preconditioned conjugate gradient methods for three dimensional linear elasticity finite element modelling of three dimensional elasticity problems give rise to large sparse matrices various preconditioning methods are developed for use in preconditioned conjugate gradient iterative solution techniques incomplete factorizations based on levels of fill drop tolerance and a two level hierarchical basis are developed various techniques for ensuring that the incomplete factors have positive pivots are presented computational tests are carried out for problems generated using unstructured tetrahedral meshes quadratic basis functions are used the performance of the iterative methods is compared to a standard direct sparse matrix solver problems with up to 70 000 degrees of freedom and small 1 element aspect ratio are considered

convergence to equilibrium of gradient flows defined on planar curves we consider the evolution of open planar curves by the steepest descent flow of a geometric functional with different boundary conditions we prove that if any set of stationary solutions with fixed energy is finite then a solution of the flow converges to a stationary solution as time goes to infinity we also present a few applications of this result

on birational macaulayfications and cohen macaulay canonical modules abstract the aim of the paper is twofold at first there is a characterization of those local domains admitting a birational macaulayfication it turns out that a local domain a quotient of a gorenstein ring possesses a cohen macaulay birational extension b if and only if its canonical module k a is a cohen macaulay module in this situation it follows that b is isomorphic to the endomorphism ring of k a the second part of the paper is devoted to the question when the canonical module of a local ring is a cohen macaulay module there are applications on sequentially cohen macaulay rings a large class of rings whose canonical module are cohen macaulay is given by the simplicial affine semigroup rings as a technical tool there is a spectral sequence related to the duality with respect to the dualizing complex

cohomological dimension cofiniteness and abelian categories of cofinite modules abstract let r be a commutative noetherian ring i j be ideals of r and m be a finitely generated r module in this paper it is shown that q j m q i m cd j m i m furthermore it is shown that for any ideal i of r and any finitely generated r module m with q i m 1 the local cohomology modules h i i m are i cofinite for all integers i 0 as a consequence of this result it is shown that if q i r 1 then for any finitely generated r module m the local cohomology modules h i i m are i cofinite for all integers i 0 finally it is shown that the category of all i cofinite r modules c r i c o f is an abelian subcategory of the category of all r modules whenever r m is a complete noetherian local ring and i is an ideal of r with q i r 1 these assertions answer affirmatively two questions raised by r hartshorne in 16 in the some special cases

finiteness properties of local cohomology modules an application of d modules to commutative algebra the main goal of this paper is to establish finiteness properties of local cohomology modules in characteristic 0 that would be analogous to those proven by c huneke and r sharp in characteristicp 0 our method based on the theory of algebraicd modules seems to be the first application ofd modules to commutative algebra

on the characterizations of cofinite complexes over affine curves and hypersurfaces abstract let a be a homomorphic image of a gorenstein ring of finite krull dimension j an ideal of a of dimension one and n a bounded below complex of a modules suppose that a is complete with respect to a j adic topology in this paper we prove that n is a j cofinite complex if and only if h i n is a j cofinite module for all i the same result is also proved for principal ideals j consequently for the fourth question given by r hartshorne cf 6 fourth question p 149 we obtain an answer over the ring on affine curves and hypersurfaces

cofiniteness with respect to ideals of dimension one we prove that the category of modules cofinite with respect to an ideal of dimension one in a noetherian ring is a full abelian subcategory of the category of modules the proof is based on a criterion for cofiniteness with respect to an ideal of dimension one namely for such ideals it suffices that the two first ext modules in the definition for cofiniteness are finitely generated this criterion is also used to prove very simply that all local cohomology modules of a finitely generated module with respect to an ideal of dimension one in an arbitrary noetherian ring are cofinite with respect to the ideal

on the vanishing of local cohomology of the absolute integral closure in positive characteristic abstract the aim of this paper is to extend the main result of c huneke and g lyubeznik in adv math 210 2007 498 504 to the class of rings that are images of cohen macaulay local rings namely let r be a local noetherian domain of positive characteristic that is an image of a cohen macaulay local ring we prove that all local cohomology of r below the dimension maps to zero in a finite extension of the ring as a direct consequence we obtain that the absolute integral closure of r is a big cohen macaulay algebra since every excellent local ring is an image of a cohen macaulay local ring this result is a generalization of the main result of m hochster and huneke in ann of math 135 1992 45 79 with a simpler proof

