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

integrable modules for affine lie superalgebras irreducible nonzero level modules with finite dimensional weight spaces are discussed for nontwisted affine lie superalgebras a complete classification of such modules is obtained for superalgebras of type a m n and c n using mathieu s classification of cuspidal modules over simple lie algebras in other cases the classification problem is reduced to the classification of cuspidal modules over finite dimensional cuspidal lie superalgebras described by dimitrov mathieu and penkov based on these results a complete classification of irreducible integrable in the sense of kac and wakimoto modules is obtained by showing that any such module is of highest weight in which case the problem was solved by kac and wakimoto

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

generalized verma modules over sln 2 induced from u hn free sln 1 modules abstract a class of generalized verma modules over sl n 2 are constructed from sl n 1 modules which are u h n free modules of rank 1 the necessary and sufficient conditions for these sl n 2 modules to be simple are determined this leads to a class of new simple sl n 2 modules

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

coherent families of weight modules of lie superalgebras and an explicit description of the simple admissible sl n 1 1 modules abstract let g be a classical lie superalgebra of type i we introduce coherent families of weight g modules with bounded weight multiplicities and establish a correspondence between cuspidal and highest weight submodules of these families by extending mathieu s work ann inst fourier 50 2000 537 this enables us to reduce the description of the g 0 module structure of arbitrary simple weight g modules with bounded weight multiplicities to the g 0 module structure of highest weight modules with bounded weight multiplicities we then construct tensor coherent families for g sl n 1 1 which yield an explicit description of the g 0 structure of an arbitrary simple weight module with bounded weight multiplicities in particular we show that for g sl n 1 1 the maximal length of an indecomposable g 0 submodule of a simple weight module with bounded multiplicities equals 5

on the variety of a highest weight module 1 2 let u g be the enveloping algebra of g detailed information on the primitive spectrum prim u g of u g was summarized by conjecture 7 4 of 181 here a main question was to relate prim u g with the springer correspondence parts i ii of this conjecture were established in i ii and a partial solution to iii was given in 21 sufficient e g to treat g of type a meanwhile borho and brylinski 5 have established iii for induced ideals while barbasch and vogan 1 2 have established a version of iii in which gr i i e prim u g is replaced by a certain wavefront set known to be contained in y gr 1 this last work involves some case by case analysis and gives little indication as to why the springer correspondence should arise in the study of prim u g

three dimensional unstructured multigrid for the euler equations the three dimensional euler equations are solved on unstructured tetrahedral meshes using a multigrid strategy the driving algorithm consists of an explicit vertex based finite element scheme which employs an edge based data structure to assemble the residuals the multigrid approach employs a sequence of independently generated coarse and fine meshes to accelerate the convergence to steady state of the fine grid solution variables residuals and corrections are passed back and forth between the various grids of the sequence using linear interpolation the addresses and weights for interpolation are determined in a preprocessing stage using an efficient graph traversal algorithm the preprocessing operation is shown to require a negligible fraction of the cpu time required by the overall solution procedure whereas gains in overall solution efficiencies greater than an order of magnitude are demonstrated on meshes containing up to 350 000 vertices solutions using globally regenerated fine meshes as well as adaptively refined meshes are given

singular gelfand tsetlin modules of gl n abstract the classical gelfand tsetlin formulas provide a basis in terms of tableaux and an explicit action of the generators of gl n for every irreducible finite dimensional gl n module these formulas can be used to define a gl n module structure on some infinite dimensional modules the so called generic gelfand tsetlin modules the generic gelfand tsetlin modules are convenient to work with since for every generic tableau there exists a unique irreducible generic gelfand tsetlin module containing this tableau as a basis element in this paper we initiate the systematic study of a large class of non generic gelfand tsetlin modules the class of 1 singular gelfand tsetlin modules an explicit tableaux realization and the action of gl n on these modules is provided using a new construction which we call derivative tableaux our construction of 1 singular modules provides a large family of new irreducible gelfand tsetlin modules of gl n and is a part of the classification of all such irreducible modules for n 3

