a new approach to hom lie bialgebras abstract in this paper we introduce a new definition of a hom lie bialgebra which is equivalent to a manin triple of hom lie algebras we also introduce a notion of an o operator and then construct solutions of the classical hom yang baxter equation in terms of o operators and hom left symmetric algebras

a new cloud based hp finite element method a hybrid computational method for solving boundary value problems is introduced which combines features of the meshless hp cloud methods with features of conventional finite elements the method admits straightforward nonuniform hp type approximations easy implementation of essential boundary conditions is robust under severe distortions of the mesh and can deliver exponential rates of convergence results of numerical experiments are presented

general charge conjugation operators in simple lie groups a description of particular elements charge conjugation operators found in any compact simple lie group k is presented such elements ri transform a physical state weight vector of a basis of a representation space into others with opposite charge ith component of the weight sometime changing also the sign of the state it is demonstrated that exploitation of these elements and the finite subgroup n of k generated by them offer new powerful methods for computing with representations of the lie group their application to construction of bases in representation spaces is considered in detail it represents a completely new direction to the problem

combined extended and superimposed finite element method for cracks a combination of the extended finite element method xfem and the mesh superposition method s version fem for modelling of stationary and growing cracks is presented the near tip field is modelled by superimposed quarter point elements on an overlaid mesh and the rest of the discontinuity is implicitly described by a step function on partition of unity the two displacement fields are matched through a transition region the method can robustly deal with stationary crack and crack growth it simplifies the numerical integration of the weak form in the galerkin method as compared to the s version fem numerical experiments are provided to demonstrate the effectiveness and robustness of the proposed method copyright 2004 john wiley sons ltd

mesh independent p orthotropic enrichment using the generalized finite element method this paper is aimed at presenting a simple yet effective procedure to implement a mesh independent p orthotropic enrichment in the generalized finite element method the procedure is based on the observation that shape functions used in the gfem can be constructed from polynomials defined in any co ordinate system regardless of the underlying mesh or type of element used numerical examples where the solution possesses boundary or internal layers are solved on coarse tetrahedral meshes with isotropic and the proposed p orthotropic enrichment copyright 2002 john wiley sons ltd

hom lie 2 algebras abstract in this paper we introduce the notions of hom lie 2 algebras which is the categorification of hom lie algebras hl algebras which is the hom analogue of l algebras and crossed modules of hom lie algebras we prove that the category of hom lie 2 algebras and the category of 2 term hl algebras are equivalent we give a detailed study on skeletal hom lie 2 algebras in particular we construct the hom analogues of the string lie 2 algebras associated to any semisimple involutive hom lie algebras we also proved that there is a one to one correspondence between strict hom lie 2 algebras and crossed modules of hom lie algebras we give the construction of strict hom lie 2 algebras from hom left symmetric algebras and symplectic hom lie algebras

locally adapted tetrahedral meshes using bisection we present an algorithm for the construction of locally adapted conformal tetrahedral meshes the algorithm is based on bisection of tetrahedra a new data structure is introduced which simplifies both the selection of the refinement edge of a tetrahedron and the recursive refinement to conformity of a mesh once some tetrahedra have been bisected we prove that repeated application of the algorithm leads to only finitely many tetrahedral shapes up to similarity and we bound the amount of additional refinement that is needed to achieve conformity numerical examples of the effectiveness of the algorithm are presented

quasi hom lie algebras central extensions and 2 cocycle like identities this paper introduces the notion of a quasi hom lie algebra or simply a qhl algebra which is a natural generalization of hom lie algebras introduced in a previous paper j t hartwig d larsson s d silvestrov deformations of lie algebras using sigma derivations math qa 0408064 quasi hom lie algebras include also as special cases color lie algebras and superalgebras and can be seen as deformations of these by maps twisting the jacobi identity and skew symmetry the natural realm for these quasi hom lie algebras is generalizations deformations of the witt algebra delta of derivations on the laurent polynomials c t t 1 we also develop a theory of central extensions for qhl algebras which can be used to deform and generalize the virasoro algebra by centrally extending the deformed witt type algebras constructed here in addition we give a number of other interesting examples of quasi hom lie algebras among them a deformation of the loop algebra less

