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

mappings of finite distortion discreteness and openness we establish a sharp integrability condition on the partial derivatives of a mapping with l p integrable distortion for some p n 1 to guarantee discreteness and openness we also show that a mapping with exponentially integrable distortion and integrable jacobian determinant is either constant or both discrete and open we give an example demonstrating the preciseness of our criterion

on monodromy groups of fixed genus abstract let g be the monodromy group of a cover of the riemann sphere by a compact connected surface of genus g assume that g is a primitive group with elementary abelian normal subgroup n it is shown that for any finite genus g 0 there exist only finitely many such groups by establishing a bound linear in g on the order pe of n for genus g 0 there are only finitely many such groups besides the well known groups given by zariski and ritt

mappings of finite distortion a new proof for discreteness and openness we give a new and elementary proof of the known result a mapping of finite distortion f r r is discrete and open provided that its distortion function k lloc if n 2 k lploc for some p n 1 if n 3

regularity properties of deformations with finite energy the purpose of this paper is to point out some regularity properties of a class of functions which play an important role in nonlinear elasticity

improved error estimation of dynamic finite element methods for second order parabolic equations dynamic finite element schemes are analyzed for second order parabolic problems these schemes permit different finite element spaces at different time levels in order to efficiently capture time changing localized phenomena such as moving sharp fronts or layers the dynamical change of grids and interpolation polynomials is necessary and essential to many large scale transient problems standard characteristic and mixed finite element methods with dynamic function spaces are considered for linear and nonlinear problems in a unified framework with improved a priori error estimates and convergence results

slow mappings of finite distortion we examine mappings of finite distortion from euclidean spaces into riemannian manifolds we use integral type isoperimetric inequalities to obtain liouville type growth results under mild assumptions on the distortion of the mappings and the geometry of the manifolds

mappings of finite distortion monotonicity and continuity we study mappings f f1 fn rn in the sobolev space w loc r n where is a connected open subset of rn with n 2 thus for almost every x we can speak of the linear transformation d f x rn rn called differential of f at x its norm is defined by d f x sup d f x h h sn 1 we shall often identify d f x with its matrix and denote by j x f det d f x the jacobian determinant thus using the language of differential forms we can write

endomorphism algebras of jacobians very little is known about the existence of curves and families of curves whose jacobians are acted on by large rings of endomorphisms in this paper we show the existence of curves x with an injection k hom jac x jac x z q where k is a subfield of even index at most 10 in a primitive cyclotomic field q p or a subfield of index 2 in q pq or q p this result generalizes previous work of brumer mestre and tautz top and verberkmoes our curves are constructed as branched covers of the projective line and the endomorphisms arise as quotients of double coset algebras of the galois groups of these coverings in certain cases we show the existence of continuous families of jacobians admitting the desired endomorphism algebras at the end we raise some questions about upper bounds for endomorphism algebras of the type described here and ask about a non galois version of hurwitz s theorem

contact between nonlinearly elastic bodies we study the contact between nonlinearly elastic bodies by variational methods after the formulation of the mechanical problem we provide existence results based on polyconvexity and on quasiconvexity then we derive the euler lagrange equation as a necessary condition for minimizers here clarke s generalized gradients are the essential tool to treat the nonsmooth obstacle condition

approximation properties of the h p version of the finite element method abstract the paper surveys the approximation and convergence properties of the h p version of the finite element method the basic theoretical results together with the main ideas of the proofs numerical examples are given

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

a characteristic mixed method with dynamic finite element space for convection dominated diffusion problems abstract a method for numerically solving convection dominated diffusion problems using a combination of the method of characteristics with the mixed finite element method is presented and analyzed in this paper different numbers of elements and different basis functions are implemented at different time levels in order to accurately resolve time changing critical features this new method provides the ability to obtain approximations of the solution and its gradient simultaneously adopt large time steps and make grid refinements and basis function adjustments at any time necessary convergence analysis and error estimates are established furthermore comments are made on the efficiency and capability of the method for accurately solving transient problems with localized phenomena such as fronts shocks and boundary layers

a moving mesh numerical method for hyperbolic conservation laws we show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in l1 r to within o n 2 by a piecewise linear function with o fn nodes the nodes are moved according to the method of characteristics we also show that a previous method of dafermos which uses piecewise constant approxima tions is accurate to o n 1 these numerical methods for conservation laws are the first to have proven convergence rates of greater than o fn 1 2 1 introduction it is well known that the solution of the hyperbolic conservation law

