solving elliptic eigenvalue problems on polygonal meshes using discontinuous galerkin composite finite element methods in this paper we introduce a discontinuous galerkin method on polygonal meshes this method arises from the discontinuous galerkin composite finite element method dgfem for source problems on domains with micro structures in the context of the present paper the flexibility of dgfem is applied to handle polygonal meshes we prove the a priori convergence of the method for both eigenvalues and eigenfunctions for elliptic eigenvalue problems numerical experiments highlighting the performance of the proposed method for problems with discontinuous coefficients and on convex and non convex polygonal meshes are presented

on the definition of the strategic stability of equilibria a new definition of strategic stability is shown to satisfy all of the requirements given by elon kohlberg and jean francois mertens 1986 the definition follows the general form of the original definition of kohlberg and mertens but rather than working with perturbations of the payoffs or strategy space works directly with perturbations to the best reply correspondence with the appropriate topology on this space of perturbations the resulting definition does satisfy all of the requirements given by kohlberg and mertens it is shown that one does not have much freedom in the topology one uses

irreducible characters of general linear superalgebra and super duality we develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra including all the finite dimensional modules by directly relating the problem to the classical kazhdan lusztig theory furthermore we prove that certain parabolic bgg categories over the general linear algebra and over the general linear superalgebra are equivalent we also verify a parabolic version of a conjecture of brundan on the irreducible characters in the bgg category of the general linear superalgebra

hp version composite discontinuous galerkin methods for elliptic problems on complicated domains in this paper we introduce the hp version discontinuous galerkin composite finite element method for the discretization of second order elliptic partial differential equations this class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features or microstructures while standard numerical methods can be devised for such problems the computational effort may be extremely high as the minimal number of elements needed to represent the underlying domain can be very large in contrast the minimal dimension of the underlying composite finite element space is independent of the number of geometric features the key idea in the construction of this latter class of methods is that the computational domain omega is no longer resolved by the mesh instead the finite element basis or shape functions are adapted to the geometric details present in omega in this paper we extend these ideas to the discontinuous galer

composite finite elements for the approximation of pdes on domains with complicated micro structures usually the minimal dimension of a finite element space is closely related to the geometry of the physical object of interest this means that sometimes the resolution of small micro structures in the domain requires an inadequately fine finite element grid from the viewpoint of the desired accuracy this fact limits also the application of multi grid methods to practical situations because the condition that the coarsest grid should resolve the physical object often leads to a huge number of unknowns on the coarsest level we present here a strategy for coarsening finite element spaces independently of the shape of the object this technique can be used to resolve complicated domains with only few degrees of freedom and to apply multi grid methods efficiently to pdes on domains with complex boundary in this paper we will prove the approximation property of these generalized fe spaces

goal oriented adaptive composite discontinuous galerkin methods for incompressible flows in this article we consider the application of goal oriented mesh adaptation to problems posed on complicated domains which may contain a huge number of local geometrical features or micro structures here we exploit the composite variant of the discontinuous galerkin finite element method based on exploiting finite element meshes consisting of arbitrarily shaped element domains adaptive mesh refinement is based on constructing finite element partitions of the domain consisting of agglomerated elements which belong to different levels of an underlying hierarchical tree data structure as an example of the application of these techniques we consider the numerical approximation of the incompressible navier stokes equations numerical experiments highlighting the practical performance of the proposed refinement strategy will be presented

character formula for infinite dimensional unitarizable modules of the general linear superalgebra the fock space of m p bosonic and n q fermionic quantum oscillators forms a unitarizable module of the general linear superalgebra glm p n q its tensor powers decompose into direct sums of infinite dimensional irreducible highest weight glm p n q modules we obtain an explicit decomposition of any tensor power of this fock space into irreducibles and develop a character formula for the irreducible glm p n q modules arising in this way

two scale composite finite element method for dirichlet problems on complicated domains in this paper we define a new class of finite elements for the discretization of problems with dirichlet boundary conditions in contrast to standard finite elements the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro structures for the proposed finite element method we prove the optimal order approximation up to logarithmic terms and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re entering corners of the domain boundary numerical experiments confirm the theoretical results and show the potential of our proposed method

