a priori l 2 error estimates for finite element methods for nonlinear diffusion equations with memory this paper studies finite element approximations to the solutions of the nonlinear diffusion equations with memory an elliptic projection with memory associated with our equations is defined and then used in the derivations of optimal error estimates for semidiscrete and crank nicolson finite element approximations

stallings foldings and subgroups of free groups we re cast in a more combinatorial and computational form the topological approach of j stallings to the study of subgroups of free groups

time discretization of an integro differential equation of parabolic type the stability and convergence properties of two time discretizations of an integro differential equation of parabolic type are studied the methods reduce to the backward euler and crank nicolson methods if the integral term is absent the integral term is approximated in each case by a quadrature rule with relatively high order truncation error so that a relatively large time step can be used for the quadrature in order to reduce the memory and computational requirements of the method

maximum norm stability and error estimates in galerkin methods for parabolic equations in one space variable maximum norm stability and error estimates of best approximation and nonsmooth data types are derived for the approximate solution of a parabolic equation in one space variable using the continuous in time galerkin method based on piecewise polynomial approximating functions on a quasi uniform mesh

sharp l 2 error estimates and superconvergence of mixed finite element methods for non fickian flows in porous media a sharper l2 error estimate is obtained for the non fickian flow of fluid in porous media by means of a mixed ritz volterra projection instead of the mixed ritz projection used in r e ewing y lin and j wang acta math univ comenian n s 70 2001 pp 75 84 moreover local l2 superconvergence for the velocity along the gauss lines and for the pressure at the gauss points is derived for the mixed finite element method via the ritz volterra projection and global l2 superconvergence for the velocity and the pressure is also investigated by virtue of an interpolation postprocessing technique on the basis of the superconvergence estimates some useful a posteriori error estimators are presented for this mixed finite element method

two splitting positive definite mixed finite element methods for parabolic integro differential equations abstract two novel mixed finite element procedures are established for parabolic integro differential equations which can be split into two independent symmetric positive definite sub schemes and do not need to solve a coupled system of equations without requiring the lbb consistency condition the convergence analysis shows that both methods lead to the optimal order l 2 norm error estimate for u and optimal h div norm error estimate for a numerical example is presented to illustrate the theoretical analysis

finite element methods of the two nonlinear integro differential equations abstract transformation of dependent variables such as the kirchhoff transformation is a classical tool for solving nonlinear partial differential equations in larsson et al this approach was used in connection with the finite element method zlamal used it in galerkin approximation in space the euler method in time and applied it to the solution of nonlinear heat equations here we give justification of the methods for the two nonlinear integro heat equations in the case that the discretization is carried through by piecewise linear polynomials in space and by the implicit euler method in time with quadrature rules which require the storage of relatively few values of the solution the estimate of the discretization error in the maximum norm is introduced the results depend on superconvergence in the gradient for a related elliptic problem which is shown to hold only for very special meshes

discontinuous galerkin method for an integro differential equation modeling dynamic fractional order viscoelasticity an integro differential equation modeling dynamic fractional order viscoelasticity with a mittag leffler type convolution kernel is considered a discontinuous galerkin method based on piecewise constant polynomials is formulated for temporal semidiscretization of the problem stability estimates of the discrete problem are proved that are used to prove optimal order a priori error estimates the theory is illustrated by a numerical example

a two grid finite difference scheme for nonlinear parabolic equations we present a two level finite difference scheme for the approximation of nonlinear parabolic equations discrete inner products and the lowest order raviart thomas approximating space are used in the expanded mixed method in order to develop the finite difference scheme analysis of the scheme is given assuming an implicit time discretization in this two level scheme the full nonlinear problem is solved on a coarse grid of size h the nonlinearities are expanded about the coarse grid solution and an appropriate interpolation operator is used to provide values of the coarse grid solution on the fine grid in terms of superconvergent node points the resulting linear but nonsymmetric system is solved on a fine grid of size h some a priori error estimates are derived which show that the discrete l infty l2 and l2 h1 errors are o h 2 h 4 d 2 delta t where d geq 1 is the spatial dimension