formulation and solution of hierarchical finite element equations the paper presents a general hierarchical formulation applicable to both elliptic and hyperbolic problems static and eigenvalue linear elastic problems as well as convection diffusion problems are studied the hierarchical formulation is well suited for adaptive procedures for the convection diffusion problem the hierarchical approximation is made in time only different hierarchical functions are proposed for different types of problems both weighted residual and least squares formulations are applied a combination of these two gives a penalty method with a constraint equation corresponding to the least squares method a whole class of time integration formulae is obtained these are all suitable for adaptive procedures owing to the hierarchical approximation in the time domain if a linear discontinuous hierarchical base function is used in the galerkin weak formulation the method so obtained corresponds to the discontinuous galerkin method in time and is especially suited for convection dominated problems the streamline diffusion method is found to be the aforementioned penalty method this paper also examines the sequence of nested equation systems that results from a hierarchical finite element formulation properties of these systems arising from static problems are investigated the paper presents some new possibilities for iterative solution of hierarchic element equations and different procedures are compared in a numerical example finally a simple id convection diffusion problem clearly shows that the proposed hierarchical formulation in time gives a stable and accurate solution even for convection dominated flow

a comparison of preconditioned iterative techniques using rapid operator application against direct solution methods the p version of the finite element method which uses high order hierarchic basis functions within an element allows for ease of adaptive computation and high accuracy with relatively few degrees of freedom however the solution time and storage requirements for large 3 d problems can be very high when direct methods are used to factor global matrices iterative techniques with rapid operator application have minimal storage requirements because neither element nor global matrices are formed but may not converge rapidly enough we study an approach that attempts to combine the two methods by using direct factorization of the matrix corresponding to low level polynomials as a preconditioner to an iterative method at a higher polynomial level this resulting solution scheme converges more rapidly than an unpreconditioned scheme but has much lower storage requirements than direct methods we illustrate the effectiveness of this scheme on some large three dimensional problems in structural analysis

cofiniteness of local cohomology modules for ideals of small dimension abstract let r be a commutative noetherian ring and let m be a non zero finitely generated r module let i be an ideal of r and t a non negative integer such that dim supp h i i m 1 for all i t it is shown that the r modules h i 0 m h i 1 m h i t 1 m are i cofinite and the r module hom r r i h i t m is finitely generated this immediately implies that if i has dimension one i e dim r i 1 then h i i m is i cofinite for all i 0 this is a generalization of the main results of delfino and marley d delfino t marley cofinite modules and local cohomology j pure appl algebra 121 1997 45 52 and yoshida k i yoshida cofiniteness of local cohomology modules for ideals of dimension one nagoya math j 147 1997 179 191 for an arbitrary noetherian ring r also we prove that if r is local and dim supp h i i m 2 for all i t then the r modules ext r j r i h i i m and hom r r i h i t m are weakly laskerian for all i t and all j 0 as a consequence it follows that the set of associated primes of h i i m is finite for all i 0 whenever dim r i 2

