multi grid methods for hamilton jacobi bellman equations in this paper we develop multi grid algorithms for the numerical solution of hamilton jacobi bellman equations the proposed schemes result from a combination of standard multi grid techniques and the iterative methods used by lions and mercier in 11 a convergence result is given and the efficiency of the algorithms is illustrated by some numerical examples

inexact schwarz algebraic multigrid preconditioners for crack problems modeled by extended finite element methods summary traditional algebraic multigrid amg preconditioners are not well suited for crack problems modeled by extended finite element methods xfem this is mainly because of the unique xfem formulations which embed discontinuous fields in the linear system by addition of special degrees of freedom these degrees of freedom are not properly handled by the amg coarsening process and lead to slow convergence in this paper we proposed a simple domain decomposition approach that retains the amg advantages on well behaved domains by avoiding the coarsening of enriched degrees of freedom the idea was to employ a multiplicative schwarz preconditioner where the physical domain was partitioned into healthy or unfractured and cracked subdomains first the healthy subdomain containing only standard degrees of freedom was solved approximately by one amg v cycle followed by concurrent direct solves of cracked subdomains this strategy alleviated the need to redesign special amg coarsening strategies that can handle xfem discretizations numerical examples on various crack problems clearly illustrated the superior performance of this approach over a brute force amg preconditioner applied to the linear system copyright 2011 john wiley sons ltd

formality for the nilpotent cone and a derived springer correspondence abstract recall that the springer correspondence relates representations of the weyl group to perverse sheaves on the nilpotent cone we explain how to extend this to an equivalence between the triangulated category generated by the springer perverse sheaf and the derived category of differential graded modules over a dg ring related to the weyl group

a non nested galerkin multi grid method for solving linear and nonlinear solid mechanics problems abstract a non nested galerkin multi grid method is proposed in this paper for the iterative solution of general linear and nonlinear problems arising in solid mechanics as the galerkin strategy has been fully adopted in the standard multi grid method for the generation of coarse mesh equations and only the geometrical information for coarse meshes is utilised the resulting approach is relatively easy to incorporate into existing solution procedures and is particularly suitable for implementation in materially non linear cases furthermore the capability of using non nested unstructured meshes provides great flexibility to handle any problem with complex geometry key issues involved in the theoretical and practical aspects for the construction of transfer operators and efficient implementation of galerkin coarse grid approximation are extensively addressed and application to geometrically nonlinear problems is also discussed finally a selected set of numerical examples which cover major classes of problems in solid mechanics including linear elastic materially nonlinear elasto plastic in small strain geometrically nonlinear elastic and geometrically nonlinear elasto plastic problems and also include both symmetric and unsymmetric cases are presented to provide an assessment of the performance of the proposed approach

a parallel multiscale domain decomposition method for the transient dynamic analysis of assemblies with friction the objective of this work is to develop an efficient strategy f or dynamic problems with multiple contacts our approach is based on the multiscale latin method with domain decomposition this is a mixed method which deals simultaneously with the forces and velocities at the interfaces of the different subdomains this strategy has already been applied successfully to a variety of static problems here it is extended to dynamics first we show how the multiscale strategy can be extended to dynamics then we illustrate the capabilities of the method through a 3d academic example and the simulation of a heterogeneous material

the multigrid method in solid mechanics part i algorithm description and behaviour a multigrid algorithm is described that can be used to obtain the finite element solution of linear elastic solid mechanics problems the method is applied to some two dimensional problems to evaluate its strengths and weaknesses extensive studies are made to determine the convergence behaviour of the method in general this depends on many factors the number of degrees of freedom in the discretization characteristics of the algorithm poisson s ratio when it is close to 0 5 the amount of bending deformation in the problem under consideration and the degree of non uniformity in the mesh only certain values of the multigrid parameters allow a converged solution to be obtained with a computational effort proportional to the number of degrees of freedom these values include the optimum ones i e those that lead to convergence with the least computational effort the constant of proportionality is only independent of the number of degrees of freedom and still depends on the other factors listed above

cofinality spectrum theorems in model theory set theory and general topology we connect and solve two longstanding open problems in quite different areas the modeltheoretic question of whether sop2 is maximal in keisler s order and the question from general topology set theory of whether p t the oldest problem on cardinal invariants of the continuum we do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely using model theoretic methods by a cofinality spectrum problem s we essentially mean the data of a pair of models m m1 which code sufficient set theory possibly in an expanded language along with a distinguished set of formulas s which define linear orders in m1 let ts the treetops of s be the smallest regular cardinal such that one of a set of derived trees in m1 has a strictly increasing sequence with no upper bound let c s ts be the set of pairs of regular cardinals 1 2 such that 1 2 or what is equivalent such that d contains no cuts for cf is good we obtain several consequences notably existence of a minimum keisler class among the non simple theories