the stability of the set of kkm points abstract let m be the collection of all kkm mappings g and denote by f g the set of all kkm points of g in this paper we prove that there exists a dense residual subset q of m such that for each g q g is stable and there exists at least one essential component of f g for each g m

character and dimension formulae for general linear superalgebra the generalized kazhdan lusztig polynomials for the finite dimensional irreducible representations of the general linear superalgebra are computed explicitly the result is then applied to prove a conjectural character formula put forward by van der jeugt et al in the late 80s we simplify this character formula to cast it into the kac weyl form and derive from it a closed formula for the dimension of any finite dimensional irreducible representation of the general linear superalgebra

a hausdorff metric inequality with applications to the existence of essential components abstract this paper gives a lemma about a hausdorff metric inequality between two nonempty compact subsets in metric spaces as applications some existence theorems for essential components of solutions for some nonlinear problems such as kkm points solutions in ky fan s lemma as well as nash equilibrium points are deduced

strongly essential set of ky fan s points and the stability of nash equilibrium abstract in this paper a stronger perturbation of inequality functions and set valued mappings is proposed by means of the hausdorff upper semi metric which includes the perturbations defined by sup norm of function and the maximum hausdorff metric of section mapping based on this perturbation a class of strongly essential sets of ky fan s points is introduced and the existence of the strongly essential component of ky fan s points is proved as an application we use the equivalence of nash equilibrium with ky fan s points to obtain the existence of the strongly essential component of equilibrium which not only give a class of stronger essential sets but also provide a method to discuss the stability with respect to the perturbation of strategic sets

hp adaptive composite discontinuous galerkin methods for elliptic eigenvalue problems on complicated domains in this paper we develop the a posteriori error estimation of hp adaptive discontinuous galerkin composite finite element methods dgfems for the discretization of second order elliptic eigenvalue problems dgfems allow for the approximation of problems posed on computational domains which may contain local geometric features the dimension of the composite finite element space is independent of the number of geometric features this is in contrast with standard finite element methods as the minimal number of elements needed to represent the underlying domain can be very large and so the dimension of the finite element space computable upper bounds on the error for both eigenvalues and eigenfunctions are derived numerical experiments highlighting the practical application of the proposed estimators within an automatic hp adaptive refinement procedure will be presented

bem with linear complexity for the classical boundary integral operators alternative representations of boundary integral operators corresponding to elliptic boundary value problems are developed as a starting point for numerical approximations as e g galerkin boundary elements including numerical quadrature and panel clustering these representations have the advantage that the integrands of the integral operators have a reduced singular behaviour allowing one to choose the order of the numerical approximations much lower than for the classical formulations low order discretisations for the single layer integral equations as well as for the classical double layer potential and the hypersingular integral equation are considered we will present fully discrete galerkin boundary element methods where the storage amount and the cpu time grow only linearly with respect to the number of unknowns

a combined finite element method for elliptic problems posted in domains with rough boundaries abstract a combined finite element method is presented in this paper to solve the elliptic problems posted in domains with rough boundaries solving these problems numerically is difficult because resolving the boundaries usually requires very fine meshes while good quality meshes often over refine unnecessarily the interior of the domain the basic idea of the proposed method is to use a fine mesh with size h in the vicinity of oscillating boundaries and a coarse mesh with size h h for other portions of the domain to reduce some unnecessary computational effort the transmission conditions across the fine coarse mesh interface are treated by the penalty technique the key point of the method lies in the new scheme employing a weighted average in the definition of the bilinear form which avoids the affection of the ratio h h in the error estimate we prove a quasi optimal convergence in terms of elements since there is no whole h 2 regularity in the domain with rough boundaries numerical results are provided for elliptic equations in domains with non oscillating or oscillating boundaries to illustrate the theoretical results