time stepping schemes for systems of volterra integro differential equations linear volterra type systems of integro differential equations sides with kernels in the form of sums of complex exponentials are considered such systems arise for example in the semi discrete finite element model of dynamic viscoelasticity and of some dispersive wave equations and in the representation of dynamic subsystems on interfaces several new families of time stepping schemes are proposed here for the solution of this class of sides the accuracy stability and efficiency properties of these schemes are demonstrated via both theoretical analysis using a scalar model problem and numerical experiments applied to first and second order sides

stationary strategies for recursive games we study two person zero sum recursive matrix games framing them in the more general context of nonleavable games our aim is to extend a result of orkin orkin m 1972 recursive matrix games j appl probab 9 813 820 by proving that a uniformly e optimal stationary strategy is available to player i player ii in any recursive game such that the set where the value of the game is strictly greater less than its utility is finite the proof exploits some new connections between nonleavable and leavable games in particular we find sufficient conditions for the existence of e optimal stationary strategies for player i in a leavable game and we use these as a basis for constructing uniformly e optimal stationary strategies for the recursive games with which we are concerned

a second order accurate numerical method for a fractional wave equation we study a generalized crank nicolson scheme for the time discretization of a fractional wave equation in combination with a space discretization by linear finite elements the scheme uses a non uniform grid in time to compensate for the singular behaviour of the exact solution at t 0 with appropriate assumptions on the data and assuming that the spatial domain is convex or smooth we show that the error is of order k2 h2 where k and h are the parameters for the time and space meshes respectively

finite element approximation of an evolution problem modeling shear band formation abstract an error analysis is provided for the spatial approximation by the finite element method of a nonlinear time dependent problem which models the antiplane shear deformations of a thermoplastic material this problem consists of a system of two coupled partial differential equations with velocity and temperature unknowns optimal order error estimates in the l 2 norm are given for each of these

finite element approximation of diffusion equations with convolution terms approximation of solutions to diffusion equations with memory represented by convolution integral terms is considered such problems arise from modeling of flows in fissured media convergence of the method is proved and results of numerical experiments confirming the theoretical results are presented the advantages of implementation of the algorithm in a multiprocessing environment are discussed

finite element methods for partial volterra integro differential equations on two dimensional unbounded spatial domains abstract the precise and effective way for solving a partial volterra integro differential equation on unbounded spatial domain is to derive artificial boundary conditions for some feasible and prescribed computational domain thus the unsolvable problem on unbounded domain is converted to the computational problem on bounded domain which is often called reduced problem a feasible numerical method will then be applied to solve the reduced problem with the artificial boundary conditions which often exhibit nonlocal properties in this paper the finite element method for the spatial space and the recently developed discontinuous galerkin time stepping method for the temporal space are considered to solve the reduced problem the convergence rate explicitly with those interesting parameters spatial grid size h temporal mesh size k the location of the artificial boundary d and the truncation number m is obtained this paper is the first theoretical analysis of the numerical methods for volterra integro differential equations on unbounded spatial domains

numerical solution of parabolic integro differential equations by the discontinuous galerkin method the numerical solution of a parabolic equation with memory is considered the equation is first discretized in time by means of the discontinuous galerkin method with piecewise constant or piecewise linear approximating functions the analysis presented allows variable time steps which as will be shown call then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data the combination with finite element discretizat in space is also studied

discretization with variable time steps of an evolution equation with a positive type memory term we consider the linearized form of an evolution equation used to model viscoelasticity and heat conduction in materials with memory the equation is discretized in time using several schemes based on the backward euler and crank nicolson methods and incorporating appropriate quadratures for the memory term by permitting variable time steps we remove a restriction present in earlier related work by the first two authors

a second order accuracy for a full discretized time dependent navier stokes equations by a two grid scheme we study a second order two grid scheme fully discrete in time and space for solving the navier stokes equations the two grid strategy consists in discretizing in the first step the fully non linear problem in space on a coarse grid with mesh size h and time step t and in the second step in discretizing the linearized problem around the velocity u h computed in the first step in space on a fine grid with mesh size h and the same time step the two grid method has been applied for an analysis of a first order fully discrete in time and space algorithm and we extend the method to the second order algorithm this strategy is motivated by the fact that under suitable assumptions the contribution of u h to the error in the non linear term is measured in the l 2 norm in space and time and thus has a higher order than if it were measured in the h 1 norm in space we present the following results if h 2 h 3 t 2 then the global error of the two grid algorithm is of the order of h 2 the same as would have been obtained if the non linear problem had been solved directly on the fine grid