on the existence of mathfrak g mathfrak k modules of finite type let g be a reductive lie algebra over an algebraically closed field of characteristic zero and k be a subalgebra reductive in g we prove that g admits an irreducible g k module m which has finite k multiplicities and which is not a g k 0 module for any proper inclusion of reductive subalgebras k k 0 g if and only if k contains its centralizer in g the main point of the proof is a geometric construction of g k modules which is analogous to cohomological induction for g gl n we show that whenever k contains its centralizer there is an irreducible g k module m of finite type over k such that k coincides with the subalgebra of all g 2 g which act locally finitely on m finally for a root subalgebra k gl n we describe all possibilities for the subalgebra l k of all elements acting locally finitely on some m math subject classification 2000 primary 17b10 secondary 22e46

involutive varieties with smooth support abstract we prove a structure theorem for conic involutive varieties of the cotangent bundle of a smooth algebraic varietyxwhose projection onxis smooth the paper includes an example of a conic involutive variety of t xwhich is not the characteristic variety of any d x module

cuspidal representations of sl n 1 abstract in this paper we study the subcategory of cuspidal modules of the category of weight modules over the lie algebra sl n 1 our main result is a complete classification and an explicit description of indecomposable cuspidal modules

generalized verma modules induced from sl 2 c and associated verma modules abstract with each generalized verma module induced from a well embedded parabolic subalgebra of a lie algebra with triangular decomposition we associate a verma module over the same algebra in a natural way in the case when the semisimple part of the levi factor of the parabolic subalgebra is isomorphic to sl 2 c and the generalized verma module is induced from an infinite dimensional simple module we prove that the associated verma module is simple if and only if the original generalized verma module is simple

new singular gelfand tsetlin mathfrak gl n modules of index 2 singular gelfand tsetlin modules of index 2 are modules whose tableaux bases may have singular pairs but no singular triples of entries on each row in this paper we construct singular gelfand tsetlin modules for arbitrary singular character of index 2 explicit bases of derivative tableaux and the action of the generators of mathfrak gl n are given for these modules our construction leads to new families of irreducible gelfand tsetlin modules and also provides tableaux bases for some simple verma modules

root fernando kac subalgebras of finite type abstract let g be a finite dimensional lie algebra and m be a g module the fernando kac subalgebra of g associated to m is the subset g m g of all elements g g which act locally finitely on m a subalgebra l g for which there exists an irreducible module m with g m l is called a fernando kac subalgebra of g a fernando kac subalgebra of g is of finite type if in addition m can be chosen to have finite jordan holder l multiplicities under the assumption that g is simple i penkov has conjectured an explicit combinatorial criterion describing all fernando kac subalgebras of finite type which contain a cartan subalgebra in the present paper we prove this conjecture for g e 8

on the structure of weight modules given any simple lie superalgebra g we investigate the structure of an arbitrary simple weight 0 module we introduce two invariants of simple weight modules the shadow and the small weyl group generalizing results of fernando and futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a levi subsuperalgebra then we classify the cuspidal levi subsuperalgebras of all simple classical lie superalgebras and of the lie superalgebra w n most of them are simply levi subalgebras of go in which case the classification of all finite cuspidal representations has recently been carried out by one of us mathieu our results reduce the classification of the finite simple weight modules over all classical simple lie superalgebras to classifying the finite cuspidal modules over certain lie superalgebras which we list explicitly

complete reducibility of torsion free cn modules of finite degree abstract we show that every torsion free weight module with finite dimensional weight spaces over a symplectic complex lie algebra which is different from sp 2 c is completely reducible

extensions of the conformal representations for orthogonal lie algebras abstract the conformal transformations with respect to the metric defining o n c give rise to an inhomogeneous polynomial representation of o n 2 c using shen s technique of mixed product we generalize the above representation to an inhomogeneous representation of o n 2 c on the tensor space of any finite dimensional irreducible o n c module with the polynomial space where a hidden central transformation is involved moreover we find a condition on the constant value taken by the central transformation such that the generalized representation is irreducible in our approach pieri s formulas invariant operators and the idea of kostant s characteristic identities play key roles the result could be useful in understanding higher dimensional conformal field theory with the constant value taken by the central transformation as the central charge our representations virtually provide natural extensions of the conformal transformations on a riemannian manifold to its vector bundles

simple sl_ n 1 module structures on u h abstract we study the category m consisting of u sl n 1 modules whose restriction to u h is free of rank 1 in particular we classify isomorphism classes of objects in m and determine their submodule structure this leads to new sl n 1 modules for n 1 we also find the central characters and derive an explicit formula for taking tensor product with a simple finite dimensional module