high performance multilevel iterative aggregation solver for large finite element structural analysis problems the present study is aimed to overcome difficulties faced with industrial applications of multilevel iterative methods to arbitrary finite element fe structural analysis problems the coarse grid concept used in multigrid methods is substituted with an aggregation coarse model based on the mechanical principle on the base of this approach together with previously developed multilevel preconditioner an efficient iterative equation solver feags was developed for using in standard comprehensive finite element software systems numerical examples for analyses of three dimensional 3 d frame structures demonstrate the efficiency of the method comparison with incomplete cholesky conjugate gradient iccg method and direct methods is presented

analysis and improvements of global local enrichments for the generalized finite element method abstract partition of unity methods like the gfem rely on closed form analytical enrichment functions however only limited a priori knowledge about the solution is available for many problems of engineering relevance control of discretization errors in this class of problems requires mesh refinement which offsets some of the advantages associated with these methods this has led to the development of the gfem with global local enrichments gfem gl which uses ideas of the classical global local finite element method to numerically build enrichment functions for the gfem it involves the solution of local boundary value problems using boundary conditions from the solution of the global problem discretized with a coarse mesh these local solutions are in turn used to enrich the solution space of the global problem with the help of the partition of unity framework this paper presents an a priori error estimate accounting for the effect of inexact boundary conditions applied to local problems two strategies to control the error of the gfem gl solution due to these boundary conditions are also presented the first one is based on the use of a buffer or over sampling zone in the local problems the second strategy investigated is based on multiple global local iterations several numerical experiments performed on three dimensional fracture mechanics problems illustrate the effectiveness of these strategies in controlling the error of the gfem gl solution the problems are selected such that the boundary conditions on the local problems range from smooth to singular functions

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

a parallel multigrid method for history dependent elastoplasticity computations abstract a multigrid method is described for solving history dependent elastoplastic solid mechanics problems using the finite element method a multigrid algorithm that solves linear matrix equations is combined with the newton raphson iteration procedure the method is implemented on shared memory parallel computers by employing an element by element approach to computing matrix vector products some two and three dimensional fracture mechanics problems are used to study the behavior of the algorithm the convergence of the linear multigrid solver deteriorates as plastic flow increases this is a result of the effect of near incompressibility on the smoothing effect of the relaxation procedure employed however reasonable performance can still be achieved the solution of some large scale problems demonstrates the speed storage requirements and parallel performance of the algorithm for example a three dimensional problem with 215 000 degrees of freedom and 10 load steps was solved in 69 100 cpu s with 108 mbytes of storage on a single processor of a convex c240 maximum speed ups of 2 96 and 4 13 were measured on a 4 processor convex and an 8 processor alliant respectively for the nonlinear algorithm

an efficient multilevel solution scheme for large scale non linear systems we present a new approach for the fast efficient solution of large systems of non linear equations arising from the finite element discretization the proposed non linear solver builds on the advantages of the popular methods of solution that are currently being employed while eliminating most of their undesirable features it combines the well known bfgs method with the fas version of the multigrid method introduced by brandt 1 to form a fast efficient solution method for non linear problems we present numerical performance studies that are indicative of the convergence properties as well as the stability of the new method

rate of convergence for some constraint decomposition methods for nonlinear variational inequalities some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset one of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition and multigrid methods for domain decomposition and multigrid methods the rate of convergence for the algorithms for obstacle problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems in order to analyse the convergence rate we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying some constraints a special nonlinear interpolation operator is introduced for decomposing the functions

multi time step and two scale domain decomposition method for non linear structural dynamics in this paper we propose a method to improve the means of taking into account the specific time scale and space scale characteristics in time dependent non linear problems this method enables the use of arbitrary time steps in each subdomain these can be coupled by prescribing continuous velocities at the interfaces which are modelled using a dual schur formulation for certain subdomains in space we adopt a two scale resolution technique inspired by the multigrid methods in order to obtain the part of the solution related to small variation lengths on a refined scale and the part corresponding to large variation lengths on a coarse scale for non linear problems we propose an algorithm with a single iteration level to deal with both the non linear equilibrium and the two space scales thanks to a two grid method in which the relaxation steps are performed using a non linear preconditioned conjugate gradient algorithm finally we present an example which demonstrates the feasibility of the method copyright 2003 john wiley sons ltd