local cohomology of certain rees and form rings ii recently several authors have studied rees rings i ona0 i and form rings gr z r z z r z n zn zn of particular ideals z of a noetherian ring r the most important motivation to look at these rings is of geometric nature namely to study the local structure of conormal cones blow up and their exceptional divisors to be explicit recall that for an algebraic variety v and an ideal y of its structure sheaf the conormal cone of x is given by the scheme spec gr x whereas the blow up of v at y is defined by bly y proj x tiv p v the exceptional fiber or divisor w v s i of the blow up is given by proj gr y therefore spec cp resp spec gr x is the attine or projecting cone over bl p resp g x so looking at r y and at g y means to study arithmetic properties of the projective v varieties bl y and s y the simplest case is the one in which 4 is locally generated by regular sequences as then spec gr cp fibers locally trivial over v 4 this result whose ring theoretic version is given by rees 191 is of fundamental significance so it is not astonishing that the only really specific results on m z and gr z have been obtained for such pairs z c r which closely imitate the situation of a complete intersection in a cohen macaulay cm ring from this point of view the basic case is the situation studied by valla 20 where r is a cm ring and z is a power of an ideal generated by a regular sequence due to the veronese embeddings zh 4 r z the case where z is generated by a regular sequence is fundamental an essential step towards more generality has been made by huneke 151 where z is assumed to be only generated by a so called d sequence most recently herzog et al 13 obtained new essential results in this direction even under weaker assumptions on i on the other side in a much more special situation goto and shimoda 71 get very specific results which among other things allow one to characterize those local rings r m for which m z is cm whenever z is generated by a system of parameters recently schenzel 171 generalized this result in an interesting direction which relates to the methods of 151 457 0021 8693 84 3 00

a gradient flow for the p elastic energy defined on closed planar curves we study the evolution of closed inextensible planar curves under a second order flow that decreases the p elastic energy a short time existence result for p in 1 infty is obtained via a minimizing movements method for p 2 that is in the case of the classic elastic energy long time existence is retrieved

on formal local cohomology and connectedness let a denote an ideal of a local ring r m let m be a finitely generated r module there is a systematic study of the formal cohomology modules lim hmi m anm i z we analyze their r module structure the upper and lower vanishing and non vanishing in terms of intrinsic data of m and its functorial behavior these cohomology modules occur in relation to the formal completion of the punctured spectrum specr v m as a new cohomological data there is a description on the formal grade fgrade a m defined as the minimal non vanishing of the formal cohomology modules there are various exact sequences concerning the formal cohomology modules among them a mayer vietoris sequence for two ideals it applies to new connectedness results there are also relations to local cohomological dimensions

finiteness of cousin cohomologies he notion of the cousin complex of a module was given by sharp in 1969 it wasn t known whether its cohomologies are finitely generated until recently in 2001 dibaei and tousi showed that the cousin cohomologies of a finitely generated a module m are finitely generated if the base ring a is local has a dualizing complex m satisfies serre s s 2 condition and is equidimensional in the present article the author improves their result he shows that the cousin cohomologies of m are finitely generated if a is universally catenary all the formal fibers of all the localizations of a are cohen macaulay the cohen macaulay locus of each finitely generated a algebra is open and all the localizations of m are equidimensional as a consequence of this he gives a necessary and sufficient condition for a noetherian ring to have an arithmetic macaulayfication

on a characterization of cofinite complexes theorem 1 let a be a ring which is a homomorphic image of a gorenstein ring of finite krull dimension j an ideal of a suppose that a is complete with respect to a jadic topology let n be a complex of a modules in d a where d a is the derived category consisting of complexes bounded below if the ideal j is of dimension one then the complex n is j cofinite if and only if h i n is in m a j cof for all i where m a j cof is the category of j cofinite modules see definition 3 below theorem 2 let a and n be as in theorem 1 let j be an ideal of a if j is principal up to radical then the complex n is j cofinite if and only if h i n is in m a j cof for all i

on a category of cofinite modules which is abelian let a be a noetherian local ring and i an ideal of a of dimension one our purpose of this paper is to prove that the category mathcal m a i cof of cofinite modules is abelian consequently this assertion answers affirmatively the questions given by hartshorne invent math 9 147 1970 cf sect 2 four questions and one theorem for an ideal of dimension one over a local ring

bass numbers of local cohomology modules let a be a regular local ring of positive characteristic this paper is concerned with the local cohomology modules of a itself but with respect to an arbitrary ideal of a the results include that all the bass numbers of all such local cohomology modules are finite that each such local cohomology module has finite set of associated prime ideals and that whenever such a local cohomology module is artinian then it must be injective this last result had been proved earlier by hartshorne and speiser under the additional assumptions that a is complete and contains its residue field which is perfect the paper ends with some low dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true

uniform annihilators of local cohomology of excellent rings abstract the main purpose of this paper is to give an affirmative answer to a conjecture of huneke c huneke uniform bounds in noetherian rings invent math 107 1992 203 223 conjecture 2 13