time discretization via laplace transformation of an integro differential equation of parabolic type we consider the discretization in time of an inhomogeneous parabolic integro differential equation with a memory term of convolution type in a banach space setting the method is based on representing the solution as an integral along a smooth curve in the complex plane which is evaluated to high accuracy by quadrature using the approach in recent work of lopez fernandez and palencia this reduces the problem to a finite set of elliptic equations with complex coefficients which may be solved in parallel the method is combined with finite element discretization in the spatial variables to yield a fully discrete method the paper is a further development of earlier work by the authors which on the one hand treated purely parabolic equations and on the other an evolution equation with a positive type memory term

fem ids for a second order strongly damped wave equation with memory abstract a linear triangular finite element method implicit difference scheme fem ids for the solution of second order strongly damped wave equation sdwe with memory on domain with interfaces is proposed sufficient conditions that guarantee the existence of a unique solution are given the finite element discretization is such that the arbitrary but smooth interface is first approximated by a polygon with a boundary whose vertices all lie on the interface with the interface being at distance from the approximate interface linear interpolation operator incorporating the influence of isproved this together with ritz volterra operator and some auxiliary error estimates in the neighborhood of the interface are used to obtain the convergence estimate of the proposed scheme the scheme is applied to some test examples and the numerical results confirm the computational efficiency of the method

a numerical method for a partial integro differential equation a method is considered for the integration in time of a partial integro differential equation the discretization technique employed is patterned after an idea of ch lubich error bounds are derived for both smooth and nonsmooth initial data

two grid scheme for characteristics finite element solution of nonlinear convection diffusion problems abstract a two grid scheme for characteristics finite element solution of nonlinear convection dominated diffusion equations was constructed the l 2 and h 1 error estimates of the scheme were derived a numerical example was also presented the scheme involves solving one small nonlinear problem on the coarse grid system one linear problem on the fine grid system the error estimates and the numerical example show that the new scheme is efficient to the nonlinear convection dominated diffusion equations

on the discretization in time for a parabolic integrodifferential equation with a weakly singular kernel ii nonsmooth initial data abstract backward euler methods are considered for a parabolic integrodifferential equation with a memory term containing a weakly singular kernel the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in error analyses the stability and convergence properties of the time discretizations are derived and are applied to the semidiscrete equations built by the galerkin finite element methods in the space variables

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

acceleration of a two grid method for eigenvalue problems this paper provides a new two grid discretization method for solving partial differential equation or integral equation eigenvalue problems in 2001 xu and zhou introduced a scheme that reduces the solution of an eigenvalue problem on a finite element grid to that of one single linear problem on the same grid together with a similar eigenvalue problem on a much coarser grid by solving a slightly different linear problem on the fine grid the new algorithm in this paper significantly improves the theoretical error estimate which allows a much coarser mesh to achieve the same asymptotic convergence rate numerical examples are also provided to demonstrate the efficiency of the new method

analysis of fully discrete approximations to parabolic problems with rough or distribution valued initial data we consider fully discrete approximations to a parabolic initial boundary value problem with rough or distribution valued initial data in two space dimensions for discretization in time and space we apply single step methods and the standard galerkin method with piecewise linear test functions respectively for spatial discretization of the initial condition we are however forced to use more involved constructions our main result is stability and error estimates of the discrete solutions

long time numerical solution of a parabolic equation with memory long time stability and convergence properties of two time discreti zation methods for an integro differential equation of parabolic type are studied the methods are based on the standard backward euler and second order back ward differencing methods the memory term is approximated by a quadrature rule with emphasis on such rules with reduced computational memory require ments discretization of the spatial partial differential operators by the finite element method is also considered