the composite mini element coarse mesh computation of stokes flows on complicated domains we introduce a new finite element method the composite mini element for the mixed discretization of the stokes equations on two and three dimensional domains that may contain a huge number of geometric details in standard finite element discretizations of the stokes problem such as the classical mini element the approximation quality is determined by the maximal mesh size of the underlying triangulation while the computational effort is determined by its number of elements if the physical domain is very complicated then the minimal number of simplices which are necessary to resolve the domain can be very large and distributed in a nonoptimal way with respect to the approximation quality in contrast to that the minimal dimension of the composite mini element space is independent of the number of geometric details instead of a geometric resolution of the domain and the boundary condition by the finite element mesh the shape of the finite element functions is adapted to the geometric details this approach allows low dimensional approximations even for problems with complicated geometric details such as holes or rough boundaries we prove its linear optimal order approximability and its inf sup stability further we will be able to control the nonconformity in the space without increasing the space dimension in such a way that the a priori error estimate mathbf u mathbf u mathrm cme 1 omega p p mathrm cme 0 omega lesssim h mathbf f 0 omega holds thereby in contrast to the classical methods the choice of the mesh size parameter h is not constrained by the size of geometric details in addition it turns out that the method can be viewed as a coarse scale generalization of the classical mini element approach i e it reduces the computational effort while the approximation quality depends on the coarse mesh size in the usual way

kazhdan lusztig polynomials and character formulae for the lie superalgebra q n abstract we compute the characters of the finite dimensional irreducible representations of the lie superalgebra q n and formulate a precise conjecture for the other irreducible characters in category o the guiding principle is that the representation theory of q n should be described by lusztig s canonical basis for tensor space over the quantum group of type b

character and dimension formulae for queer lie superalgebra closed formulae are constructed for the characters and dimensions of the finite dimensional simple modules of the queer lie superalgebra mathfrak q n this is achieved by refining brundan s algorithm for computing simple mathfrak q n characters

a mumford shah level set approach for geometric image registration a new method for nonrigid registration of multimodal images is presented due to the large interdependence of segmentation and registration the approach is based on simultaneous segmentation and edge alignment the two processes are directly coupled and thus benefit from using complementary information of the entire underlying data set the approach is formulated as a bivariate variational free discontinuity problem in the mumford shah framework a geometric variable describing the contour set and a functional variable which represents the underlying deformation are simultaneously identified the contour set is represented by a level set function we derive a regularized gradient flow and describe an efficient numerical implementation using finite element discretization and multigrid techniques finally we illustrate the method in several applications such as multimodal intrapatient registration and reconstruction by registration to a reference object

stability of solutions for ky fan s section theorem with some applications abstract in this paper we prove that most of problems in ky fan s section theorem in the sense of baire category are essential and that for any problem in ky fan s section theorem there exists at least one essential component of its solution set as applications we deduce both the existence of essential components of the set of ky fan s points based on ky fan s minimax inequality theorem and the existence of essential components of the set of nash equilibrium points for general n person non cooperative games with non concave payoffs

finite dimensional half integer weight modules over queer lie superalgebras we give a new interpretation of representation theory of the finite dimensional half integer weight modules over the queer lie superalgebra mathfrak q n it is given in terms of the brundan s work on finite dimensional integer weight mathfrak q n modules by means of lusztig s canonical basis using this viewpoint we compute the characters of the finite dimensional half integer weight irreducible modules for a large class of irreducible modules whose highest weights are of special types i e totally connected or totally disconnected we derive closed form character formulas that are reminiscent of the kac wakimoto character formula for basic lie superalgebras

an unfitted hp adaptive finite element method based on hierarchical b splines for interface problems of complex geometry generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries the paper at hand introduces a b spline finite element method which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement which adds several levels of local basis functions in the close vicinity of interfaces but unfitted to their exact location so that a simple regular grid of knot span elements can be maintained numerical experiments show that an hp refinement strategy which simultaneously increases the polynomial degree of the b spline basis and the levels of refinement around interfaces achieves exponential rates of convergence despite the presence of discontinuities it is also demonstrated that the hierarchical b spline fem can be used to transfer the recently introduced finite cell concept to geometrically nonlinear problems its computational performance imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one two and three dimensions and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite

risk averse shape optimization risk averse optimization has attracted much attention in finite dimensional stochastic programming in this paper we propose a risk averse approach in the infinite dimensional context of shape optimization we consider elastic materials under stochastic loading as measures of risk awareness we investigate the expected excess and the excess probability the developed numerical algorithm is based on a regularized gradient flow acting on an implicit description of the shapes based on level sets we incorporate topological derivatives to allow for topological changes in the shape optimization procedure numerical results in two dimensions demonstrate the impact of the risk averse modeling on the optimal shapes and on the cost distribution over the set of scenarios

kac wakimoto character formula for ortho symplectic lie superalgebras abstract we classify finite dimensional tame modules over the ortho symplectic lie superalgebras for these modules we show that their characters are given by the kac wakimoto character formula thus establishing the kac wakimoto conjecture for the ortho symplectic lie superalgebras we further relate the kac wakimoto formula to the super jacobi polynomials of sergeev and veselov and show that these polynomials up to a sign give the super characters for these tame modules

reduction method for representations of queer lie superalgebras we develop a reduction procedure which provides an equivalence from an arbitrary block of the bgg category for the queer lie superalgebra x1d52e n to a s weights s block of a bgg category for finite direct sum of queer lie superalgebras we give descriptions of blocks we also establish equivalences between certain maximal parabolic subcategories for x1d52e n and blocks of atypicality one of the category of finite dimensional modules for x1d524 x1d529 l n l

supercharacters of queer lie superalgebras let g g0 g1 be the queer lie superalgebra and let l be a finite dimensional non trivial irreducible g module restricting the g action on l to g0 we show that the space of g0 invariants lg0 is trivial as a consequence we establish a conjecture first formulated by gorelik grantcharov and mazorchuk on the triviality of the supercharacter of irreducible g modules in the case when the modules are finite dimensional

stability of solutions for variational relation problems with applications abstract in this paper we prove that most of problems in variational relations in the sense of baire category are essential and that for any problem in variational relations there exists at least one essential component of its solution set as applications we deduce the existence of essential components of the set of ky fan s points based on ky fan s minimax inequality theorem the existence of essential components of the set of nash equilibrium points for general n person non cooperative games the existence of essential component of the set of solutions for vector ky fan s minimax inequality the existence of essential components of the set of kkm points and the existence of essential components of the set of solutions for ky fan s section theorem

on multiscale fe analyses of heterogeneous structures from homogenization to multigrid solvers heterogeneous structures like composites often need a fine scale resolution of micro effects which influence the macroscopic overall response this is of particular relevance in the fully non linear range of large strains and inelastic material response of the constituents suitable solution methods introduce a multifield scenario of hierarchically superimposed states on different length scales for big differences of micro and macro scales the argument of scale separation induces the application of homogenization methods such types of physical multiscale approaches can be treated by nested multilevel finite element analyses that discretize both the fine scale micro structure as well as the macroscopic boundary value problem in contrast small scale differences require full resolution of the heterogeneous structure effective solution methods for the resulting large scale problems with strongly oscillating properties are suitably designed geometric multigrid techniques which may be considered as numerical multiscale approaches in both scenarios a key ingredient is the suitable formulation of scale bridging algorithms that govern the transfer between different scales the paper outlines new mesh bridging techniques in a deformation driven context for fully non linear response which exploit in a non trivial manner weak constraints on the average deformation in typical finite element patches the framework is based on an incremental variational structure of finite inelasticity the proposed new formulations provide variational based homogenization algorithms for physical multiscale scenarios and problem dependent optimal finite element grid transfers for numerical multiscale scenarios of heterogeneous materials copyright 2007 john wiley sons ltd