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

dynamic domain decomposition and grid modification for parabolic problems abstract a dynamic grid modification and domain decomposition method is given and analyzed for parabolic problems this method allows one to apply different domain decompositions and different grids and interpolation polynomials on the subdomains at different time levels when necessary the procedure relies on an implicit galerkin method in the subdomains and explicit flux calculation on the inter domain boundaries in addition a dynamic finite element scheme is proposed and analyzed which is applicable to general parabolic problems these methods are well suited to large scale time dependent problems involving localized phenomena such as sharp fronts or layers which also change with time convergence and stability analyses in the l 2 norm are given numerical experiments are provided to check the performance of the methods and make comparison with other methods

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

analysis of some moving space time finite element methods two space time finite element methods for solving time dependent partial differential equations are defined and analyzed the methods are based on the use of isoparametric finite elements to implicitly define the time discretization on a moving mesh in the space dimensions one method allows for adding and deleting knots in a continuous fashion while the other allows for discontinuous changes in the mesh static rezone a detailed convergence analysis for a model parabolic equation with a possibly large convection term is presented here the authors obtain symmetric best approximation error estimates similar to those obtained by dupont math comp 39 1982 pp 85 107 for the semidiscrete case

mappings with integrable dilatation in higher dimensions let f w 1 n loc r be a mapping with nonnegative jacobian jf x detdf x 0 for a e x in a domain r n the dilatation of f is defined almost everywhere in by the formula k x df x jf x iwaniec and sverak is have conjectured that if p n 1 and k lploc then f must be continuous discrete and open moreover they have confirmed this conjecture in the two dimensional case n 2 in this article we verify it in the higher dimensional case n 2 whenever p n 1

mappings of finite distortion the degree of regularity this paper investigates the self improving integrability properties of the so called mappings of finite distortion let k x 1 be a measurable function defined on a domain rn n 2 and such that exp k x lloc1 0 we show that there exist two universal constants c1 n c2 n with the following property let f be a mapping in wloc1 1 rn with df x n k x j x f for a e x and such that the jacobian determinant j x f is locally in l1log c1 n l then automatically j x f is locally in l1logc2 n l this result constitutes the appropriate analog for the self improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion namely we obtain novel results on the size of removable singularities for bounded mappings of finite distortion and on the area distortion under this class of mappings

adaptive finite element methods for parabolic problems v long time integration we continue our previous work on adaptive finite element methods for parabolic problems now with particular emphasis on long time integration for semidefinite problems

an analysis of the moving finite element procedure the moving finite element procedure of miller siam j numer anal 18 1981 pp 1019 1057 is analysed and broken down into a number of steps the central step is a mapping from element velocities to nodal velocities which is exact for a class of one dimensional scalar nonlinear partial differential equation pdes for more general pdes the mapping is preceded by an element by element projection step in higher dimensions the mapping is followed by node by node projection for first order pdes the procedure gives rise to ordinary differential equation systems approximating the method of characteristics for second order operators the behaviour of the nodes can only be described qualitatively the central mapping is a legendre transformation from the data to the nodal velocities while the full procedure is an approximate legendre transformation the same transformation can be used to construct a dual pde which may be numerically easier to solve the structure is used to analyse nodal movement and to giv

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

adaptive finite element methods for parabolic problems ii optimal error estimates in l l 2 and l l optimal error estimates are derived for a complete discretization of linear parabolic problems using space time finite elements the discretization is done first in time using the discontinuous galerkin method and then in space using the standard galerkin method the underlying partitions in time and space need not be quasi uniform and the partition in space may be changed from time step to time step the error bounds show in particular that the error may be controlled globally in time on a given tolerance level by controlling the discretization error on each individual time step on the same given level i e without error accumulation effects the derivation of the estimates is based on the orthogonality of the galerkin procedure and the use of strong stability estimates the particular and precise form of these error estimates makes it possible to design efficient adaptive methods with reliable automatic error control for parabolic problems in the norms under consideration