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

the coset poset and probabilistic zeta function of a finite group abstract we investigate the topological properties of the poset of proper cosets xh in a finite group g of particular interest is the reduced euler characteristic which is closely related to the value at 1 of the probabilistic zeta function of g our main result gives divisibility properties of this reduced euler characteristic

a two grid method based on newton iteration for the navier stokes equations in this paper we consider a two grid method for resolving the nonlinearity in finite element approximations of the equilibrium navier stokes equations we prove the convergence rate of the approximation obtained by this method the two grid method involves solving one small nonlinear coarse mesh system and two linear problems on the fine mesh which have the same stiffness matrix with only different right hand side the algorithm we study produces an approximate solution with the optimal asymptotic in h and accuracy for any reynolds number numerical example is given to show the convergence of the method

non fickian convection diffusion models in porous media in this paper we propose a numerical scheme to approximate the solution of a non fickian coupled model that describes e g miscible transport in porous media the model is defined by a system of a quasilinear elliptic equation which governs the fluid pressure and a quasilinear integro differential equation which models the convection diffusion transport process the numerical scheme is based on a conforming piecewise linear finite element method for the discretization in space the fully discrete approximations is obtained with an implicit explicit method estimates for the continuous in time and the fully discrete methods are derived showing that the numerical approximation for the concentrations and the pressure are second order convergent in a discrete l 2 norm and in a discrete h 1 norm respectively

borel stochastic games with lim sup payoff we consider two person zero sum stochastic games with limit superior payoff function and borel measurable state and action spaces the games are shown to have a value and the value function is calculated by transfinite iteration of an operator and proved to be upper analytic the paper extends results of our earlier article 17 in which the same class of games was considered for countable state spaces and finite action sets

two grid methods for characteristic finite volume element solution of semilinear convection diffusion equations abstract two grid methods for characteristic finite volume element solutions are presented for a kind of semilinear convection dominated diffusion equations the methods are based on the method of characteristics two grid method and the finite volume element method the nonsymmetric and nonlinear iterations are only executed on the coarse grid with grid size h and the fine grid solution with grid size h can be obtained by a single symmetric and linear step it is proved that the coarse grid can be much coarser than the fine grid the two grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h 1 3

existence of value and randomized strategies in zero sum discrete time stochastic dynamic games two players with conflicting objectives are simultaneously controlling a discrete time stochastic system the goal of this paper is to analyze such zero sum discrete time stochastic systems when the two players are allowed to use randomized strategies previous results have been restricted to systems with finite or compact state spaces such restrictions are usually untenable from the point of view of applications since many applications frequently use either the integers or mathbb r n as a state space our results are proved for complete separable metric spaces which are very useful for applications all previously known results emerge as special cases of our results in addition a variety of conjectures and open problems are resolved regarding the existence of a value function its properties such as borel measurability or continuity and the existence for either or both players of optimal or varepsilon optimal stationary strategies

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

finite element methods for parabolic and hyperbolic partial integro differential equations on formule plusieurs methodes de galerkin a temps discret pour des equations integrodifferentielles paraboliques

finite element and dg analysis for neutral type volterra integro differential equations with weakly singular kernels abstract in this paper we analyze the numerical solution of volterra integro differential equations of neutral type with weakly singular kernels we establish a priori error estimations for the finite element method semi discretization of the given problem by defining a suitable ritz volterra projection operator here the key point in the proof is the fact that the definition of the ritz volterra projection operator that is not related to the neutral term we then discuss the discontinuous galerkin time stepping method for the semi discretized equation together with a fully discretized form

concurrent reachability games we consider concurrent two player games with reachability objectives in such games at each round player 1 and player 2 independently and simultaneously choose moves and the two choices determine the next state of the game the objective of player 1 is to reach a set of target states the objective of player 2 is to prevent this these are zero sum games and the reachability objective is one of the most basic objectives determining the set of states from which player 1 can win the game is a fundamental problem in control theory and system verification there are three types of winning states according to the degree of certainty with which player 1 can reach the target from type 1 states player 1 has a deterministic strategy to always reach the target from type 2 states player 1 has a randomized strategy to reach the target with probability 1 from type 3 states player 1 has for every real e 0 a randomized strategy to reach the target with probability greater than email protected we show that for finite state spaces all three sets of winning states can be computed in polynomial time type 1 states in linear time and type 2 and type 3 states in quadratic time the algorithms to compute the three sets of winning states also enable the construction of the winning and spoiling strategies