general topology meets model theory on p and t cantor proved in 1874 cantor g 1874 j reine angew math 77 258 262 that the continuum is uncountable and hilbert s first problem asks whether it is the smallest uncountable cardinal a program arose to study cardinal invariants of the continuum which measure the size of the continuum in various ways by godel godel k 1939 proc natl acad sci usa 25 4 220 224 and cohen cohen p 1963 proc natl acad sci usa 50 6 1143 1148 hilbert s first problem is independent of zfc zermelo fraenkel set theory with the axiom of choice much work both before and since has been done on inequalities between these cardinal invariants but some basic questions have remained open despite cohen s introduction of forcing the oldest and perhaps most famous of these is whether which was proved in a special case by rothberger rothberger f 1948 fund math 35 29 46 building on hausdorff hausdorff 1936 fund math 26 241 255 in this paper we explain how our work on the structure of keisler s order a large scale classification problem in model theory led to the solution of this problem in zfc as well as of an a priori unrelated open question in model theory

multi scale domain decomposition method for large scale structural analysis with a zooming technique application to plate assembly this article is concerned with a multi scale domain decomposition method ddm based on the feti dp solver for large scale structural elastic analysis and suited to problems that exhibit structural heterogeneities such as plate assemblies in the presence of structural details in this approach once a partition of the global fine mesh into subdomains has been performed all subdomains possess a fine mesh and to optimize the computational time the fine mesh is preserved only in the zones of interest with local phenomena due to discontinuity hole etc while the remaining subdomains are replaced by numerical homogenized coarse elements indeed the multi scale aspect is introduced by the description of subdomains with either a fine or a coarse scale mesh as a result an extension of the feti dp ddm is proposed in this article called herein feti dp micro macro that allows the simultaneous usage of different discretizations fine microscopic mesh for subdomains in zones of interest and coarse macroscopic or homogenized mesh for the complementary part of the structure using this strategy raises the problem of the determination of the stiffness of coarse subdomains and of the incompatible finite element connection between fine and coarse subdomains two approaches collocation and mortar are presented and compared the article ends with patch tests and some numerical examples in 2d and 3d the obtained numerical results exemplify the efficiency and capability of the feti dp micro macro approach and reveal that the mortar approach is more accurate at constant cost than the collocation approach copyright 2009 john wiley sons ltd

model theory and the cardinal numbers p and t modern mathematical logic is a multifaceted subject which concerns itself with the strengths and limitations of formal proofs and algorithms and the relationship between language and mathematical structure modern mathematical logic also addresses foundational issues that arise in mathematics this commentary summarizes the groundbreaking results of malliaris and shelah 1 recently published in pnas 2 relating two branches of logic model theory and set theory

on multi grid methods for variational inequalities we consider here a general class of algorithms for the numerical solution of variational inequalities a convergence proof is given and in particular a multi grid method is described numerical results are presented for the finite difference discretization of an obstacle problem for minimal surfaces

a full 3d finite element analysis using adaptive refinement and pcg solver with back interpolation in this paper adaptive refinement finite element analyses were carried out for full 3d problems in order to achieve an optimal computation cost and to eliminate the effects of singular points the adaptive refinement procedure was applied in conjunction with a back interpolation for the construction of a good initial guess for the solution of the linear equations system the combined use of the adaptive refinement procedure the back interpolation scheme and the preconditioned conjugate gradient pcg solver lead to a significant reduction in the operations for the solution of the simultaneous linear equations in the adaptive refinement analysis in many cases the number of iterations needed by the pcg solver to reach a converged solution is independent of the number of equations in the global system the numerical results obtained indicated that for a series of carefully designed adaptive meshes the computational cost required to solve the linear system could be made only proportional to the number of degrees of freedom in the mesh to our knowledge this is the first time that the global stiffness equations in 3d stress analysis have been solved with this efficiency the same set of numerical results also showed that in general the total computational cost of the adaptive refinement procedure can usually be minimized by gradually reducing the target relative error of the solution during successive refinements rather than employing a constant target relative error throughout the whole adaptive analysis