analysis of a fully discrete finite element method for a nonlinear magnetic field problem in this paper nonlinear stationary magnetic field problems and their approximation by finite elements are studied as a new aspect estimates for the error caused by the approximation of the material function are given a technique for the interpolation of a pointwise given material function that preserves monotonicity is described and error estimates are presented

deformations of lie algebras using σ derivations in this article we develop an approach to deformations of the witt and virasoro algebras based on sigma derivations we show that sigma twisted jacobi type identity holds for generators of such deformations for the sigma twisted generalization of lie algebras modeled by this construction we develop a theory of central extensions we show that our approach can be used to construct new deformations of lie algebras and their central extensions which in particular include naturally the q deformations of the witt and virasoro algebras associated to q difference operators providing also corresponding q deformed jacobi identities c 2005 elsevier inc all rights reserved

quasi optimal rates of convergence for the generalized finite element method in polygonal domains we consider a mixed boundary value interface problem for the elliptic operator p i j i a i j j u f on a polygonal domain r 2 with straight sides we endowed the boundary of partially with dirichlet boundary conditions u 0 on d and partially with neumann boundary conditions i j i a i j j u 0 on n the coefficients a i j are piecewise smooth with jump discontinuities across the interface which is allowed to have singularities and cross the boundary of in particular we consider triple junctions and even multiple junctions our main result is to construct a sequence of generalized finite element spaces s n that yield h m quasi optimal rates of convergence m 1 for the galerkin approximations u n s n of the solution u more precisely we prove that u u n c dim s n m 2 f h m 1 where c depends on the data for the problem but not on f u or n and dim s n our construction is quite general and depends on a choice of a good sequence of approximation spaces s n on a certain subdomain w that is at some distance to the vertices in case the spaces s n are generalized finite element spaces then the resulting spaces s n are also generalized finite element spaces

a computational approach to handle complex microstructure geometries in multiscale analysis of components there is usually a need to solve microstructures with complex geometries in this paper we use the extended finite element method x fem to solve scales involving complex geometries the x fem allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach for material interfaces this is achieved by introducing a new enrichment strategy although the mesh does not need to conform to the physical surfaces it needs to be fine enough to capture the geometry of these surfaces a simple algorithm is described to adaptively refine the mesh to meet this geometrical requirement numerical experiments on the periodic homogenization of two phase complex cells demonstrate the accuracy and simplicity of the x fem c 2003 elsevier b v all rights reserved

a generalized finite element method for polycrystals with discontinuous grain boundaries summary we present a generalized finite element method for the analysis of polycrystals with explicit treatment of grain boundaries grain boundaries and junctions understood as loci of possible displacement discontinuity are inserted into finite elements by exploiting the partition of unity property of finite element shape functions consequently the finite element mesh does not need to conform to the polycrystal topology the formulation is outlined and a numerical example is presented to demonstrate the potential and accuracy of the approach the proposed methodology can also be used for branched and intersecting cohesive cracks and comparisons are made to a related approach int j numer meth engng 2000 48 1741 copyright 2006 john wiley sons ltd

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

a new dual mortar method for curved interfaces 2d elasticity dual mortar method formulations have shown to be a very effective and efficient way for interfacing e g tying contacting dissimilar meshes on the other hand we have recently found that they can sometimes perform quite poorly when applied to curved surfaces in some solid mechanics applications a new modified two dimensional dual mortar method for piecewise linear finite elements is developed that overcomes this deficiency and is demonstrated on a model problem furthermore mathematical analysis is provided to demonstrate the optimal convergence and stability of the new method copyright 2005 john wiley sons ltd

generalized finite element method using mesh based handbooks application to problems in domains with many voids this paper describes a new version of the generalized finite element method originally developed int j numer methods engrg 47 2000 1401 comput methods appl mech engrg 181 2000 43 the design and implementation of the generalized finite element method ph d thesis texas am comput methods appl mech engrg 190 2001 4081 which is well suited for problems set in domains with a large number of internal features e g voids inclusions cracks etc the main idea is to employ handbook functions constructed on subdomains resulting from the mesh discretization of the problem domain the proposed new version of the gfem is shown to be robust with respect to the spacing of the features and is capable of achieving high accuracy on meshes which are rather coarse relative to the distribution of the features

