathematical conclusions

a posteriori error estimates in the maximum norm for parabolic problems we derive a posteriori error estimates in the l infty 0 t l infty omega norm for approximations of solutions to linear parabolic equations using the elliptic reconstruction technique introduced by makridakis and nochetto and heat kernel estimates for linear parabolic problems we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations we then establish a posteriori bounds for a fully discrete backward euler finite element approximation the elliptic reconstruction technique greatly simplifies our development by allowing the straightforward combination of heat kernel estimates with existing elliptic maximum norm error estimators

microscopic analyses of the dynamical density functional equation of dense fluids we briefly review the recent microscopic derivation of the generalized dynamical density functional equation with some modification as well as the time evolution equation of the configuration distribution function we then describe our attempt to analyze the microscopic memory kernel operator that enters these equations in particular we give a matrix continued fraction recursion formula of the memory kernel operator

maximum norm a posteriori error estimates for singularly perturbed elliptic reaction diffusion problems residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polyhedral domains standard finite element approximations are considered the error constants are independent of the diameters of mesh elements and the small perturbation parameter in our analysis we employ sharp bounds on the green s function of the linearized differential operator numerical results are presented that support our theoretical findings

on the non existence of path integrals let a b be two classes of paths so that if x in a and y in b then the integral int xdy is appropriately defined then despite the fact that int xdy exists with probability one whenever x y are independent brownian motions a cap b has weiner measure zero