an l2 quotient algorithm for finitely presented groups abstract the paper develops algorithmic methods to enumerate all normal subgroups of a finitely presented group such that the factor groups are either isomorphic to psl 2 p n or to pgl 2 p n the case of two generators is treated in detail a range of examples starting with the free group on two generators and ending with groups having only finitely many normal subgroups of this type is discussed

algorithmically insoluble problems about finitely presented solvable groups lie and associative algebras 1 1 in 1955 novikov 111 constructed a finitely presented group with an insoluble word problem cf miller lo and also boone 8 shortly afterwards first adyan l 4 and then rabin 13 143 used this group of novikov to prove that for every one of a host of group theoretic properties there is no algorithm whereby one can decide whether any finitely presented group has such a property in particular there is no algorithm whereby one can decide whether any finitely presented group is of order 1 thus the isomorphism problem for finitely presented groups is algorithmically insoluble these negative results were followed in 1959 by a number of similar ones concerned with elements and subgroups of finitely presented groups baumslag boone and neumann s thus for example there is no algorithm whereby one can decide whether or not any element in a suitably chosen finitely presented group is a proper power in 1981 harlampovich 9 constructed a finitely presented solvable group of derived length 3 with an insoluble word problem this then raised the possibility that many of the results cited above for finitely presented groups can be carried over to finitely presented solvable groups it is not possible to carry over these results en masse to finitely presented solvable groups since it is easy to decide whether any finitely presented solvable group is of order 1 indeed there is an algorithm whereby one can determine whether any finitely presented solvable group is polycyclic baumslag cannonito and miller 6 again no such algorithm exists for finitely presented groups as a whole adyan 2 the methods employed previously cannot be applied to solvable groups because the constructions

on some closure properties of the non abelian tensor product abstract we prove that the class of nilpotent by finite solvable by finite polycyclic by finite nilpotent of nilpotency class n and supersolvable groups are closed under the formation of the non abelian tensor product we provide necessary and sufficient conditions for the non abelian tensor product of finitely generated groups to be finitely generated

isoperimetric inequalities and the homology of groups an isoperimetric function for a finitely presented group g bounds the number of defining relations needed to prove that a word w g 1 in terms of the length of w suppose g f r is a finitely presented group where f is a finitely generated free group and r is the normal closure of a finite set of relators for each w r we define the area w to be the least number of uses of the given relators needed to prove that w g 1 the isoperimetric function g n is defined to be the maximum of the areas of all the words in r of length n one compares the size of such functions using a suitable order 1 on functions different presentations give rise to equivalent isoperimetric functions where is the equivalence relation determined by 1 interest in such functions was stimulated by gromov 13 who characterized word hyperbolic groups as those with linear isoperimetric function that is g n n further information was obtained by cannon et al 8 who showed that automatic groups satisfy g 1 n and that asynchronously automatic groups satisfy g 1 2 they also showed the free nilpotent group of class 2 and rank 2 has g n n gersten 10 used geometric arguments to obtain interesting lower bounds of exponential type for certain one relator groups he exploited the fact that for a torsion free one relator groupg the usual 2 dimensional complex having fundamental group isomorphic to g is aspherical the general technique for giving upper bounds for g n is to describe a solution to the word problem for g and analyze its use of defining relations it seems quite difficult to establish lower bounds for g n in this manner in this paper we begin a more algebraic study of isoperimetric inequalities and introduce new methods for establishing lower bounds for g instead of trying to analyze r directly we consider some of its quotient groups for which additional machinery is available for instance we consider for each w r its image w r f in r r f and define a suitable centralized isoperimetric function g n such that g n 1 g n using the hopf formula it can be shown that r r f is a direct sum of the second homology group h2 g z with a free abelian

a posteriori error estimators for the upstream weighting mixed methods for convection diffusion problems abstract in this paper we design a posteriori error estimators for the mixed finite element approximation of the convection diffusion problems with dominant convection which exhibit solutions with internal layers and boundary layers a discontinuous upstream weighting scheme in association with the mixed method is used for the convection term the a posteriori error estimator yields global upper and local lower bounds on the error measured in the l2 norm we present several numerical results to show the performance of the error estimator we find that our estimator is not only reliable and efficient but also robust for various numerical examples