multiscale computations with a combination of the h and p versions of the finite element method the numerical treatment of structural problems is often difficult if the global system behavior is affected by phenomena on different length scales when simulating multiscale problems it is essential to find a discretization that equally reflects all aspects of the problem with sufficient accuracy the present paper is concerned with the formulation of a finite element procedure which combines the p method to resolve global features of a problem with an h method to resolve local features thus the advantageous properties of both standard procedures can be optimally exploited the implementation provides the further advantage that the numerical treatment of the problems on the different length scales can be done with independent finite element discretizations to solve the overall problem an iteration scheme with an adaptive solution strategy is developed benchmark problems and an example with relevance to soil mechanics are presented

high order discontinuous galerkin methods for the elastodynamics equation on polygonal and polyhedral meshes abstract we propose and analyze a high order discontinuous galerkin finite element method for the approximate solution of wave propagation problems modeled by the elastodynamics equations on computational meshes made by polygonal and polyhedral elements we analyze the well posedness of the resulting formulation prove h p version error a priori estimates and present a dispersion analysis showing that polygonal meshes behave as classical simplicial quadrilateral grids in terms of dispersion properties the theoretical estimates are confirmed through various two dimensional numerical verifications

existence and stability of solutions for maximal element theorem on hadamard manifolds with applications abstract in this paper maximal element theorem on hadamard manifolds is established first we prove the existence of solutions for maximal element theorem on hadamard manifolds further we prove that most of problems in maximal element theorem on hadamard manifolds in the sense of baire category are essential and that for any problem in maximal element theorem on hadamard manifolds there exists at least one essential component of its solution set as applications we study existence and stability of solutions for variational relation problems on hadamard manifolds and existence and stability of weakly pareto nash equilibrium points for n person multi objective games on hadamard manifolds

an unfitted finite element method using discontinuous galerkin in this paper we present a new approach to simulations on complex shaped domains the method is based on a discontinuous galerkin dg method using trial and test functions defined on a structured grid essential boundary conditions are imposed weakly via the dg formulation this method offers a discretization where the number of unknowns is independent of the complexity of the domain we will show numerical computations for an elliptic scalar model problem in 2 and 3 convergence rates for different polynomial degrees are studied

super duality and irreducible characters of ortho symplectic lie superalgebras we formulate and establish a super duality which connects parabolic categories o for the ortho symplectic lie superalgebras and classical lie algebras of bcd types this provides a complete and conceptual solution of the irreducible character problem for the ortho symplectic lie superalgebras in a parabolic category o which includes all finite dimensional irreducible modules in terms of classical kazhdan lusztig polynomials

on constrained newton linearization and multigrid for variational inequalities we consider the fast solution of a class of large piecewise smooth minimization problems for lack of smoothness usual newton multigrid methods cannot be applied we propose a new approach based on a combination of convex minization with constrained newton linearization no regularization is involved we show global convergence of the resulting monotone multigrid methods and give polylogarithmic upper bounds for the asymptotic convergence rates efficiency is illustrated by numerical experiments

type a blocks of super category o we show that every block of category oo for the general linear lie superalgebra glm n k glm n k is equivalent to some corresponding block of category oo for the queer lie superalgebra qm n k qm n k this implies the truth of the kazhdan lusztig conjecture for the so called type a blocks of category oo for the queer lie superalgebra as formulated by cheng kwon and wang

finite element methods for the stokes problem on complicated domains abstract it is a standard assumption in the error analysis of finite element methods that the underlying finite element mesh has to resolve the physical domain of the modeled process in case of complicated domains which appear in many applications such as ground water flows this requirement sometimes becomes a bottleneck the resolution condition links the computational complexity to the number and size of geometric details although the accuracy requirements possibly are moderate and would allow a locally coarse mesh width therefore even the coarsest available discretization can lead to a huge number of unknowns the composite mini element is a remedy to this dilemma because the degrees of freedom are not linked to the number of geometric details the basic concept for the stokes problem with uniform no slip boundary conditions has been introduced and analyzed in d peterseim s sauter the composite mini element coarse mesh computation of stokes flows on complicated domains sinum 46 6 2008 3181 3206 here we generalize the composite mini element to slip leak and neumann boundary conditions so that it becomes applicable to this much larger and more important problem class the main results are a the algorithmic concept remains unchanged and the new boundary conditions can be implemented as a weighted quadrature rule b the stability and convergence can be proved under very mild assumption on the domain geometries c the analysis is far from trivial and requires the development of substantially new tools compared to the simple case of uniform no slip boundary conditions