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

error estimates on a new nonlinear galerkin method based on two grid finite elements a new nonlinear galerkin method based on finite element discretization is presented in this paper for semilinear parabolic equations the new scheme is based on two different finite element spaces defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h nonlinearity and time dependence are both treated on the coarse space and only a fixed stationary equation needs to be solved on the fine space at each time with linear finite element discretizations it is proved that the difference between the new nonlinear galerkin solution and the standard galerkin solution in h 1 omega norm is of the order of h 3

lp error estimates of two grid schemes of expanded mixed finite element methods abstract we present a scheme for solving two dimensional semi linear reaction diffusion equations using an expanded mixed finite element method to linearize the mixed method equations we use a two grid scheme based on the newton iteration method we give the l p error estimate to the variable p in the two grid solution it is shown that the algorithm achieve asymptotically optimal approximation applying the two grid methods as long as the mesh sizes satisfy h o h 1 3 for semi linear problems

superconvergence of numerical solutions to weakly singular volterra integro differential equations we discuss the application of a class of spline collocation methods to first order volterra integro differential equations vides which contain a weakly singular kernel t s with 0 1 it will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and graded meshes the grading exponents of graded meshes used are not greater thanm the polynomial degree which is independent of this is in contrast to the theories of spline collocation methods for volterra or fredholm integral equation of the second kind numerical examples are given to illustrate the theoretical results

superconvergence analysis of two grid fem for maxwell s equations with a thermal effect abstract based on two grid algorithm we develop the superconvergence analysis of fully discrete scheme with the lowest order nedelec element and crank nicolson scheme for the magneto heat coupling model which is also considered as maxwell s equations with a thermal effect then main process of numerical analysis has two parts on one hand we utilize newton type taylor expansion on superconvergent solutions for those nonlinear terms on coarse mesh differing from the numerical solution which makes our given two grid method successful on the other hand we prove the solutions on the fine mesh to be higher accuracy by the postprocessing interpolation technique such a design is conducive to improving the computational accuracy in space and decreasing time consumption simultaneously by employing the skill we can obtain the convergent rate of o t 2 h 2 h 3 which means that the space mesh size should satisfy h o h 3 2 finally the numerical example is presented to verify the theoretical results

subgroup graph methods for presentations of finitely generated groups and the connectivity of associated simplicial complexes abstract in this article we generalize the theory of subgroup graphs of subgroups of free groups developed by i kapovich and a myasnikov based on a work by j stallings to finite index subgroups of finitely generated groups given a presentation of a finitely generated group g x r and a finite connected x regular graph which fulfills the relators r we associate to a finite index subgroup h of g conversely the schreier coset graph of h with respect to x and g is such a graph firstly we study and prove various properties of h in relation to the graph secondly we prove that for many finitely generated infinite groups the order and nerve complexes that we associate to g are contractible in particular this is the case for free and free abelian groups fuchsian groups of genus g 2 infinite right angled coxeter groups artin and pure braid groups infinite virtually cyclic groups baumslag solitar groups as well as the free product of at least two of these and all finite index subgroups of these groups

a new expanded mixed method for parabolic integro differential equations a new expanded mixed scheme is studied and analyzed for linear parabolic integro differential equations the proposed method s gradient belongs to the simple square integrable space replacing the classical h div space the new expanded mixed projection is introduced the existence and uniqueness of solution for semi discrete scheme are proved and the fully discrete error estimates based on both backward euler scheme are obtained moreover the optimal a priori error estimates in l 2 and h 1 norm for the scalar unknown u and the error results in l 2 norm for its gradient and its flux the coefficients times the negative gradient are derived finally some numerical results are calculated to verify our theoretical analysis

a two grid finite difference method for the primitive equations of the ocean abstract a two level method is proposed for the finite difference approximation of a class of dissipative systems including the primitive equations of the ocean the two level or multilevel method amounts to solving a nonlinear problem on the single coarsest grid subsequent approximations are generated on a succession of refined meshes by solving a linearized problem about the solution on the previous level the stability analysis for an euler type scheme shows that the multilevel method can allow larger time steps than the traditional one level methods some numerical simulations of the primitive equations of the ocean using this new multi scale algorithm are presented to illustrate the method

order complexes of coset posets of finite groups are not contractible abstract we show that the order complex of the poset of all cosets of all proper subgroups of a finite group g is never f 2 acyclic and therefore never contractible this settles a question of k s brown