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

a posteriori error estimates for the mortar mixed finite element method several a posteriori error estimators for mortar mixed finite element discretizations of elliptic equations are derived a residual based estimator provides optimal upper and lower bounds for the pressure error an efficient and reliable estimator for the velocity and mortar pressure error is also derived which is based on solving local element problems in a higher order space the interface flux jump term that appears in the estimators can be used as an indicator for driving an adaptive process for the mortar grids only

adaptive multigrid method using duality in plane elasticity this work presents an adaptive multigrid method for the mixed formulation of plane elasticity problems first a mixed hybrid formulation is introduced where the continuity of the normal components of the stress tensor is indirectly imposed using a lagrange multiplier two different numerical approximations naturally associated with the primal problem and the dual problem are then proposed the complementary energy principle provides an a posteriori error estimate for the effective solving of both systems of equations a non standard multigrid algorithm has been designed that allows us to solve the two problems dual and primal with reasonable cost and in an integrated way finally a significant numerical application is presented to check the efficiency of the error estimator and the good performance of the algorithm copyright 2001 john wiley sons ltd

a priori and a posteriori analysis of mixed finite element methods for nonlinear elliptic equations we study the mixed finite element approximation of the second order elliptic problem with gradient nonlinearities in two and three space dimensions existence and uniqueness of the approximate solution are proved and optimal order a priori error estimates in l m omega are obtained also reliable and efficient a posteriori error estimators measured in the l m omega norm are derived numerical examples are provided to illustrate the performance of the proposed estimator

minimum hellinger distance estimation for multivariate location and covariance abstract the hellinger distance between a nonparametric density estimator and a model family is minimized to produce estimates of location and covariance in multivariate data with suitable restrictions on the density estimators and the model family these minimum hellinger distance estimators mhde s are shown to be affine invariant consistent and asymptotically normal the robustness of the mhde as measured by the breakdown point compares favorably against the previously studied m estimators monte carlo results suggest that the mhde s are an attractive robust alternative to the usual sample means and covariance matrix

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

some computations of non abelian tensor products of groups on considere le produit tensoriel generalise g h des groupes g h on presente le calcul de g g

ideals in computable rings abstract we show that the existence of a nontrivial proper ideal in a commutative ring with identity which is not a field is equivalent to wkl 0 over rca 0 and that the existence of a nontrivial proper finitely generated ideal in a commutative ring with identity which is not a field is equivalent to aca 0 over rca 0 we also prove that there are computable commutative rings with identity where the nilradical is 1 0 complete and the jacobson radical is 2 0 complete respectively

multilevel schwarz methods we consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations a class of multilevel algorithms are studied using the additive schwarz framework we establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels this is an improvement on dryja and widlund s result on a multilevel additive schwarz algorithm as well as bramble pasciak and xu s result on the bpx algorithm some multiplicative variants of the multilevel methods are also considered we establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one which is independent of the number of levels for a proper ordering the iteration operators correspond to the error propagation operators of certain v cycle multigrid methods using gauss seidel and damped jacobi methods as smoothers respectively

chain conditions in computable rings friedman simpson and smith showed that over rca 0 the statements every ring has a maximal ideal and every ring has a prime ideal are equivalent to aca 0 and wkl 0 respectively more recently downey lempp and mileti have shown that over rca 0 the statement every ring that is not a field contains a nontrivial ideal is equivalent to wklo in this article we explore the reverse mathematical strength of the classic theorems from commutative algebra which say that every artinian ring is noetherian and every artinian ring is of finite length in particular we show that over rca 0 the former implies wkl 0 and is implied by aca 0 while over rca 0 b 2 the latter is equivalent to aca 0

computable algebra and group embeddings this work grew out of our investigations of the infinitely generated subgroups of finitely presented groups a useful technique for embedding countable groups with sufficiently nice local properties into finitely presented groups was introduced in l this technique was then applied in 1 21 to prove that every countable locally polycyclic group and every countable metabelian group can be embedded in a finitely presented group it became apparent from 2 that effective methods in commutative algebra could be extended to group rings of polycyclic groups and then used to embed certain groups in finitely presented groups in this paper these extensions are carried out with the consequent application that every countable locally abelian by nilpotent group can be embedded in a finitely presented group much of the effective commutative algebra we require seems to be known see for instance 6 9 14 171 unfortunately a systematic account is not available in particular we need a proof of the hilbert basis theorem which

a basic norm equivalence for the theory of multilevel methods subspace decompositions of finite element spaces based onl 2 like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods recently several authors have proved the equivalence of the associated discrete norms with theh 1 norm the present paper gives an elementary self contained derivation of this result which is based on the use ofk functionals known from the theory of interpolation spaces