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

lie stacks and their enveloping algebras abstract a lie stack is an algebra morphisms a a bwhereaandbare finite dimensional c algebras withbbeing augmented local we construct the enveloping algebrau s of a lie stack and show that it is an irreducible hopf algebra domain with a poincare birkhoff witt basis we recover the enveloping algebrasu g of lie algebras as special instances we give conditions such thatu s is neither commutative nor cocommutative and we give such examples forbbeing any non commutative augmented local algebra and for beingathe path algebra of a suitable bipartite quiver by studying orbit closes in the variety of lie stacks of fixed dimension one obtains in this way deformations of enveloping algebras of lie algebras

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

supports of weight modules over witt algebras as the first step towards a classification of simple weight modules with finite dimensional weight spaces over witt algebras w n we explicitly describe the supports of such modules we also obtain

irreducible completely pointed modules of quantum groups of type a abstract we give a classification of all irreducible completely pointed u q sl n 1 modules over a characteristic zero field in which q is not a root of unity this generalizes the classification result of benkart britten and lemire in the non quantum case we also show that any infinite dimensional irreducible completely pointed u q sl n 1 can be obtained from some irreducible completely pointed module over the quantized weyl algebra a n 1 q

finite weight modules over twisted affine lie superalgebras abstract this work provides the first step toward the classification of irreducible finite weight modules over twisted affine lie superalgebras we divide the class of such modules into two subclasses called hybrid and tight we reduce the classification of hybrid irreducible finite weight modules to the classification of cuspidal modules of finite dimensional cuspidal lie superalgebras which is discussed in a work of dimitrov mathieu and penkov

three dimensional adaptive grid embedding euler technique a new three dimensional adaptive grid euler procedure is presented that automatically detects high gradient regions in the flow and locally subdivides the computational grid in these regions to provide a uniform high level of accuracy over the entire domain a tunable semistructured data system is utilized that provides global topological unstructured grid flexibility along with the efficiency of a local structured grid system in addition this data structure allows for the flow solution algorithm to be executed on a wide variety of parallel vector computing platforms an explicit lime marching control volume procedure is used to integrate the euler equations to steady state

a koszul category of representations of finitary lie algebras abstract we find for each simple finitary lie algebra g a category t g of integrable modules in which the tensor product of copies of the natural and conatural modules is injective the objects in t g can be defined as the finite length absolute weight modules where by absolute weight module we mean a module which is a weight module for every splitting cartan subalgebra of g the category t g is koszul in the sense that it is antiequivalent to the category of locally unitary finite dimensional modules over a certain direct limit of finite dimensional koszul algebras we describe these finite dimensional algebras explicitly we also prove an equivalence of the categories t o and t sp corresponding respectively to the orthogonal and symplectic finitary lie algebras o sp

a combinatorial description of blocks in o p λ associated with sl 2 induction abstract we study the category o p where is an admissible category of dense weight sl 2 modules we give a combinatorial description of projectively stratified algebras arising from o p and prove a double centralizer property moreover we determine the characters of tilting modules in o p and prove that the finite dimensional algebra associated with the principal block of our o p is its own ringel dual

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

galois orders in skew monoid rings abstract we introduce a new class of noncommutative rings galois orders realized as certain subrings of invariants in skew semigroup rings and develop their structure theory the class of galois orders generalizes classical orders in noncommutative rings and contains many important examples such as the generalized weyl algebras the universal enveloping algebra of the general linear lie algebra associated yangians and finite w algebras

classification of irreducible nonzero level modules with finite dimensional weight spaces for affine lie algebras abstract we classify the irreducible weight affine lie algebra modules with finite dimensional weight spaces on which the central element acts nontrivially in particular we show that any such module is a quotient of a generalized verma module the classification of such irreducible modules is reduced to the classification of irreducible torsion free modules for algebras of type a n and c n which is due to o mathieu forthcoming