koszul duality for kac moody groups and characters of tilting modules we establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic p in terms of p kazhdan lusztig polynomials for p h the coxeter number using results of andersen one may deduce a character formula for simple modules if p 2h 2 our results are a consequence of an extension to modular coefficients of a monoidal koszul duality equivalence established by bezrukavnikov and yun

multigrid for the galerkin least squares method in linear elasticity abstract in siam j numer anal 28 1991 1680 1697 franca and stenberg developed several galerkin least squares methods for the solution of the problem of linear elasticity that work concerned itself only with the error estimates of the method it did not address the related problem of finding effective methods for the solution of the associated linear systems in this work we prove the convergence of a multigrid method this multigrid is robust in that the convergence is uniform as the parameter goes to 1 2 computational experiments are included

shear band capturing using a multiscale extended digital image correlation technique abstract finite elements have been used recently to solve the optical flow conservation principle invoked to determine displacement fields by digital image correlation inspired by these recent advances and by the computational effort that has been accomplished during the past 10 years for the simulation of discontinuities by the extended finite element method xfem an extended correlation technique is introduced for capturing shear band like discontinuities from images of real mechanical tests because of the specific information at hand i e gray level images this extended finite element approach is included in a non linear multi grid solver the performances of the proposed approach are discussed on two examples namely the analysis of a bolted assembly and the formation of piobert luders bands

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

constructing regular ultrafilters from a model theoretic point of view this paper contributes to the set theoretic side of understanding keisler s order we consider properties of ultralters which aect saturation of unstable theories the lower conality lcf 0 d of 0 modulo d saturation of the minimum unstable theory the random graph exi bility goodness goodness for equality and realization of symmetric cuts we work in zfc except when noted as several constructions appeal to complete ultralters thus assume a measurable car dinal the main results are as follows first we investigate the strength of exibility known to be detected by non low theories assuming 0 is measurable we construct a regular ultralter on 2 which is exible but not good and which moreover has large lcf 0 but does not even saturate models of the random graph this implies a that exibility alone cannot characterize saturation of any theory however b by separating exibility from goodness we remove a main obstacle to proving non low does not imply maximal since exible is precisely ok this also shows that c from a set theoretic point of view consistently ok need not imply good addressing a prob lem from dow 1985 second under no additional assumptions we prove that there is a loss of saturation in regular ultrapowers of unstable theories and also give a new proof that there is a loss of saturation in ultrapowers of non simple theories more precisely for d regular onand m a model of an unstable theory m d is not 2 saturated and for m a model of a non simple theory and m d is not saturated in the third part of the paper we investigate realization and omission of symmetric cuts signicant both because of the maximality of the strict order property in keisler s order and by recent work of the authors on sop2 we prove that if d is a complete ultralter on any ultrapower of a suciently saturated model of linear or der will have no cuts and that ifd is also normal it will have a cut we apply this to prove that for any n assuming the existence of n measurable cardinals below there is a regular ultralter d onsuch that any d ultrapower of a model of linear order will haven alternations of cuts as dened below moreover d will saturate all stable theories but will not 2 saturate any unstable theory whereis the smallest measurable cardinal used in the construction

the multigrid method in solid mechanics part ii practical applications no proportionality is observed between the total number of operations to convergence and the problem size despite such behaviour the multigrid algorithm proves to be an effective matrix equation solver for solid mechanics problems it is orders of magnitude faster than a direct factorization method and yields converged solutions several times faster than the jacobi preconditioned conjugate gradient method

kostant section universal centralizer and a modular derived satake equivalence we prove analogues of fundamental results of kostant on the universal centralizer of a connected reductive algebraic group for algebraically closed fields of positive characteristic with mild assumptions and for integral coefficients as an application we use these results to obtain a mixed modular analogue of the derived satake equivalence of bezrukavnikov finkelberg

a nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity a robust optimal order multigrid method for the pure traction problem in two dimensional linear elasticity is studied the finite element discretization is a variant of a mixed method proposed by falk where the displacement is approximated by nonconforming piecewise linear functions and the pressure is approximated by piecewise constant functions on a coarser grid full multigrid convergence is obtained the performance of this multigrid algorithm does not deteriorate as the material becomes nearly incompressible

a multigrid method for the generalized symmetric eigenvalue problem part i algorithm and implementation a multigrid method is described that can solve the generalized eigenvalue problem encountered in structural dynamics the algorithm combines relaxation on a fine mesh with the solution of a singular equation on a coarse mesh a sequence of coarser meshes may be used to quickly solve this singular equation using another multigrid method the hierarchy of increasingly finer meshes can be further exploited using a nested iteration scheme whereby initial approximations to the fine mesh eigenvectors are computed using interpolated coarse mesh eigenvectors the solution of some simple plate problems on a convex c240 demonstrates the efficiency of a vectorized version of the multigrid algorithm