on a new stopping rule for stochastic approximation a new stopping rule for the robbins monro process based on an f statistic criterion is proposed and its asymptotic behavior established on the basis of evidence obtained through experimental sampling the procedure seems to work well over a wide variety of situations a two stage procedure coupling the new rule with an earlier one proposed by sielken 1973 is recommended for practical use

the algorithmic theory of polycyclic by finite groups a group is said to be polycyclic by finite or a pf group for short if it has a polycyclic normal subgroup of finite index equivalently pf groups are exactly the groups which have a series of finite length whose infinite factors are cyclic by a well known theorem of p hall every pf group is finitely presented and in fact pf groups form the largest known sectionclosed class of finitely presented groups it is this fact that makes pf groups natural objects of study from the algorithmic standpoint the general aim of the algorithmic theory of pf groups can be described as the collection of information about a pf group which can in principle

proper divisibility in computable rings abstract we study divisibility in computable integral domains we develop a technique for coding 2 0 binary trees into the divisibility relation of a computable integral domain we then use this technique to prove two theorems about non atomic integral domains in every atomic integral domain the divisibility relation is well founded we show that this classical theorem is equivalent to aca 0 over rca 0 in every computable non atomic integral domain there is a 3 0 infinite sequence of proper divisions we show that this upper bound cannot be improved to 2 0 in general

the computability definability and proof theory of artinian rings abstract we show that in the context of reverse mathematics wk l 0 weak konig s lemma implies the statement ar t 0 which says that every artinian ring is noetherian over rc a 0 recursive comprehension axiom to achieve this goal we prove a general computable full structure theorem for computable artinian rings similar to the classical version found in most algebra texts

adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems we consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations by a well known postprocessing technique the discrete problem is equivalent to a modified nonconforming discretization which is solved by preconditioned cg iterations using a multilevel preconditioner in the spirit of bramble pasciak and xu designed for standard nonconforming approximations local refinement of the triangulations is based on an a posteriori error estimator which can be easily derived from superconvergence results the performance of the preconditioner and the error estimator is illustrated by several numerical examples

a posteriori error estimates for lowest order mixed finite element discretizations of convection diffusion reaction equations we establish residual a posteriori error estimates for lowest order raviart thomas mixed finite element discretizations of convection diffusion reaction equations on simplicial meshes in two or three space dimensions the upwind mixed scheme is considered as well and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusion dispersion tensor and on a possible convection dominance global upper bounds for the approximation error in the energy norm are derived where in particular all constants are evaluated explicitly so that the estimators are fully computable our estimators give local lower bounds for the error as well and they hold from the cases where convection or reaction are not present to convection or reaction dominated problems we prove that their local efficiency depends only on local variations in the coefficients and on the local peclet number moreover the developed general framework allows for asymptotic exactness and full robustness with respect to inhomogeneities and anisotropies the main idea of the proof is a construction of a locally postprocessed approximate solution using the mean value and the flux in each element known in the mixed finite element method and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem numerical experiments confirm the guaranteed upper bound and excellent efficiency and robustness of the derived estimators

multilevel methods for mixed finite elements in three dimensions in this paper we consider second order scalar elliptic boundary value problems posed over three dimensional domains and their discretization by means of mixed raviart thomas finite elements 18 this leads to saddle point problems featuring a discrete flux vector field as additional unknown following ewing and wang 26 the proposed solution procedure is based on splitting the flux into divergence free components and a remainder it leads to a variational problem involving solenoidal raviart thomas vector fields a fast iterative solution method for this problem is presented it exploits the representation of divergence free vector fields as bf curl s of the bf h bf curl conforming finite element functions introduced by nedelec 43 we show that a nodal multilevel splitting of these finite element spaces gives rise to an optimal preconditioner for the solenoidal variational problem duality techniques in quotient spaces and modern algebraic multigrid theory 50 10 31 are the main tools for the proof