on modules of bounded multiplicities for the symplectic algebras simple infinite dimensional highest weight modules having bounded weight multipicities are classified as submodules of a tensor product also it is shown that a simple torsion free module of finite degree tensored with a finite dimensional module is completely reducible

lie algebra modules with finite dimensional weight spaces i let g denote a reductive lie algebra over an algebraically closed field of characteristic zero and let 4 denote a cartan subalgebra of g in this paper we study finitely generated g modules that decompose into direct sums of finite dimensional 4 weight spaces we show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules those we call torsion free modules we also show that if g is a simple lie algebra that admits a torsion free module then g is of type a or c

modular representations of the special linear groups with small weight multiplicities abstract we classify irreducible representations of the special linear groups in positive characteristic with small weight multiplicities with respect to the group rank and give estimates for the maximal weight multiplicities for the natural embeddings of the classical groups inductive systems of representations with totally bounded weight multiplicities are classified an analogue of the steinberg tensor product theorem for arbitrary indecomposable inductive systems for such embeddings is proved

faces and maximizer subsets of highest weight modules abstract in this paper we study general highest weight modules v over a complex finite dimensional semisimple lie algebra g we present three formulas for the set of weights of a large family of modules v which include but are not restricted to all simple modules and all parabolic verma modules these formulas are direct and do not involve cancellations and were not previously known in the literature our results extend the notion of the weyl polytope to general highest weight g modules v we also show that for all simple modules the convex hull of the weights is a w j invariant polyhedron for some parabolic subgroup w j we compute its vertices faces and symmetries more generally we also do this for all parabolic verma modules and for all modules v with highest weight not on a simple root hyperplane to show our results we extend the notion of convexity to arbitrary additive subgroups a r of coefficients our techniques enable us to completely classify weak a faces of the support sets wt v in the process extending classical results of satake borel tits vinberg and casselman as well as modern variants by chari dolbin ridenour and cellini marietti to general highest weight modules

cuspidal sln modules and deformations of certain brauer tree algebras we show that the algebras describing blocks of the category of cuspidal weight resp generalized weight sln modules are one parameter resp multi parameter deformations of certain brauer tree algebras we explicitly determine these deformations both graded and ungraded the algebras we deform also appear as special centralizer subalgebras of temperley lieb algebras or as generalized khovanov algebras they show up in the context of highest weight representations of the virasoro algebra in the context of rational representations of the general linear group and schur algebras and in the study of the milnor fiber of kleinian singularities

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

smoothness improvements of algebraic surface grid generation conclusion by adapting the grid with the three gridalgorithm we reduce not only the number of cells by almost 9 10 but also the cpu time by the same amount compared to single ne grid iteration and obtain a solutionclose to the ne grid solution the reduction in cpu time is due to both fewer iterations to convergenceand fewer cells there is a gain in cpu time on the adapted grid with multigrid iteration comparedto single griditeration thecpu time is about35 longer for the single grid

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

large annihilator category o for sl o sp abstract we construct a new analogue of the bgg category o for the infinite dimensional lie algebras g sl o sp a main difference with the categories studied in 9 and 2 is that all objects of our category satisfy the large annihilator condition introduced in 5 despite the fact that the splitting borel subalgebras b of g are not conjugate one can eliminate the dependency on the choice of b and introduce a universal highest weight category ola of g modules the letters la coming from large annihilator the subcategory of integrable objects in ola is precisely the category t g studied in 5 we investigate the structure of ola and in particular compute the multiplicities of simple objects in standard objects and the multiplicities of standard objects in indecomposable injectives we also complete the annihilators in u g of simple objects of ola