multigrid solver with automatic mesh refinement for transient elastoplastic dynamic problems this paper presents an adaptive refinement strategy based on a hierarchical element subdivision dedicated to modelling elastoplastic materials in transient dynamics at each time step the refinement is automatic and starts with the calculation of the solution on a coarse mesh then an error indicator is used to control the accuracy of the solution and a finer localized mesh is created where the user prescribed accuracy is not reached a new calculation is performed on this new mesh using the non linear full approximation scheme multigrid strategy applying the error indicator and the refinement strategy recursively the optimal mesh is obtained this mesh verifies the error indicator on the whole structure the multigrid strategy is used for two purposes first it optimizes the computational cost of the solution on the finest localized mesh second it ensures information transfer among the different hierarchical meshes a standard time integration scheme is used and the mesh is reassessed at each time step copyright 2010 john wiley sons ltd

direct estimation of generalized stress intensity factors using a three scale concurrent multigrid x fem a concurrent multigrid method is devised for the direct estimation of stress intensity factors and higher order coefficients of the elastic crack tip asymptotic field the proposed method bridges three characteristic length scales that can be present in fracture mechanics the structure the crack and the singularity at the crack tip for each of them a relevant model is proposed first a truncated analytical reduced order model based on williams expansion is used to describe the singularity at the tip then it is coupled with a standard extended finite element fe method model which is known to be suitable for the scale of the crack a multigrid solver finally bridges the scale of the crack to that of the structure for which a standard fe model is often accurate enough dedicated coupling algorithms are presented and the effects of their parameters are discussed the efficiency and accuracy of this new approach are exemplified using three benchmarks

hourglass control in linear and nonlinear problems abstract mesh stabilization techniques for controlling the hourglass modes in under integrated hexahedral and quadrilateral elements are described it is shown that the orthogonal hourglass techniques previously developed can be obtained from simple requirements that insure the consistency of the finite element equations in the sense that the gradients of linear fields are evaluated correctly it is also shown that this leads to an hourglass control that satisfies the patch test the nature of the parameters which relate the generalized stresses and strains for controlling hourglass modes is examined by means of a mixed variational principle and some guidelines for their selection are discussed finally effective means of implementing these hourglass procedure in computer codes are described applications to both the laplace equation and the equations of solid mechanics in 2 and 3 dimensions are considered

a dividing line within simple unstable theories abstract we give the first zfc dividing line in keisler s order among the unstable theories specifically among the simple unstable theories that is for any infinite cardinal for which there is 2 we construct a regular ultrafilter d on so that i for any model m of a stable theory or of the random graph m d is saturated but ii if th n is not simple or not low then n d is not saturated the non saturation result relies on the notion of flexible ultrafilters to prove the saturation result we develop a property of a class of simple theories called qr 1 generalizing the fact that whenever b is a set of parameters in some sufficiently saturated model of the random graph b and 2 then there is a set a with a so that any nonalgebraic p s b is finitely realized in a in addition to giving information about simple unstable theories our proof reframes the problem of saturation of ultrapowers in several key ways we give a new characterization of good filters in terms of excellence a measure of the accuracy of the quotient boolean algebra we introduce and develop the notion of moral ultrafilters on boolean algebras we prove a so called separation of variables result which shows how the problem of constructing ultrafilters to have a precise degree of saturation may be profitably separated into a more set theoretic stage building an excellent filter followed by a more model theoretic stage building so called moral ultrafilters on the quotient boolean algebra a process which highlights the complexity of certain patterns arising from first order formulas in certain boolean algebras

equivariant sheaves on flag varieties we show that the borel equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of differential graded modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves this proves a conjecture of soergel and lunts in the case of flag varieties

existence of optimal ultrafilters and the fundamental complexity of simple theories abstract in the first edition of classification theory the second author characterized the stable theories in terms of saturation of ultrapowers prior to this theorem stability had already been defined in terms of counting types and the unstable formula theorem was known a contribution of the ultrapower characterization was that it involved sorting out the global theory and introducing nonforking seminal for the development of stability theory prior to the present paper there had been no such ultrapower characterization of an unstable class in the present paper we first establish the existence of so called optimal ultrafilters on suitable boolean algebras which are to simple theories as keisler s good ultrafilters 15 are to all first order theories then assuming a supercompact cardinal we characterize the simple theories in terms of saturation of ultrapowers to do so we lay the groundwork for analyzing the global structure of simple theories in zfc via complexity of certain amalgamation patterns this brings into focus a fundamental complexity in simple unstable theories having no real analogue in stability