infinite dimensional proper subspaces of computable vector spaces abstract this article examines and distinguishes different techniques for coding incomputable information into infinite dimensional proper subspaces of a computable vector space and is divided into two main parts in the first part we describe different methods for coding into infinite dimensional subspaces more specifically we construct several computable infinite dimensional vector spaces each of which satisfies one of the following 1 every infinite coinfinite dimensional subspace computes turing s halting set 2 every infinite cofinite dimensional proper subspace computes turing s halting set 3 there exists x v such that every infinite dimensional proper subspace not containing x computes turing s halting set 4 every infinite dimensional proper subspace computes turing s halting set vector space 4 generalizes vector spaces 1 and 2 and its construction is more complicated the same simple and natural technique is used to construct vector spaces 1 3 finally we examine the reverse mathematical implications of our constructions 1 4 in the second part we examine the limitations of our simple and natural method for coding into infinite dimensional subspaces described in the previous paragraph in particular we prove that our simple and natural coding technique cannot produce a vector space of type 4 above and that any vector space of type 4 must have densely many from a certain point of view finite dimensional computable subspaces in other words the construction of a vector space of type 4 is necessarily more complicated than the construction of vector spaces of types 1 3 we also introduce a new statement in second order arithmetic about the existence of infinite dimensional proper subspaces in a restricted class of vector spaces related to 1 3 above and show that it is implied by weak konig s lemma in the context of reverse mathematics in the context of reverse mathematics this gives rise to two statements from effective algebra about the existence of infinite dimensional proper subspaces for a certain class of vector spaces of the form v x v a v and v x v b v that each imply a c a 0 over r c a 0 but such that the seemingly weaker statement v x v a v b v is provable via w k l 0 over r c a 0 furthermore we highlight some general similarities between constructing of infinite dimensional proper subspaces of computable vector spaces and constructing solutions to computable instances of various combinatorial principles such as ramsey s theorem for pairs

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

a posteriori error estimates for elliptic problems in two and three space dimensions let u in h be the exact solution of a given selfadjoint elliptic boundary value problem which is approximated by some tilde u in mathcal s mathcal s being a suitable finite element space efficient and reliable a posteriors estimates of the error u tilde u measuring the local quality of tilde u play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh a well known class of error estimates can be derived systematically by localizing the discretized defect problem by using domain decomposition techniques in this paper we provide a guideline for the theoretical analysis of such error estimates we further clarify the relation to other concepts our analysis leads to new error estimates which are specially suited to three space dimensions the theoretical results are illustrated by numerical computations

subspaces of computable vector spaces abstract we show that the existence of a nontrivial proper subspace of a vector space of dimension greater than one over an infinite field is equivalent to wkl 0 over rca 0 and that the existence of a finite dimensional nontrivial proper subspace of such a vector space is equivalent to aca 0 over rca 0

uniform convergence and preconditioning method for mortar mixed element method for nonselfadjoint and indefinite problems abstract in this paper a mortar version of the modified mixed element method is proposed for nonselfadjoint and indefinite second order elliptic problems it is proven that the modified mortar mixed element method is equivalent to a modified mortar nonconforming element method based on this equivalence the existence uniqueness and uniform convergence of the solution for the modified mortar mixed element method are shown under minimal regularity assumption meanwhile an optimal error estimate for the modified mortar mixed element method is obtained under h2 smoothness assumption furthermore an additive schwarz preconditioning method for solving the discrete problem is provided and the nearly optimal convergence rate for the preconditioned gmres method is proven under the minimal regularity assumption

schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems part i algorithms and numerical results we describe sequential and parallel algorithms based on the schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the raviart thomas finite element spaces these lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block gauss seidel or block jacobi methods with overlapping blocks the indefiniteness requires special treatment the sub blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of dryja and widlund for standard elliptic problems if there is sufficient overlap between the subdomains the algorithm converges with a rate independent of the mesh size the number of subdomains and discontinuities of the coefficients extensions of the above algorithms to the case of local grid refinement is also described convergence theory for these algorithms will be presented in a subsequent paper

convergence and optimality of adaptive mixed finite element methods the convergence and optimality of adaptive mixed finite element methods for the poisson equation are established in this paper the main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality a quasi orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result this discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation

a posteriori error estimators for the raviart thomas element when error estimators for the rte are developed two difficulties prevent the success of the straightforward application of frequently used arguments the hdiv norm is an anisotropic norm i e it refers to differential operators of different orders moreover the traces of hdiv functions are only in h 1 2 therefore one does not obtain optimal a posteriori error estimates when using natural norms this drawback is overcome by using mesh dependent norms

a stationary stochastic approximation method abstract a problem occuring in statistics and econometrics is the minimization or root finding of a function of several parameters expressed as an integral if the integral cannot be evaluated analytically and numerical integration is required evaluation can be prohibitively costly when the dimension of integration exceeds two since monte carlo techniques are natural in such a case stochastic approximation can be viewed as combining integration and optimization while these methods require minimal conditions for convergence implementation as a practical problem solver faces difficulties in verification and stopping a stationary stochastic approximation method is proposed which is less sophisticated mathematically this method is applied to two types of problems a bayesian decision problem and the computation of minimum hellinger distance estimators

