composite media and asymptotic dirichlet forms we introduce a metric topology on the space of dirichlet forms and study compactness and closure properties of families of local and non local forms

toward hp adaptive solution of 3d electromagnetic scattering from cavities an effective hp adaptive finite element fe approach is presented for a reliable and accurate solution of 3d electromagnetic scattering problems the far field is approximated with the infinite element method this allows one to reduce the external domain discretised with finite elements to a minimum preserving the possibility of arbitrary reduction of the error as the method does not introduce modelling error the work is focused on scattering from cavity backed apertures recessed in a ground plane near optimal discretisations that can effectively resolve local rapid variations in the scattered field can be obtained adaptively by local mesh refinements so called h type refinements blended with graded polynomial enrichments p enrichments the discretisation error can be controlled by a self adaptive process which is driven by a posteriori error estimates in terms of the energy norm or in a quantity of interest the radar cross section rcs is an example of the latter h and p adaptively constructed solutions are compared to pure uniform p approximations numerical highly accurate and fairly converged solutions for generic cavities are given and compared to previously published results

deformations of principal bundles on the projective line the study of bundles on ip 1 apparently has a long history see 22 chap i sect 2 4 grothendieck proved that any principal bundle on ip with a complex reductive lie groups as structure group admits a reduction of structure group to a maximal torus unique up to weyl group action 9 harder gave a simple proof of this result which works for ip 1 over arbitrary fields 1 11 in this paper we study the deformations of principal bundles over ipl let g be a split reductive group over the field k by the result of grothendieck harder and zariski locally trivial g bundle on ip is associated to the g bundle k z 0 i p 1 by a 1 ps 2 g g let us denote this g bundle by e let e s 1 be a g bundle with an isomorphism eso e s o x lp e we then call e a deformation of ex parametrized by s s o we say that the gbundle e tends or degenerates to the g bundle e and write e e if there is a deformation e s x l p 1 of e such that in every neighbourhood of the base point socs e o e there is an s such that e e we prove theorem 7 4 that if 2 are dominant 1 ps then e e z if and only if 2 i e 2 p is a positive integral combination of simple coroots or equivalently 2 oi e2g for every fundamental weight co i see sect 2 5 note that the set of dominant such that 2 is the same as the set of dominant weights occuring in the indecomposable or irreducible if char k 0 representation of the dual group g see sect 2 6 with highest weight 2 cf 16 sect 21 3 the deformation theory of g bundles on ip seems to be much the same as the representation theory of the dual group g cf 9 p 123 it would be interesting to find a more intrinsic connection between them the g bundles e and e are said to be algebraically equivalent if there is a g bundle e s x lp 1 with s connected such that e e s and e e s for some s s es we prove theorem 7 7 that the algebraic equivalence classes of zariski locally trivial g bundles are classified by the fundamental group of g i e the quotient of the lattice of 1 ps of g by the lattice of coroots this result

convergence of the infinite element methods for the helmholtz equation in separable domains to the best knowledge of the authors this work presents the first convergence analysis for the infinite element method iem for the helmholtz equation in exterior domains the approximation applies to separable geometries only combining an arbitrary finite element fe discretization on the boundary of the domain with a spectral like approximation in the radial direction with shape functions resulting from the separation of variables the principal idea of the presented analysis is based on the spectral decomposition of the problem

γ convergence of equi coercive nonlinear energies defined on vector valued functions with non uniformly bounded coefficients abstract the present paper deals with the asymptotic behavior of equi coercive sequences n of nonlinear functionals defined over vector valued functions in w 0 1 p m where p 1 m 1 and is a bounded open set of r n n 2 the strongly local energy density f n d u of the functional n satisfies a lipschitz condition with respect to the second variable which is controlled by a positive sequence a n which is only bounded in some suitable space l r we prove that the sequence n converges for the strong topology of l p m to a functional which has a strongly local density f d u for sufficiently regular functions u this compactness result extends former results on the topic which are based either on maximum principle arguments in the nonlinear scalar case or adapted div curl lemmas in the linear case here the vectorial character and the nonlinearity of the problem need a new approach based on a careful analysis of the asymptotic minimizers associated with the functional n the relevance of the conditions which are imposed to the energy density f n d u is illustrated by several examples including some classical hyperelastic energies

on the motivic class of the stack of bundles let g be a split connected semisimple group over a field we give a conjectural formula for the motivic class of the stack of g bundles over a curve c in terms of special values of the motivic zeta function of c the formula is true if or g sln if upon applying the poincare or called the serre characteristic by some authors the formula reduces to results of teleman and atiyah bott on the gauge group if upon applying the counting measure it reduces to the fact that the tamagawa number of g over the function field of c is p1 g

de rham diagram for hp finite element spaces abstract we prove that the hp finite elements for h curl spaces introduced in 1 fit into a general de rham diagram involving hp approximations the corresponding interpolation operators generalize the notion of hp interpolation introduced in 2 and are different from the classical operators of nedelec and raviart thomas