a multigrid method for the generalized symmetric eigenvalue problem part ii performance evaluation the behaviour of the multigrid method is studied by solving some simple test problems optimum choices for some of the parameters are discussed together with effective techniques for solving the coarse mesh correction equation the effects of ill conditioning on the performance of the algorithm are examined in particular thin shells and non uniform meshes are observed to slow convergence the solution of practical large scale problems demonstrates the utility and speed of the proposed multigrid method for example the first 10 eigensolutions of a stiffened plate problem with 193 536 degrees of freedom were computed in 1 6 cpu hours using 42 mbytes of memory on a convex c240

convergence rate analysis of an asynchronous space decomposition method for convex minimization we analyze the convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive banach space this method includes as special cases parallel domain decomposition methods and multigrid methods for solving elliptic partial differential equations in particular the method generalizes the additive schwarz domain decomposition methods to allow for asynchronous updates it also generalizes the bpx multigrid method to allow for use as solvers instead of as preconditioners possibly with asynchronous updates and is applicable to nonlinear problems applications to an overlapping domain decomposition for obstacle problems are also studied the method of this work is also closely related to relaxation methods for nonlinear network flow accordingly we specialize our convergence rate results to the above methods the asynchronous method is implementable in a multiprocessor system allowing for communication and computation delays among the processors

multi grid solutions to the elastic plastic torsion problem in multiply connected domains the elastic plastic torsion problem for an elastic perfectly plastic cylinder with multiply connected cross section twisted around its longitudinal axis is formulated as an obstacle problem for an associated stress potential the obstacle being defined in terms of a generalized distance function based upon the reformulation of the obstacle problem as an equivalent linear complementarity problem the latter is discretized by means of finite difference techniques and a monotonically convergent iterative scheme for its numerical solution is developed at each step of the iteration the solution of a reduced system of discrete poisson equations is required which is done by applying multi grid techniques with respect to a hierarchy of grid point sets combined with a suitably chosen nested iteration process this results in a computationally very efficient algorithm for the approximate solution of the elastic plastic torsion problem

unstructured multigrid method for shells summary an accelerated multigrid method which exploits shell element formulation to speed up the iterative process is developed for inherently poor conditioned thin domain problems on unstructured grids its building blocks are i intergrid transfer operators based on the shell element shape functions ii heavy smoothing procedures in the form of modified incomplete cholesky factorization and iii various two and threeparameter acceleration schemes both the flat shell triangular element and the assumed strain degenerated solid shell element are considered numerical results show a remarkable robustness for a wide spectrum of span thickness ratios encountered in practical applications

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

iterative substructuring methods for spectral element discretizations of elliptic systems i compressible linear elasticity an iterative substructuring method for the system of linear elasticity in three dimensions is introduced and analyzed the pure displacement formulation for compressible materials is discretized with the spectral element method the resulting stiffness matrix is symmetric and positive definite the proposed method provides a domain decomposition preconditioner constructed from local solvers for the interior of each element and for each face of the elements and a coarse global solver related to the wire basket of the elements as in the scalar case the condition number of the preconditioned operator is independent of the number of spectral elements and grows as the square of the logarithm of the spectral degree

a multigrid algorithm for steady transonic potential flows around aerofoils using newton iteration abstract the application of multigrid relaxation to transonic potential flow calculation was investigated fully conservative potential flows around aerofoils were taken as test problems the solution algorithm was based on newton iteration in each newton iteration step multigrid relaxation was used to calculate correction potentials it was found that the iteration to the circulation has to be kept outside the multigrid algorithm in order to obtain meaningful norms of residuals to be used in termination tests of loops difference formulas with asymptotic scaling were introduced nonlinear instability problems were solved by upwind differencing using mass flux vector splitting instead of artificial viscosity or artificial density it was also found that the multigrid method cannot efficiently update shock positions due to the mainly linear character of individual multigrid relaxation cycles for subsonic flows the algorithm is quite efficient for transonic flows the algorithm was found robust it efficiency should be increased by improving the iteration on the shock positions this is a highly nonlinear process