e6 unification model building iii clebsch gordan coefficients in e6 tensor products of the 27 with higher dimensional representations e6 is an attractive group for unification model building however the complexity of a rank 6 group makes it nontrivial to write down the structure of higher dimensional operators in an e6 theory in terms of the states labeled by quantum numbers of the standard model gauge group in this paper we show the results of our computation of the clebsch gordan coefficients for the products of the 27 with irreducible representations of higher dimensionality 78 351 351 351 and 351 application of these results to e6 model building involving higher dimensional operators is straightforward

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions copyright 2000 john wiley sons ltd

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

a generalized finite element method for the simulation of three dimensional dynamic crack propagation this paper is aimed at presenting a partition of unity method for the simulation of three dimensional dynamic crack propagation the method is a variation of the partition of unity finite element method and hp cloud method in the context of crack simulation this method allows for modeling of arbitrary dynamic crack propagation without any remeshing of the domain in the proposed method the approximation spaces are constructed using a partition of unity pu and local enrichment functions the pu is provided by a combination of shepard and finite element partitions of unity this combination of pus allows the inclusion of arbitrary crack geometry in a model without any modification of the initial discretization it also avoids the problems associated with the integration of moving least squares or conventional shepard partitions of unity used in several meshless methods the local enrichment functions can be polynomials or customized functions these functions can efficiently approximate the singular fields around crack fronts the crack propagation is modeled by modifying the partition of unity along the crack surface and does not require continuous remeshings or mappings of solutions between consecutive meshes as the crack propagates in contrast with the boundary element method the proposed method can be applied to any class of problems solvable by the classical finite element method in addition the proposed method can be implemented into most finite element data bases several numerical examples demonstrating the main features and computational efficiency of the proposed method for dynamic crack propagation are presented

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

embedded interfaces by polytope fem in this paper the polytope finite element method is employed to model an embedded interface through the body independent of the background fem mesh the elements that are crossed by the embedded interface are decomposed into new polytope elements which have some nodes on the interface line the interface introduces discontinuity into the primary variable strong or into its derivatives weak both strong and weak discontinuities are studied by the proposed method through different numerical examples including fracture problems with traction free and cohesive cracks and heat conduction problems with dirichlet and dirichlet neumann types of boundary conditions on the embedded interface for traction free cracks which have tip singularity the nodes near the crack tip are enriched with the singular functions through the extended finite element method the concept of natural element coordinates necs is invoked to drive shape functions for the produced polytopes a simple treatment is proposed for concave polytopes produced by a kinked interface and also for locating crack tip inside an element prior to using the singularity enrichment the proposed method pursues some implementational details of extended generalized finite element methods for interfaces but here the additional dofs are constructed on the interface lines in contrast to x g fem which attach enriched dofs to the previously existed nodes copyright 2011 john wiley sons ltd

a discontinuous finite element based domain decomposition method in this work we introduce a somewhat unconventional finite element based nonoverlapping domain decomposition method the convenient space decomposition method csd for parallel computers that can be applied to a wide class of boundary value problems and which can only be implemented efficiently exclusively with primal variables by means of the use of discontinuous elements interesting aspects and properties are pointed out along with proofs for the most relevant results numerical experiments for purely diffusion problems are also presented in order to evaluate the effectiveness of the proposed scheme and to confirm the validity of the related theoretical results

hom lie hopf algebras abstract we studied both the double cross product and the bicrossproduct constructions for the hom hopf algebras of general type this allows us to consider the universal enveloping hom hopf algebras of hom lie algebras which are of id type we show that the universal enveloping hom hopf algebras of a matched pair of hom lie algebras form a matched pair of hom hopf algebras we observe also that the semi dualization of a double cross product hom hopf algebra is a bicrossproduct hom hopf algebra in particular we apply this result to the universal enveloping hom hopf algebras of a matched pair of hom lie algebras to obtain hom lie hopf algebras