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

a classification of simple lie modules having a 1 dimensional weight space let l denote a simple lie algebra over the complex numbers in this paper we classify and construct all simple l modules which may be infinite dimensional but have at least one 1 dimensional weight space this completes the study begun earlier by the authors for the case of l a the approach presented here relies heavily on the results of suren fernando whose dissertation dealt with simple weight modules and their weight systems 0 introduction let l be a finite dimensional simple lie algebra over the complex field c having a cartan subalgebra h and denote by c l the centralizer of h in the universal enveloping algebra u of l if x h c is a weight function of a 1 dimensional weight space mx in a simple l module m then q c l c defined by q c v cv for v e mx and c e c l is an algebra homomorphism called a mass function of m clearly x restricted to h is equal to x conversely given any algebra homomorphism q c l c one can construct a unique simple l module which admits x as a mass function 4 10 in 4 the authors determined all algebra homomorphisms q c l c for the simple lie algebras l of type an and using these classified all pointed an modules where we call a module pointed if it is simple and has at least one 1 dimensional weight space in this paper we complete the classification of all pointed l modules for arbitrary simple lie algebras the collection of all pointed l modules clearly includes the highest weight l modules and is included in the collection of all harish chandra l modules relative to the cartan subalgebra h i e simple l modules having a weight space decomposition and finite dimensional weight spaces relative to h the latter inclusion is strict since there exist examples of harish chandra a2 modules in which every weight space is two dimensional in the special case of a1 modules every harish chandra a1 module is pointed our approach to this problem makes heavy use of the results from fernando s thesis 8 in particular from fernando s results we see that in place of determining all algebra homomorphisms q c l c it suffices to find only those algebra homomorphisms x which are associated with so called pointed torsion free l received by the editors may 23 1984 this paper was presented at the 1984 canadian mathematical society s summer seminar on lie algebras and related topics held at the university of windsor 1980 mathematics subject classification primary 17b10

irreducible representations of _ with a 1 dimensional weight space in this paper we classify all irreducible linear representations of the simple lie algebra an which admit a one dimensional weight space with respect to some cartan subalgebra h of a we first show that the problem is equivalent to determining all algebra homomorphisms from the centralizer of the cartan subalgebra h in the universal enveloping algebra of an to the base field we construct all such algebra homomorphisms and provide conditions under which two such algebra homomorphisms provide inequivalent irreducible representations of an

weight modules for current algebras for any finite dimensional simple lie algebra g and commutative associative algebra s of finite type we give a complete classification of the simple weight modules of g s with bounded weight multiplicities the classification of simple weight modules is widely seen as a difficult problem in a spectacular tour de force olivier mathieu gave a complete description of such modules for finite dimensional simple lie algebras ma the crucial part of mathieu s classifi cation was understanding the simple admissible highest weight modules in the present paper we consider this question in the context of generalised current algebras l g k s where g is a finite dimensional simple lie algebra over k and s is a commutative associative and unital k algebra which is of finite type these algebras include the finite dimensional simple lie algebras g and the loop algebras g c c t t 1 of affine kac moody theory our methods work not only for the classification of simple admissible highest weight modules but also describe the simple objects in the much larger category of admissible modules weight modules whose weight multiplicities are uniformly bounded we hope that this paper will contribute to solving the harder problem of classifying all simple weight modules for these algebras the category of admissible modules includes the finite dimensional simple modules all of which are known to be evaluation modules see the introductions of l1 cfk or the more general results in nss lp l2 for details what is surprising is that this larger category of generally infinite dimensional modules can also be described in terms of evaluations we now summarize the content of this paper funding from the natural sciences and engineering research council of canada is gratefully ac knowledged

a generic cubic surface contains no involutive curve abstract in 1988 j bernstein and v lunts invent math 94 223 243 proved that a generic surface in p 3 of degree 4 or greater would not contain a curve which supported a module over the second weyl algebra it is known that quadric surfaces and planes always do this paper shows that a generic cubic surface does not

z graded oscillator representations of symplectic lie algebras abstract in this paper we study two parameter z graded oscillator representations of symplectic lie algebras obtained from the natural oscillator representation by partially swapping differential operators and multiplication operators we prove that the homogeneous subspaces with nonzero degree or unequal parameters are infinite dimensional irreducible weight submodules with finite dimensional weight subspaces when the two parameters are equal the homogeneous subspace of degree zero is exactly a direct sum of two infinite dimensional irreducible weight submodules with finite dimensional weight subspaces in generic case these irreducible modules are non unitary modules without highest weight vectors our results are extensions of howe s oscillator construction of infinite dimensional multiplicity free irreducible representations for special linear lie algebras

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

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

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

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