asymptotic behaviour of arithmetically cohen macaulay blow ups this paper addresses problems related to the existence of arithmetic macaulayfications of projective schemes let y be the blow up of a projective scheme x projr along the ideal sheaf of i r it is known that there are embeddings y projk i e c for c d i e 1 where d i denotes the maximal generating degree of i and that there exists a cohen macaulay ring of the form k i e c if and only if h 0 y oy k h i y oy 0 for i 1 dimy 1 y is equidimensional and cohen macaulay cutkosky and herzog asked when there is a linear bound on c and e ensuring that k i e c is a cohen macaulay ring we obtain a surprising compelte answer to this question namely that under the above conditions there are well determined invariants and e0 such that k i e c is cohen macaulay for all c d i e and e e0 our approach is based on recent results on the asymptotic linearity of the castelnuovo mumford regularity of ideal powers we also investigate the existence of a cohen macaulay rees algebra of the form r i e ct which provides an arithmetic macaulayfication for x if r has negative a invariant we prove that such a cohen macaulay rees algebra exists if and only if oy ox r i oy 0 for i 0 y is equidimensional and cohen macaulay especially these conditions imply the cohen macaulayness of r i e ct for all c d i e and e e0 the above results can be applied to obtain several new classes of cohen macaulay algebras

high order total variation based image restoration the total variation tv denoising method is a pde based technique that preserves edges well but has the sometimes undesirable staircase effect namely the transformation of smooth regions ramps into piecewise constant regions stairs in this paper we present an improved model constructed by adding a nonlinear fourth order diffusive term to the euler lagrange equations of the variational tv model our technique substantially reduces the staircase effect while preserving sharp jump discontinuities edges we show numerical evidence of the power of resolution of this novel model with respect to the tv model in some 1d and 2d numerical examples

on arithmetic macaulayfication of noetherian rings the rees algebra is the homogeneous coordinate ring of a blowing up the present paper gives a necessary and sufficient condition for a noetherian local ring to have a cohen macaulay rees algebra a noetherian local ring has a cohen macaulay rees algebra if and only if it is unmixed and all the formal fibers of it are cohen macaulay as a consequence of it we characterize a homomorphic image of a cohen macaulay local ring for non local rings this paper gives only a sufficient condition by using it however we obtain the affirmative answer to sharp s conjecture that is a noetherian ring having a dualizing complex is a homomorphic image of a finite dimensional gorenstein ring

curvature dependent energies the elastic case abstract we continue our analysis on functionals depending on the curvature of graphs of curves in high codimension euclidean space we deal with the elastic case corresponding to a superlinear dependence on the pointwise curvature we introduce the corresponding relaxed energy functional and prove an explicit representation formula different phenomena w r t the plastic case i e to the relaxation of the total curvature functional are observed a p curvature functional is well defined on continuous curves with finite relaxed energy and the relaxed energy is given by the length plus the p curvature the wider class of graphs of one dimensional bv functions is treated

error estimation and adaptivity in elastodynamics abstract the paper discusses adaptive procedures for elastodynamic problems using semi discrete finite elements the adaptivity in space may be done by mesh refinement h version by increase of order of the approximation polynomials p version or a combination of these hp version in time the integration is either made by mode superposition of characteristic functions or by direct integration the adaptation in time may either be made by change of the global time step h version or increase of the order of approximation polynomials p version crucial parts of the analysis are the error estimator and the error indicator which are based either on interpolation theory or on total energy obtained from postprocessed stresses for the mesh generation and regeneration an isoline technique is utilized by which the mesh can be smoothed by interpolation theory the isolines are created for some characteristic scalar function for time discretization a new simple a posteriori local error estimator for the newmark scheme is described it is derived by a post processing technique without solving new equations so the additional computational cost is small and the implementation is convenient the paper also describes a sequence of nested time integration methods for dynamic problems formulated as two first order equations in time these nested integration methods in time based on a hierarchical formulation in time are a stable schemes of order 2k and l stable schemes of order 2k 1 a considerable improvement in terms of accuracy as well as effectiveness is obtained compared to currently available methods