improved xfem an extra dof free well conditioning and interpolating xfem abstract although its unprecedentedly success in approximating the singular field around a crack tip independent of mesh discretization xfem still has been long hindered in engineering applications by its ill conditioning and the difficulty in mass lumping corresponding to extra dofs an improved xfem i xfem is developed to overcome those difficulties from three aspects 1 to eliminate the linear dependence and the ill conditioning issues of the standard and the corrected xfems 2 to get rid of extra dofs in crack tip enrichment to facilitate optimal mass lumping in dynamic analyses 3 to be interpolating at enriched nodes to enable direct essential contact boundary treatments the core of i xfem is a globally interpolating approximation based on local least squares fitting the new method is numerically examined with comprehensive comparisons with the standard xfem and the corrected xfem fries 2008 the new method s excellent performance in terms of accuracy convergence and conditioning stability are demonstrated

e6 unification model building i clebsch gordan coefficients of 27 27 in an effort to develop tools for grand unified model building for the lie group e6 in this paper we present the computation of the clebsch gordan coefficients for the product 100000 000010 where 100000 is the fundamental 27 dimensional representation of e6 and 000010 is its charged conjugate the results are presented in terms of the dominant weight states of the irreducible representations in this product these results are necessary for the group analysis of e6 operators also involving higher representations which is the next step in this project in this paper we apply the results to the construction of the operator 273

a high order generalized fem for through the thickness branched cracks this paper presents high order implementations of a generalized finite element method for through the thickness three dimensional branched cracks this approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks independently of the background finite element mesh representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies the combination of local refinement at crack fronts and high order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions copyright 2007 john wiley sons ltd

the splitting method as a tool for multiple damage analysis the paper presents a reliable and efficient method for solving problems on a large set of domains with different geometries in applications these geometries might be idealizations of a mechanical structure with multiple damage the method is designed and used to analyse large three dimensional problems the present paper formulates the method and presents the theory for a simple model problem i e the laplace equation in two dimensions in the numerical implementation of the splitting scheme the p version of the finite element method and strongly graded meshes are used this approach provides high accuracy and error control the exponential rate of convergence of the method is shown and practically exemplified a numerical example shows the effects of various choices of parameters on the accuracy of the method a complex problem related to aircraft safety which was solved using the present method is used to exemplify the generality and the efficiency of the proposed method

clebsch gordan coefficients for su 5 unification models the clebsch gordan coefficients for the product 1001 1001 where 1001 is the adjoint representation of su 5 with respect to the group basis and the subgroup basis in the reduction su 5 su 3 su 2 u 1 are computed one of the basic tools in this computation is the exhaustive use of the verma algorithm to find bases for the weight subspaces of dimension higher than 1 it allows for the construction of bases in a systematic way by using the so called verma inequalities only the coefficients for the dominant weights are calculated the other ones can be obtained by using the elements of finite order charge conjugation operators of su 5

recycling krylov subspaces for sequences of linear systems many problems in science and engineering require the solution of a long sequence of slowly changing linear systems we propose and analyze two methods that significantly reduce the total number of matrix vector products required to solve all systems we consider the general case where both the matrix and right hand side change and we make no assumptions regarding the change in the right hand sides furthermore we consider general nonsingular matrices and we do not assume that all matrices are pairwise close or that the sequence of matrices converges to a particular matrix our methods work well under these general assumptions and hence form a significant advancement with respect to related work in this area we can reduce the cost of solving subsequent systems in the sequence by recycling selected subspaces generated for previous systems we consider two approaches that allow for the continuous improvement of the recycled subspace at low cost we consider both hermitian and non hermitian problems and we analyze our algorithms both theoretically and numerically to illustrate the effects of subspace recycling we also demonstrate the effectiveness of our algorithms for a range of applications from computational mechanics materials science and computational physics

non planar 3d crack growth by the extended finite element and level sets part i mechanical model a methodology for solving three dimensional crack problems with geometries that are independent of the mesh is described the method is based on the extended finite element method in which the crack discontinuity is introduced as a heaviside step function via a partition of unity in addition branch functions are introduced for all elements containing the crack front the branch functions include asymptotic near tip fields that improve the accuracy of the method the crack geometry is described by two signed distance functions which in turn can be defined by nodal values consequently no explicit representation of the crack is needed examples for three dimensional elastostatic problems are given and compared to analytic and benchmark solutions the method is readily extendable to inelastic fracture problems copyright 2002 john wiley sons ltd