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

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

numerical solution to a mixed navier stokes darcy model by the two grid approach we study numerical methods for a coupled navier stokes darcy model which describes a fluid flow filtrating through porous media a decoupled and linearized two grid algorithm is proposed numerical analysis and experiments are presented to show the efficiency and effectiveness of the decoupled and linearized algorithm

error estimates for semidiscrete finite element methods for parabolic integro differential equations application des methodes d elements finis aux equations integro differentielles de type parabolique avec un noyau integral forme par un operateur aux derivees partielles d ordre 2

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

a two grid method for expanded mixed finite element solution of semilinear reaction diffusion equations we present a scheme for solving two dimensional semilinear reaction diffusion equations using an expanded mixed finite element method to linearize the mixed method equations we use a two grid algorithm based on the newton iteration method the solution of a non linear system on the fine space is reduced to the solution of two small one linear and one non linear systems on the coarse space and a linear system on the fine space it is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h1 3 as a result solving such a large class of non linear equation will not be much more difficult than solving one single linearized equation copyright 2003 john wiley sons ltd

optimal long time l_p 0 t stability and semidiscrete error estimates for the volterra formulation of the linear quasistatic viscoelasticity problem the purpose of this article is to show how the solution of the linear quasistatic compressible viscoelasticity problem written in volterra form with fading memory may be sharply bounded in terms of the data if certain physically reasonable assumptions are satisfied the bounds are derived by making precise assumptions on the memory term which then make it possible to avoid the gronwall inequality and use instead a comparison theorem which is more sensitive to the physics of the problem once the data stability estimates are established we apply the technique also to deriving a priori error bounds for semidiscrete finite element approximations our bounds are derived for viscoelastic solids and fluids under the small strain assumption in terms of the eigenvalues of a certain matrix derived from the stress relaxation tensor for isotropic materials we can be explicit about the form of these bounds while for the general case we give a formula for their computation

fuchsian groups coverings of riemann surfaces subgroup growth random quotients and random walks abstract fuchsian groups acting as isometries of the hyperbolic plane occur naturally in geometry combinatorial group theory and other contexts we use character theoretic and probabilistic methods to study the spaces of homomorphisms from fuchsian groups to symmetric groups we obtain a wide variety of applications ranging from counting branched coverings of riemann surfaces to subgroup growth and random finite quotients of fuchsian groups as well as random walks on symmetric groups in particular we show that in some sense almost all homomorphisms from a fuchsian group to alternating groups an are surjective and this implies higman s conjecture that every fuchsian group surjects onto all large enough alternating groups as a very special case we obtain a random hurwitz generation of an namely random generation by two elements of orders 2 and 3 whose product has order 7 we also establish the analogue of higman s conjecture for symmetric groups we apply these results to branched coverings of riemann surfaces showing that under some assumptions on the ramification types their monodromy group is almost always sn or an another application concerns subgroup growth we show that a fuchsian group has n o 1 index n subgroups where is the measure of and derive similar estimates for so called eisenstein numbers of coverings of riemann surfaces a final application concerns random walks on alternating and symmetric groups we give necessary and sufficient conditions for a collection of almost homogeneous conjugacy classes in an to have product equal to an almost uniformly pointwise our methods involve some new asymptotic results for degrees and values of irreducible characters of symmetric groups

cα regularity for non autonomous linear integrodifferential equations of parabolic type abstract linear integrodifferential equations in general banach space are studied and applications are given to linear integrodifferential partial differential equations