the twisted inverse image pseudofunctor over commutative dg rings and perfect base change abstract let k be a gorenstein noetherian ring of finite krull dimension and consider the category of cohomologically noetherian commutative differential graded rings a over k such that h 0 a is essentially of finite type over k and a has finite flat dimension over k we extend grothendieck s twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup we prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps and prove a perfect base change result for f in this setting as application we deduce a perfect derived base change result for the twisted inverse image of a map between ordinary commutative noetherian rings our results generalize and solve some recent conjectures of yekutieli

on the associated graded rings of parameter ideals in buchsbaum rings is an invariant 1 a of a not depending on the particular choice of a parameter ideal q of a where i a q and e a denote the length of the amodule a q and the multiplicity of a relative to q respectively this is equivalent to saying that every system a a ad of parameters for a is a weak a sequence i e the equality a ai a i a ai m holds for all 0 i d 1 21 thus every cohen macaulay local ring a is clearly buchsbaum with i a 0 and vice versa in this sense the notion of buchsbaum rings is an extension of that of cohen macaulay rings and is originated in an answer of vogel 29 to a problem of buchsbaum 11 notice that the concept of buchsbaum rings may be canonically generalized to that of buchsbaum modules cf e g 11 221 now let us fix a parameter ideal q of a and let g denote the associated graded ring g a ona q qn of q recall that the ring g is cohenmacaulay if a is so actually g is a polynomial ring over a q in this case then what happens when a is buchsbaum this is a motive problem in the present research and our conclusion is briefly stated

local rings with zero dimensional formal fibers abstract we study noetherian local rings whose all formal fibers are of dimension zero universal catenarity and going up property of the canonical map to the completion are considered we present several characterizations of these rings including a characterization of weierstrass preparation type a characterization of local rings with going up property by a strong form of lichtenbaum hartshorne theorem is obtained as an application we give an upper bound for dimension of formal fibers of a large class of algebras over these rings

adaptive finite element procedures for linear and non linear dynamics this paper discusses implementation and adaptivity of the discontinuous galerkin dg finite element method as applied to linear and non linear structural dynamic problems by the dg method both displacements and velocities are approximated as piecewise bilinear functions in space and time and may be discontinuous at the discrete time levels both implicit and explicit iterative algorithms for solving the resulted system of coupled equations are derived they are third order accurate and while the implicit procedure is unconditionally stable the explicit one is conditionally stable an h adaptive procedure based on the zienkiewicz zhu error estimate using the spr technique is applied numerical examples are presented to show the suitability of the dg method for both linear and non linear structural dynamic analysis copyright 1999 john wiley sons ltd

cofiniteness of extension functors of cofinite modules abstract let r be a commutative noetherian ring i an ideal of r and let m and n be non zero r modules it is shown that the r modules ext r i n m are i cofinite for all i 0 whenever m is i cofinite and n is finitely generated of dimension d 2 also we prove that the r modules ext r i n m are i cofinite for all i 0 whenever n is finitely generated and m is i cofinite of dimension d 1 this immediately implies that if i has dimension one i e dim r i 1 then ext r i n h i i m is i cofinite for all i 0 and all finitely generated r modules m and n also we prove that if r is local then the r modules ext r i n m are i weakly cofinite for all i 0 whenever m is i cofinite and n is finitely generated of dimension d 3 finally it is shown that the r modules ext r i n m are i weakly cofinite for all i 0 whenever r is local n is finitely generated and m is i cofinite of dimension d 2

an example of an infinite set of associated primes of a local cohomology module let r m be a local noetherian ring let i subset r be any ideal and let m be a finitely generated r module in 1990 craig huneke conjectured that the local cohomology modules h i i m have finitely many associated primes for all i in this paper i settle this conjecture by constructing a local cohomology module of a local k algebra with an infinite set of associated primes and i do this for any field k

associated primes of local cohomology modules and matlis duality abstract let r m be a commutative noetherian local ring of dimension d and i an ideal of r we show that the set of associated primes of the local cohomology module h i 2 r is finite whenever r is regular also it is shown that if x 1 x d is a system of parameters for r then d h x 1 x i i r has infinitely many associated prime ideals for all i d 1 where d hom r e denotes the matlis dual functor and e e r r m is the injective hull of the residue field r m finally we explore a counterexample of grothendieck s conjecture by showing that if d 3 then the r module hom r r i h i d 1 r is not finitely generated where i x 1 x 2 x d

global analysis of the generalised helfrich flow of closed curves immersed in r n in this paper we consider the evolution of regular closed elastic curves immersed in r n equipping the ambient euclidean space with a vector eld