hom novikov algebras we study a twisted generalization of novikov algebras called hom novikov algebras in which the two defining identities are twisted by a linear map it is shown that hom novikov algebras can be obtained from novikov algebras by twisting along any algebra endomorphism all algebra endomorphisms on complex novikov algebras of dimensions 2 or 3 are computed and their associated hom novikov algebras are described explicitly another class of hom novikov algebras is constructed from hom commutative algebras together with a derivation generalizing a construction due to dorfman and gel fand two other classes of hom novikov algebras are constructed from hom lie algebras together with a suitable linear endomorphism generalizing a construction due to bai and meng

a hom version of the affinizations of balinskii novikov and novikov superalgebras in this paper we introduce the notions of hom balinskii novikov and hom novikov superalgebras in which the defining identities are twisted by homomorphisms then we construct some infinite dimensional hom lie superalgebras by the affinizations of the above two hom superalgebras moreover we apply the bilinear forms with some invariance conditions on the hom balinskii novikov and hom novikov superalgebras to construct central extensions of the infinite dimensional hom lie superalgebras

clebsch gordan coefficients for e6 and so 10 unification models we illustrate here a new method for computing clebsch gordan coefficients cgc for e6 by computing cgc for the product 27 27 of the irreducible representation 100000 of e6 with itself these cgc are calculated thrice once in a weight vector basis independent of any semisimple subgroup then in a basis which refers to so 10 e6 and finally in a basis referring to su 5 so 10 e6

the generalized finite element method an example of its implementation and illustration of its performance the generalized finite element method gfem was introduced in reference 1 as a combination of the standard fem and the partition of unity method the standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data the geometry of the domain the loads and the boundary conditions the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation the essential boundary conditions can be imposed exactly as in the standard fem adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation this paper gives an example of how the gfem can be developed for the laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the gfem versus the standard fem copyright 2000 john wiley sons ltd

a simple algorithm for localized construction of non matching structural interfaces a simple and effective algorithm for the modular construction of non matched interfaces is presented for the partitioned solution of large scale structural problems the formulation is based on a recently developed four field variational principle which introduces a connection frame between the interfaced partitions a key result of the present study is a frame nodal placement criterion that uniquely determines the frame discretization into piecewise linear elements so that the interface patch test condition is satisfied a priori the method is demonstrated with several 2d and 3d example problems

e6 unification model building ii clebsch gordan coefficients of 78 78 we have computed the clebsch gordan coefficients for the product 000 001 000 001 where 000 001 is the adjoint 78 dimensional representation of e6 the results are presented for the dominant weights of the irreducible representations in this product as a simple application we express the singlet operator in 27 78 27 in terms of multiplets of the standard model gauge group

analysis and applications of a generalized finite element method with global local enrichment functions abstract this paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the hp cloud method the procedure combines classical global local finite element method concepts with the partition of unity approach it involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization the local solutions are in turn used to enrich the global space using the partition of unity framework the computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently the effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper we also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution key aspects of the computational implementation in particular the numerical integration of generalized fem approximations built with global local enrichment functions are presented the method is applied to fracture mechanics problems with multiple cracks in the domain our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global local enrichment functions

p version of the generalized fem using mesh based handbooks with applications to multiscale problems in this paper we analyse the p convergence of a new version of the generalized finite element method generalized fem or gfem which employs mesh based handbook functions which are solutions of boundary value problems in domains extracted from vertex patches of the employed mesh and are pasted into the global approximation by the partition of unity method pum we show that the p version of our gfem is capable of achieving very high accuracy for multiscale problems which may be impossible to solve using the standard fem we analyse the effect of the main factors affecting the accuracy of the method namely a the data and the buffer included in the handbook domains and b the accuracy of the numerical construction of the handbook functions we illustrate the robustness of the method by employing as model problem the laplacian in a domain with a large number of closely spaced voids similar robustness can be expected for problems of heat conduction and elasticity set in domains with a large number of closely spaced voids cracks inclusions etc copyright 2004 john wiley sons ltd