parallel adaptive subspace correction schemes with applications to elasticity in this paper we give a survey on the three main aspects of the efficient treatment of pdes i e adaptive discretization multilevel solution and parallelization we emphasize the abstract approach of subspace correction schemes and summarize its convergence theory then we give the main features of each of the three distinct topics and treat the historical background and modern developments furthermore we demonstrate how all three ingredients can be put together to give an adaptive and parallel multilevel approach for the solution of elliptic pdes and especially of linear elasticity problems we report on numerical experiments for the adaptive parallel multilevel solution of some test problems namely the poisson equation and lame s equation here we emphasize the parallel efficiency of the adaptive code even for simple test problems with little work to distribute which is achieved through hash storage techniques and space filling curves

a posteriori error analysis for elliptic pdes on domains with complicated structures the discretisation of boundary value problems on complicated domains cannot resolve all geometric details such as small holes or pores the model problem of this paper consists of a triangulated polygonal domain with holes of a size of the mesh width at most and mixed boundary conditions for the poisson equation reliable and efficient a posteriori error estimates are presented for a fully numerical discretisation with conforming piecewise affine finite elements emphasis is on technical difficulties with the numerical approximation of the domain and their influence on the constants in the reliability and efficiency estimates

quantum group of type a and representations of queer lie superalgebra abstract we establish a maximal parabolic version of the kazhdan lusztig conjecture 10 conjecture 5 10 for the bgg category o k of q n modules of weights where k n and c 1 2 z as a consequence the irreducible characters of these q n modules in this maximal parabolic category are given by the kazhdan lusztig polynomials of type a lie algebras as an application closed character formulas for a class of q n modules resembling polynomial and kostant modules of the general linear lie superalgebras are obtained

the representation theory of the exceptional lie superalgebras f 4 and g 3 this thesis is a resolution of three related problems proposed by yu i manin and v kac for the so called exceptional lie superalgebras f 4 and g 3 the first problem posed by kac 1978 is the problem of finding character and superdimension formulae for the simple modules the second problem posed by kac 1978 is the problem of classifying all indecomposable representations the third problem posed by manin 1981 is the problem of constructing the superanalogue of borel weil bott theorem

cohomology of generalized supergrassmannians and character formulae for basic classical lie superalgebras we use geometrical methods adapted from borel weil bott theory to compute the character of every finite dimensional simple module over a basic classical lie superalgebra the answer is given by a combinatorial algorithm in terms of weight and cap diagrams which were defined by brundan and stroppel for the general linear superalgebra ams subject classification 20g05 14f05

shape optimization under uncertainty a stochastic programming perspective we present an algorithm for shape optimization under stochastic loading and representative numerical results our strategy builds upon a combination of techniques from two stage stochastic programming and level set based shape optimization in particular usage of linear elasticity and quadratic objective functions permits us to obtain a computational cost which scales linearly in the number of linearly independent applied forces which often is much smaller than the number of different realizations of the stochastic forces numerical computations are performed using a level set method with composite finite elements both in two and in three spatial dimensions

multiscale methods for problems with complex geometry we propose a multiscale method for elliptic problems on complex domains e g domains with cracks or complicated boundary for local singularities this paper also offers a discrete alternative to enrichment techniques such as xfem we construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account the corrections only need to be computed in regions surrounding fine scale geometric features we achieve linear convergence rate in the energy norm for the multiscale solution moreover the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh the analytical findings are verified in a series of numerical experiments

hook young diagrams with applications to combinatorics and to representations of lie superalgebras definition et etude des tableaux k l semi standards applications a la combinatoire et aux representations des superalgebres de lie des diagrammes de young a crochets