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

coarse fine mesh preconditioners for the iterative solution of finite element problems a class of preconditioners built around a coarse fine mesh framework is presented the proposed method involves the reconstruction of the stiffness equations using a coarse fine mesh idealization with relative degrees of freedom derived from the element shape functions this approach leads naturally to effective preconditioners for iterative solvers which only require a factorization involving coarse mesh variables a further extension is the application of the proposed method to super elements in conjunction with substructuring domain decomposition techniques the derivation of the coarse fine mesh discretization via the use of transformation matrices allows a straightforward implementation of the proposed techniques as well as multigrid type procedures within an existing finite element system

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

curve shortening straightening flow for non closed planar curves with infinite length abstract we consider a motion of non closed planar curves with infinite length the motion is governed by a steepest descent flow for the geometric functional which consists of the sum of the length functional and the total squared curvature we call the flow shortening straightening flow in this paper first we prove a long time existence result for the shortening straightening flow for non closed planar curves with infinite length then we show that the solution converges to a stationary solution as time goes to infinity moreover we give a classification of the stationary solution

nilpotence of frobenius actions on local cohomology and frobenius closure of ideals abstract the study of frobenius actions on local cohomology modules over a local ring of prime characteristic has interesting connections with the theory of tight closure this paper establishes new connections by developing the notion of relative frobenius actions on local cohomology as an application we show that a ring has f nilpotent singularities if and only if the tight closure of every parameter ideal is equal to its frobenius closure

on macaulayfication of noetherian schemes the macaulayfication of a noetherian scheme x is a birational proper morphism from a cohen macaulay scheme to x in 1978 faltings gave a macaulayfication of a quasi projective scheme if its non cohen macaulay locus is of dimension 0 or 1 in the present article we construct a macaulayfication of noetherian schemes without any assumption on the non cohen macaulay locus of course a desingularization is a macaulayfication and in 1964 hironaka already gave a desingularization of an algebraic variety over a field of characteristic 0 our method however to construct a macaulayfication is independent of the characteristic

hilbert coefficients and partial euler poincare characteristics of koszul complexes of d sequences abstract we construct in a finitely generated module over a cohen macaulay local ring several subquotient modules in terms of multiplicities of these subquotients we give precise formulas computing all the partial euler poincare characteristics of the koszul complex and the hilbert coefficients of the module relative to an almost p standard system of parameters a very strict subclass of d sequences on the module the formulas enable us to establish some comparison between the partial euler poincare characteristics and the hilbert coefficients

local cohomology and serre subcategories the membership of the local cohomology modules h a n m of a module m in certain serre subcategories of the category of modules is studied from below i n generalizations of depth and regular

cofiniteness and associated primes of local cohomology modules abstract let r be a d dimensional regular local ring i an ideal of r and m a finitely generated r module of dimension n we prove that the set of associated primes of ext r i r i h i j m is finite for all i and j in the following cases a dim m 3 b dim r 4 c dim m im 2 and m satisfies serre s condition s n 3 d dim m im 3 ann r m 0 r is unramified and m satisfies s n 3 in these cases we also prove that h i i m p is i p cofinite for all but finitely many primes p of r additionally we show that if dim r i 2 and spec r i m i is disconnected then h i d 1 r is not i cofinite generalizing a result due to huneke and koh