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

ritz volterra projections to finite element spaces and applications to integrodifferential and related equations the object of this paper is to investigate the convergence of finite element approximations to solutions of parabolic and hyperbolic integrodifferential equations and also of equations of sobolev and viscoelasticity type the concept of ritz volterra projection will be seen to unify much of the analysis for the different types of problems optimal order error estimates are obtained in l p for 2 leqq p infty and almost optimal order pointwise results given

maximum norm stability and maximal l p lp regularity of fems for parabolic equations with lipschitz continuous coefficients in this paper we study the semi discrete galerkin finite element method for parabolic equations with lipschitz continuous coefficients we prove the maximum norm stability of the semigroup generated by the corresponding elliptic finite element operator and prove the space time stability of the parabolic projection onto the finite element space in l infty q t l qt and l p 0 t l q omega lp 0 t lq 1

a two grid method for finite volume element approximations of second order nonlinear hyperbolic equations the two grid method is studied for solving a two dimensional second order nonlinear hyperbolic equation using finite volume element method the method is based on two different finite element spaces defined on one coarse grid with grid size h and one fine grid with grid size h respectively the nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step it is proved that the coarse grid can be much coarser than the fine grid a prior error estimate in the h 1 norm is proved to be o h h 3 lnh for the two grid semidiscrete finite volume element method with these proposed techniques solving such a large class of second order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation finally a numerical example is presented to validate the usefulness and efficiency of the method

two grid finite volume element method for linear and nonlinear elliptic problems two grid finite volume element discretization techniques based on two linear conforming finite element spaces on one coarse and one fine grid are presented for the two dimensional second order non selfadjoint and indefinite linear elliptic problems and the two dimensional second order nonlinear elliptic problems with the proposed techniques solving the non selfadjoint and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem on the fine space and solving the non selfadjoint and indefinite elliptic problem on a much smaller space solving a nonlinear elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic problem on a much smaller space convergence estimates are derived to justify the efficiency of the proposed two grid algorithms a set of numerical examples are presented to confirm the estimates

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

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

on the numerical approximation of an evolution problem in nonlinear viscoelasticity abstract an error analysis is furnished for the spatial approximation via the finite element method of certain nonlinear time dependent problems emanating in the theory of viscoelastic solids numerical examples are given first for a model equation and then for numerical equilibrium solutions in cases of a nonconvex double well potential and a convex potential the latter derived from the nonconvex one via relaxation by an equal area construction

a remark on split least squares mixed element procedures for pseudo parabolic equations abstract in this paper we introduce two novel split least squares mixed element procedures for pseudo parabolic equations by selecting the least squares functional properly each procedure can be split into two independent symmetric positive definite sub procedures one of sub procedures is for the primitive unknown variable u which is the same as the standard galerkin finite element procedure and the other is for the introduced flux variable optimal order error estimates are developed a numerical example is given to show the efficiency of the introduced schemes

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

a meshless method on non fickian flows with mixing length growth in porous media based on radial basis functions a comparative study the present study aims to introduce a solution for parabolic integro differential equations arising in heat conduction in materials with memory which naturally occur in many applications two radial basis functions rbfs collocation schemes are employed for solving this equation the first method tested is an unsymmetric method and the second one which appears to be more efficient is a symmetric one the convergence of these two schemes is accelerated as we use the cartesian nodes as the center nodes

error estimates with sharp constants for a fading memory volterra problem in linear solid viscoelasticity the problem characterizing nonageing linear isothermal quasi static isotropic compressible solid viscoelasticity in the time interval 0 t is described this is essentially a volterra equation of the second kind arrived at by adding smooth fading memory to the elliptic linear elasticity equations we analyze the errors resulting from replacing the relaxation functions with practical approximations in a semidiscrete finite element approximation and in a fully discrete scheme derived by replacing the hereditary integral with the trapezoidal rule for numerical integration the error estimates are sharp in the sense that if certain bounds on the data are independent of t then so also are the constants involved in them this is a consequence of bypassing the usual gronwall lemmas with arguments that are more sensitive to the fading memory of the physical problem

fully nonlinear integrodifferential equations in general banach space on etudie des equations integrodifferentielles non lineaires dans un espace de banach qui peuvent etre considerees comme la version abstraite d equations integrodifferentielles partielles non lineaires on considere un probleme aux valeurs limites et initiales