adaptive finite element methods in computational mechanics we present a general approach to adaptivity for finite element methods and give applications to linear elasticity non linear elasto plasticity and nonlinear conservation laws including numerical results

a residual based error estimator for mortar finite element discretizations a residual based error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite element methods is introduced and analyzed in particular we consider mortar finite element techniques restricting ourselves to geometrically conforming domain decomposition methods using p1 approximations in each subdomain additionally a residual based error estimator for crouzeix raviart elements of lowest order is presented and compared with the error estimator obtained in the more general mortar situation it is shown that the computational effort of the error estimator can be considerably reduced if the special structure of the lagrange multiplier is taken into account

the non abelian tensor product of finite groups is finite si m et n sont des groupes finis ou p groupes agissant l un sur l autre et si leurs actions sont compatibles alors m n est fini ou un p groupe

the algorithmic theory of finitely generated metabelian groups algorithms are constructed which when an explicit presentation of a finitely generated metabelian group g in the variety x 2 is given produce finitary presentations for the derived subgroup g the centre z g the fitting subgroup fit g and the frattini subgroup 0 g additional algorithms of independent interest are developed for commutative algebra which construct the associated set of primes ass m of a finitely generated module m over a finitely generated commutative ring r and the intersection pr m of the maximal submodules of m

schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems part ii convergence theory in this paper we discuss bounds for the convergence rates of several domain decomposition algorithms to solve symmetric indefinite linear systems arising from mixed finite element discretizations of elliptic problems the algorithms include schwarz methods and iterative refinement methods on locally refined grids the implementation of schwarz and iterative refinement algorithms have been discussed in part i a discussion on the stability of mixed discretizations on locally refined grids is included and quantiative estimates for the convergence rates of some iterative refinement algorithms are also derived

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

error reduction and convergence for an adaptive mixed finite element method an adaptive mixed finite element method amfem is designed to guarantee an error reduction also known as saturation property after each refinement step the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms this error reduction property is established here for the raviart thomas finite element method with a reduction factor p 1 uniformly for the l 2 norm of the flux errors our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels the adaptive algorithm surprisingly does not require any particular mesh design unlike the conforming finite element method the new arguments are a discrete local efficiency and a quasi orthogonality estimate the proof does not rely on duality or on regularity

asymptotic properties of least squares estimates in stochastic regression models on etablit la consistance forte des estimations des moindres carres dans des modeles de regression stochastique sous l hypothese que le modele peut etre reparametrise de telle facon que les nouveaux vecteurs soient faiblement correles

unified primal formulation based a priori and a posteriori error analysis of mixed finite element methods we derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second order elliptic problems it is based on the classical primal weak formulation the postprocessing of the potential proposed in t arbogast and z chen on the implementation of mixed methods as nonconforming methods for second order elliptic problems math comp 64 1995 943 972 and the discrete friedrichs inequality our analysis in particular avoids any explicit use of the uniform discrete inf sup condition and in a straightforward manner and under minimal necessary assumptions known convergence and superconvergence results are recovered the same framework then turns out to lead to optimal a posteriori energy error bounds in particular estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived they give a guaranteed and fully computable upper bound on the energy error represent error local lower bounds and are robust under some conditions on the diffusion dispersion tensor they are thus suitable for both overall error control and adaptive mesh refinement moreover the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method we finally prove that in parallel and simultaneously in converse to galerkin finite element methods under some circumstances the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the h 1 0 space and also establish several links between mixed finite element approximations and some generalized weak solutions

multilevel iterative methods for mixed finite element discretizations of elliptic problems for solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed the methods are within the general framework of the product multiplicative scheme for operators in a hilbert space proposed recently by bramble pasciak wang and xu 5 6 26 27 and make use of certain multilevel decomposition of the corresponding spaces for the flux variable

a comparison of a posteriori error estimators for mixed finite element discretizations by raviart thomas elements we consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations in particular we present and analyze four different kinds of error estimators a residual based estimator a hierarchical one error estimators relying on the solution of local subproblems and on a superconvergence result respectively finally we examine the relationship between the presented error estimators and compare their local components