a hierarchical iterative procedure for the analysis of composite laminates a multilevel recursively defined preconditioner for use with the preconditioned conjugate gradient pcg algorithm is developed and applied to solve thin and thick laminated plate problems the preconditioner is constructed from a sequence of hierarchical vector spaces arising from the p version of the finite element fe method numerical studies have been conducted using quadrilateral elements for two dimensional problems up to p 8 and serendipity brick elements for three dimensional problems up to p 5 the efficiency of the iterative procedure is illustrated using bending response of laminated plates the effects of element span ratios convergence tolerance polynomial order material properties and plate span ratios on the number of iterations required for convergence has been investigated results indicate that the preconditioner developed herein can be used to produce an efficient iterative solver for two and three dimensional problems in structural mechanics

solution techniques for the p version of the adaptive finite element method this paper investigates the performance of a class of methods based on the preconditioned conjugate gradient pcg method for the solution of large and sparse systems of linear equations which arise from the p version of the adaptive finite element method the hierarchical type of preconditioning used in the past is extended to the ssor and incomplete factorization type preconditioning and applied with a global handling of the stiffness matrix the paper also presents some new ideas on domain decomposition dd matrix handling techniques whereby the explicit formulation of the resulting schur complement is avoided by utilizing the pcg method with efficient preconditioners and by processing separately on the coarse and the fine mesh the suitability of the methods to the p version of the adaptive finite element method is demonstrated through extensive numerical tests in two dimensions

an adaptive procedure for eigenvalue problems using the hierarchical finite element method an adaptive procedure for the solution of the generalized linear eigenvalue problem within the hierarchical finite element method is described the problem of finding for a given discretization an upper limit eigenvalue that is accurate within a prescribed tolerance is especially studied an error estimator is presented and a recomputational scheme for improved solutions is proposed a numerical example is included

on gorenstein projective injective and flat dimensions a functorial description with applications gorenstein homological dimensions are refinements of the classical homological dimensions and finiteness singles out modules with amenable properties reflecting those of modules over gorenstein rings as opposed to their classical counterparts these dimensions do not immediately come with practical and robust criteria for finiteness not even over commutative noetherian local rings in this paper we enlarge the class of rings known to admit good criteria for finiteness of gorenstein dimensions

l 2 flow of elastic curves with clamped boundary conditions abstract in this paper we investigate the l 2 flow of elastic curves with clamped boundary conditions in n dimensional euclidean spaces the l 2 flow corresponds to a fourth order parabolic equation in the case of closed curves the long time existence of solutions of this evolution equation has been derived in the literature we extend this result to the case of open i e non closed curves with clamped boundary conditions

on the structure of sequentially generalized cohen macaulay modules abstract a finitely generated module m over a local ring is called a sequentially generalized cohen macaulay module if there is a filtration of submodules of m m 0 m 1 m t m such that dim m 0 dim m 1 dim m t and each m i m i 1 is generalized cohen macaulay the aim of this paper is to study the structure of this class of modules many basic properties of these modules are presented and various characterizations of sequentially generalized cohen macaulay property by using local cohomology modules theory of multiplicity and in terms of systems of parameters are given we also show that the notion of dd sequences defined in n t cuong d t cuong dd sequences and partial euler poincare characteristics of koszul complex j algebra appl 6 2 2007 207 231 is an important tool for studying this class of modules