quasiconformal 4 manifolds for any pseudo group of homeomorphism of euclidean space one can define the corresponding category of manifolds the most familiar examples in topology are the full pseudo group of homeomorphisms giving rise to the theory of topological manifolds and the subgroup of smooth differomorphisms giving rise to the theory of c manifolds in this paper we discuss an intermediate category quasiconformal homeomorphisms and manifolds recall that a homeomorphism 9 d r n from a domain d in r n to its image 9 d is k quasiconformal if for all x in d

mappings of finite distortion hausdorff measure of zero sets we prove that a for a mapping f of finite distortion k in l p n p the n p hausdorff measure of any point preimage is zero provided j f is integrable df in l s with s p and the multiplicity function of f is essentially bounded as a consequence for p n 1 we obtain that the mapping is then open and discrete

optimal assumptions for discreteness there exists in every dimension n geq 3 a lipschitz mapping of finite distortion such that the outer distortion satisfies k in l n 1 but it maps a line segment to a point

mixed methods with dynamic finite element spaces for miscible displacement in porous media abstract miscible displacement in porous media is modeled by a nonlinear coupled system of two partial differential equations we approximate the pressure equation which is elliptic and the concentration equation which is parabolic but normally convection dominated by the mixed methods with dynamic finite element spaces i e different number of elements and different basis functions are adopted at different time levels and the approximate concentration is projected onto the next finite element space in weighted l 2 norm for starting a new time step this allows us to make local grid refinements or unrefinements and basis function improvements two fully discrete schemes are presented and analysed error estimates show that these methods have optimal convergent rate in some sense the efficiency and capability of the dynamic finite element method are commented for accurately solving time dependent problems with localized phenomena such as fronts shocks and boundary layers

grid modification for second order hyperbolic problems a family of galerkin finite element methods is presented to accurately and efficiently solve the wave equation that includes sharp propagating wave fronts the new methodology involves different finite element discretizations at different time levels thus at any time level relatively coarse grids can be applied in regions where the solution changes smoothly while finer grids can be employed near wave fronts the change of grid from time step to time step need not be continuous and the number of grid points at different time levels can be arbitrarily different the formulation is applicable to general second order hyperbolic equations stability results are proved and a priori error estimates are established for several boundary conditions our error estimates consist of three parts the time finite difference discretization error the spatial finite element discretization error and the error due to the projections of the approximated solution from old grids onto new grids

distortion of mappings and lq p cohomology we study some relation between some geometrically defined classes of diffeomorphisms between manifolds and the l q p cohomology of these manifolds we apply these results to the l q p cohomology of a manifold with a cusp

simulation of miscible displacement in porous media by a modified uzawa s algorithm combined with a characteristic method abstract miscible displacement in porous media is modeled by a nonlinear coupled system of two partial differential equations the pressure equation which is elliptic and the concentration equation which is parabolic but usually convection dominated in this paper we focus on some computational difficulties in the simulation of this problem in order to obtain accurate approximation for velocity we implement a modified uzawa s algorithm for the pressure equation with a full permeability tensor including strong heterogeneity and anisotropy for handling the convection dominance over diffusion we employ a modified method of characteristics for the concentration equation numerical simulation results show that the combination of the modified method of characteristics and the modified uzawa s algorithm is a good choice for the problem in particular the modified method of characteristics dramatically reduces numerical diffusion and non physical oscillation in the approximate concentration and the modified uzawa s algorithm leads to positive definite linear systems with velocity unknowns only and exhibits the accuracy of mixed finite element methods

grid modification for the wave equation with attenuation the wave equation with attenuation due to a linear friction is approximated by a new mixed finite element method which allows one to use different grids and basis functions at different times when necessary this method enables one to track sharp moving wave fronts more efficiently and accurately error estimates with optimal convergent rates are established unconditional stability is also proved for this method

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

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

different domain decompositions at different times for capturing moving local phenomena abstract a conservative galerkin domain decomposition method for time dependent problems is given and analyzed this method allows one to apply different domain decompositions at different time levels when necessary in order to capture time changing local phenomena such as propagating fronts or moving layers error estimates in the energy and l 2 norms are established numerical results are presented

an application of gauge theory to four dimensional topology soit x une u variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee q est definie positive alors cette forme est equivalente sur les entiers a la forme diagonale standard soit dans une base q u 1 u 2 u 2 u 2 1 u 2 2 u 2 2