axioms of adaptivity this paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of discrete lower bounds inexact solvers inhomogeneous boundary data or the use of equivalent error estimators solely four axioms guarantee the optimality in terms of the error estimators compared to the state of the art in the temporary literature the improvements of this article can be summarized as follows first a general framework is presented which covers the existing literature on optimality of adaptive schemes the abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method second efficiency of the error estimator is neither needed to prove convergence nor quasi optimal convergence behavior of the error estimator in this paper efficiency exclusively characterizes the approximation classes involved in terms of the best approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant third some general quasi galerkin orthogonality is not only sufficient but also necessary for the r linear convergence of the error estimator which is a fundamental ingredient in the current quasi optimality analysis due to stevenson 2007 finally the general analysis allows for equivalent error estimators and inexact solvers as well as different non homogeneous and mixed boundary conditions

a multiresolution approach to homogenization and effective modal analysis of complex boundary value problems we apply multiresolution techniques to study the effective properties of bound ary value problems conditions under which boundary values are preserved in the effective ho mogenized formulation are developed and discussed relations between the eigenfunctions and eigenvalues of the generic formulation and those of the effective formulation are explored and their dependence on the micro scale is studied applications to the construction of effective green func tions in a complex lamination are discussed the analytic results are demonstrated via numerical computations

on the complexity of radicals in noncommutative rings this article expands upon the recent work by downey et al 2007 3 who classified the complexity of the nilradical and jacobson radical of commutative rings in terms of the arithmetical hierarchy let r be a computable not necessarily commutative ring with identity then it follows from the definitions that the prime radical of r is 11 and the levitzki radical of r is 20 we show that these upper bounds for the complexity of the prime and levitzki radicals are optimal by constructing two noncommutative computable rings with identity such that the prime radical of one is 11 complete while the levitzki radical of the other is 20 complete

convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics abstract a nonlinear system of two coupled partial differential equations models miscible displacement of one incompressible fluid by another in a porous medium conservation of mass for the mixture leads to an elliptic equation for pressure and conservation for the displacing fluid yields a convectiondominated parabolic equation for the concentration of that fluid a sequential implicit time stepping procedure is defined in which the pressure and darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the concentration is approximated by a combination of a galerkin finite element method and the method of characteristics optimal order convergence in l2 is proved time truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the concentration equation temporal and spatial errors are lessened by direct computation of the velocity in the mixed method as opposed to differentiation of the pressure several extensions of these results are outlined

a posteriori error estimation and adaptive mesh refinement techniques abstract we analyse three different a posteriori error estimators for elliptic partial differential equations they are based on the evaluation of local residuals with respect to the strong form of the differential equation on the solution of local problems with neumann boundary conditions and on the solution of local problems with dirichlet boundary conditions we prove that all three are equivalent and yield global upper and local lower bounds for the true error thus adaptive mesh refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities some numerical examples prove the efficiency of the error estimators and the mesh refinement techniques

on a hierarchical basis multilevel method with nonconforming p1 elements we develop a hierarchical basis multilevel method for symmetric second order elliptic boundary value problems in two dimensional polygonal domains based on nonconforming p1 triangular elements the main result is that the condition number of the hierarchical discretization is bounded byo k wherek is the number of refinement levels for comparison yserentant s result yields the sharp boundo k 2 for the corresponding hierarchical discretization with conforming linear elements

optimization using stochastic approximation and monte carlo simulation with application to harvesting of atlantic menhaden summary stochastic approximation is a sequential technique which can be used for optimizing the mean of a random variable y x in terms of a parameter vector x when the mean is unknown but y x can be observed for any value of x this paper is intended to introduce and illustrate stochastic approximation to those statisticians unfamiliar with the technique the atlantic menhaden is a commercially important fish in the herring family clupeidae stochastic approximation is used in conjunction with a stochastic simulation model of the menhaden population to determine optimal values for the two parameters in a proposed class of menhaden harvesting policies the objective is to maximize the expected utility of the random catch over a 25 year period optimization is a response surface problem although some of the more familiar sequential response surface methods might have been successfully applied the nonsequential response surface techniques would have been too expensive for estimation of a specific feature of the surface such as the location of the maximum the results of using stochastic approximation are encouraging atlantic menhaden are migratory fish caught from florida to new england mainly by purse seining menhaden are not palatable to humans but they are processed into two important products fish oil and fish meal the oil is used in the manufacture of paints lubricants and cosmetics and the meal is a high protein ingredient in animal feed together atlantic and gulf menhaden support the largest u s commercial fishery measured in pounds landed and excluding shellfish menhaden catches rank next to those of tuna and salmon in monetary value at present the atlantic menhaden fishery is subjected to some intrastate regulation but it is not managed on a fishery wide basis catches of menhaden experienced a rapid decline after 1962 and have only partially recovered but some biologists feel that proper management could increase future catches substantially to explore the