modeling of electromagnetic absorption scattering problems using hp adaptive finite elements a model problem for the steady state form of maxwell s equations is considered the problem is formulated in a weak form using a lagrange multiplier laying down a foundation for a general class of novel hp adaptive fe approximations a convergence proof for affine elements is presented the proposed method is illustrated and verified with a series of 2d experiments including elements with curved boundaries and nonhomogeneous media

on the formulation of hybrid finite element and boundary integral methods for 3 d scattering this paper studies in detail a variety of formulations for the hybrid finite element and boundary integral fe bi method for three dimensional 3 d electromagnetic scattering by inhomogeneous objects it is shown that the efficiency and accuracy of the fe bi method depends highly on the formulation and discretization of the boundary integral equation bie used a simple analysis of the matrix condition number identifies the efficiency of the different fe bi formulations and an analysis of weighting functions shows that the traditional fe bi formulations cannot produce accurate solutions a new formulation is then proposed and numerical results show that the resulting solution has a good efficiency and accuracy and is completely immune to the problem of interior resonance finally the multilevel fast multipole algorithm mlfma is employed to significantly reduce the memory requirement and computational complexity of the proposed fe bi method

two transparent boundary conditions for the electromagnetic scattering from two dimensional overfilled cavities we consider electromagnetic scattering from two dimensional 2d overfilled cavities embedded in an infinite ground plane the unbounded computational domain is truncated to a bounded one by using a transparent boundary condition tbc proposed on a semi ellipse for overfilled rectangular cavities with homogeneous media another tbc is introduced on the cavity apertures which produces a smaller computational domain the existence and uniqueness of the solutions of the variational formulations for the transverse magnetic and transverse electric polarizations are established in the exterior domain the 2d scattering problem is solved in the elliptic coordinate system using the mathieu functions in the interior domain the problem is solved by a finite element method numerical experiments show the efficiency and accuracy of the new boundary conditions

on eigenvalue pinching in positive ricci curvature we shall show that for manifolds with ric n 1 the radius is close to iff the n 1 st eigenvalue is close to n this extends results of cheng and croke which show that the diameter is close to iff the first eigenvalue is close to n we shall also give a new proof of an important theorem of colding to the effect that if the radius is close to then the volume is close to that of the sphere and the manifold is gromov hausdorff close to the sphere from work of cheeger and colding these conditions imply that the manifold is diffeomorphic to a sphere

solution of 3d laplace and helmholtz equations in exterior domains using hp infinite elements abstract this work is devoted to a convergence study for infinite element discretizations for laplace and helmholtz equations in exterior domains the proposed approximation applies to separable geometries only combining an hp fe discretization on the boundary of the domain with a spectral like representation resulting from the separation of variables in the radial direction the presentation includes a convergence proof for the laplace equation and a stability analysis for the variational formulation of the helmholtz equation in weighted sobolev spaces the theoretical investigations are verified and illustrated with numerical examples for the exterior spherical domain

a posteriori finite element bounds for linear functional outputs of elliptic partial differential equations we present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order coercive linear partial differential equations in two space dimensions the method is based upon the construction of an augmented lagrangian in which the objective is a quadratic energy reformulation of the desired output and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements the bounds on the output for a suitably fine truth mesh discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid flux candidate lagrange multipliers generated by a k element coarser working mesh approximation independent of the form of the original partial differential equation the computation on the truth mesh is reduced to k decoupled subdomain local symmetric neumann problems the technique is illustrated for the convection diffusion and linear elasticity equations

a new div curl result applications to the homogenization of elliptic systems and to the weak continuity of the jacobian in this paper a new div curl result is established in an open set omega of mathbb r n n geq 2 for the product of two sequences of vector valued functions which are bounded respectively in l p omega n and l q omega n with 1 p 1 q 1 1 n 1 and whose respectively divergence and curl are compact in suitable spaces we also assume that the product converges weakly in w 1 1 omega the key ingredient of the proof is a compactness result for bounded sequences in w 1 q omega based on the imbedding of w 1 q s n 1 into l p s n 1 s n 1 the unit sphere of mathbb r n through a suitable selection of annuli on which the gradients are not too high in the spirit of de giorgi and manfredi the div curl result is applied to the homogenization of equi coercive systems whose coefficients are equi bounded in l rho omega for some rho n 1 over 2 if n 2 or in l 1 omega if n 2 it also allows us to prove a weak continuity result for the jacobian for bounded sequences in w 1 n 1 omega satisfying an alternative assumption to the l infty strong estimate of brezis and nguyen two examples show the sharpness of the results

a three dimensional self adaptive hp finite element method for the characterization of waveguide discontinuities abstract we propose the use of a highly accurate three dimensional 3d fully automatic hp adaptive finite element method fem for the characterization of rectangular waveguide discontinuities these discontinuities are either the unavoidable result of mechanical electrical transitions or deliberately introduced in order to perform certain electrical functions in modern communication systems the proposed numerical method combines the geometrical flexibility of finite elements with an accuracy that is often superior to that provided by semi analytical methods it supports anisotropic refinements on irregular meshes with hanging nodes and isoparametric elements it makes use of hexahedral elements compatible with high order h curl discretizations the 3d hp adaptive fem is applied for the first time to solve a wide range of 3d waveguide discontinuity problems of microwave communication systems in which exponential convergence of the error is observed