preconditioning and two level multigrid methods of arbitrary degree of approximation let h be a mesh parameter corresponding to a finite element mesh for an elliptic problem we describe preconditioning methods for two level meshes which for most prob lems solved in practice behave as methods of optimal order in both storage and computa tional complexity namely per mesh point these numbers are bounded above by relatively small constants for all h ho where ho is small enough to cover all but excessively fine meshes we note that in practice multigrid methods are actually solved on a finite often even a fixed number of grid levels in which case also these methods are not asymptotically optimal as h 0 numerical tests indicate that the new methods are about as fast as the best implementations of multigrid methods applied on general problems variable coefficients general domains and boundary conditions for all but excessively fine meshes furthermore most of the latter methods have been implemented only for difference schemes of second order of accuracy whereas our methods are applicable to higher order approximations we claim that our scheme could be added fairly easily to many existing finite element codes 1 introduction consider the numerical solution of elliptic boundary value prob lems discretized by finite element methods we assume that the boundary is polygonal or consists of planes we note that in practical problems one often has a fine enough grid already after the definition of the boundary and the minimal number of vertices needed for a first coarse triangulation anyhow if not so in most cases one makes only a few steps of mesh refinement hence the power of multigrid methods their optimal order of computational complexity is most often not achieved fully because optimality requires a large number of recursively defined meshes for details see e g 4 and for further references see 7 hence one might as well consider other methods perhaps simpler and more effective on a fixed mesh but which are not asymptotically optimal here we shall describe a method which uses only a fixed mesh but for which one nevertheless achieves a low order of computational complexity and of seemingly optimal order except for from a practical viewpoint excessively small meshes to be more precise the computational cost per mesh point is bounded by clog n for n no where n is the number of mesh points no is large enough to cover most applications and c is small enough that the method is competitive with multigrid methods as is well known the latter need recursion and the usual smoothing followed by corrections of the solutions on the different mesh levels we claim that the new method is more suitable for implementation in existing finite element packages in fact most packages for the multigrid methods are only for second order difference methods

attached primes of local cohomology modules under localization and completion let r m be a noetherian local ring and m a finitely generated r module following i g macdonald 8 the set of all attached primes of the artinian local cohomology module hmi m is denoted by attr hmi m in 13 theorem 3 7 r y sharp proved that if r is a quotient of a gorenstein local ring then the shifted localization principle holds true for any local cohomology modules hmi m i e 1 attrp hprpi dim r p mp qrp q attrhmi m q p for any p spec r in this paper we improve sharp s result as follows the shifted localization principle holds true if and only if r is universally catenary and all its formal fibers are cohen macaulay if and only if the shifted completion principle 2 attr hmi m p attr hmi m assr r pr holds true for any local cohomology module hmi m this also improves the main result of the paper by t d m chau and the first author 2

macaulayfication of noetherian schemes to reduce to resolving cohen macaulay singularities faltings initiated the program of macaulayfying a given noetherian scheme x for a wide class of x kawasaki built the sought for cohen macaulay modifications with a crucial drawback that his blowups did not preserve the locus cm x x where x is already cohen macaulay we extend kawasaki s methods to show that every quasi excellent noetherian scheme x has a cohen macaulay x with a proper map x x that is an isomorphism over cm x this completes faltings s program reduces the conjectural resolution of singularities to the cohen macaulay case and implies that every proper smooth scheme over a number field has a proper flat cohen macaulay model over the ring of integers

the łojasiewicz simon gradient inequality for open elastic curves abstract in this paper we consider the elastic energy for open curves in euclidean space subject to clamped boundary conditions and obtain the ojasiewicz simon gradient inequality for this energy functional thanks to this inequality we can prove that a suitably reparametrized solution to the associated l 2 gradient flow converges for large time to an elastica that is to a critical point of the functional

adaptive solution strategy for solving large systems of p type finite element equations a solution strategy is proposed and implemented for taking advantage of the hierarchical structure of linear equation sets arising from the p type finite element method using a hierarchical basis function set the algorithm dynamically branches to either direct or iterative solution methods in the iterative solution branch the substructure of the finite element equation set is used to generate a lower order preconditioner for a preconditioned conjugate gradient pcg method the convergence rate of the pcg algorithm is monitored to improve the heuristics used in the choice of the preconditioner the robustness and efficiency of the method are demonstrated on a variety of three dimensional examples utilizing both hexahedral and tetrahedral mesh discretizations this strategy has been implemented in a p version finite element code which has been used in an industrial environment for over two years to solve mechanical design problems

local cohomology modules via certain flat extension rings abstract let r m be a noetherian local ring and m a finitely generated r module let p spec r and p p r then the natural map r p r p is a flat local homomorphism in this paper we provide some relations between the two sets of attached primes of the artinian local cohomology modules h p r p i r p m p r p r p and h p r p i m p where i 0 is an integer and r p dim r p p r p then we compute the dimension and multiplicity of h p r p i r p m p r p r p in terms of that of h p r p i m p respectively as applications we give connections between the cohen macaulayness in dimension s of m p and that of m p for an integer s 1

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100