adaptive multiresolution antenna modeling using hierarchical mixed order tangential vector finite elements hierarchical mixed order tangential vector finite elements tvfes for tetrahedra are attractive for accurate and efficient finite element or hybrid finite element boundary integral fe bi simulation of complicated electromagnetic problems they provide versatility in the geometrical modeling of physical structures guarantee solutions free of spurious modes and allow a local increase of resolution by combination of mixed order tvfes of different orders within a computational domain regions with higher order mixed order tvfes can be selected a priori or be found adaptively via methods for a posteriori error estimation or indication this paper demonstrates the merits of various explicit residual methods for a posteriori error indication via adaptive refinement of fe bi solutions of electromagnetic radiation problems using hierarchical mixed order tvfes for tetrahedra

borel weil bott theory on the moduli stack of g bundles over a curve let g be a semi simple group and m the moduli stack of g bundles over a smooth complex projective curve using representation theoretic methods i prove the pure dimensionality of sheaf cohomology for certain evaluation vector bundles over m twisted by powers of the fundamental line bundle this result is used to prove a borel weil bott theorem conjectured by g segal for certain generalized flag varieties of loop groups along the way the homotopy type of the group of algebraic maps from an affine curve to g and the homotopy type the hodge theory and the picard group of m are described one auxiliary result in appendix a is the alexander cohomology theorem conjectured in gro2 a self contained account of the uniformization theorem of ls for the stack m is given including a proof of a key result of drinfeld and simpson in characteristic 0 a basic survey of the simplicial theory of stacks is outlined in appendix b

nonlocal effects in two dimensional conductivity the paper deals with the asymptotic behaviour as e 0 of a two dimensional conduction problem whose matrix valued conductivity ae is e periodic and not uniformly bounded with respect to e we prove that only under the assumptions of equi coerciveness and l1 boundedness of the sequence ae the limit problem is a conduction problem of same nature this new result points out a fundamental difference between the two dimensional conductivity and the three dimensional one indeed under the same assumptions of periodicity equi coerciveness and l1 boundedness it is known that the high conductivity regions can induce nonlocal effects in three or greater dimensions

an infinite element for maxwell s equations a new infinite element ie approximation for steady state maxwell s equations is proposed the ie is compatible with a new class of hp finite element fe discretizations presented in 13 32 following 12 a convergence analysis for the proposed coupled fe ie approximations is presented

cycle groups for artin stacks we construct an algebraic homology functor for artin stacks of finite type over a field and we develop intersection theoretic properties

an hp adaptive finite element method for electromagnetics part 3 a three dimensional infinite element for maxwell s equations this paper is a continuation of reference 26 cecot demkowicz and rachowicz computer methods in applied mechanics and engineering 2000 188 625 643 and describes an implementation of the infinite element for three dimensional time harmonic maxwell s equations proposed in reference 15 demkowicz and pal computer methods in applied mechanics and engineering 1998 164 77 94 the element is compatible with the hp finite element discretizations for maxwell s equations in bounded domains reported in references 16 18 computer methods in applied mechanics and engineering 1998 152 103 124 1999 169 331 344 2000 187 307 337 copyright 2003 john wiley sons ltd

determination of the closure of the set of elasticity functionals we determine the closure for mosco convergence in l2 3 of the set of elasticity functionals we prove that this closure coincides with the set of all non negative lower semicontinuous quadratic functionals which are objective i e which vanish for rigid motions the result is still valid if we consider only the set of isotropic elasticity functionals which have a prescribed poisson coefficient this shows that a very large family of materials can be reached when homogenizing a composite material with highly contrasted rigidity coefficients

homogenisation of a class of fourth order equations with application to incompressible elasticity synopsis upon formalising an analogy between two dimensional stokes flow and two dimensional isotropic conductivity we exhibit a class of fourth order equations which behave isomorphically like isotropic conductivity from the standpoint of homogenisation and from that of corresponding bounding methods on possible effective behaviours in particular lipton s result on the g closure problem for mixtures of two incompressible elastic materials is recovered in the two dimensional case

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

a novel hybridization of higher order finite element and boundary integral methods for electromagnetic scattering and radiation problems a novel hybridization of the finite element fe and boundary integral methods is presented for an efficient and accurate numerical analysis of electromagnetic scattering and radiation problems the proposed method derives an adaptive numerical absorbing boundary condition abc for the finite element solution based on boundary integral equations unlike the standard finite element boundary integral approach the proposed method is free of interior resonance and produces a purely sparse system matrix which can be solved very efficiently unlike the traditional finite element absorbing boundary condition approach the proposed method uses an arbitrarily shaped truncation boundary placed very close to the scatterer radiator to minimize the computational domain and more importantly the method produces a solution that converges to the true solution of the problem to demonstrate its great potential the proposed method is implemented using higher order curvilinear vector elements a mixed functional is designed to yield both electric and magnetic fields on an integration surface without numerical differentiation to be used in the calculation of the adaptive abc the required evaluation of boundary integrals is carried out using the multilevel fast multipole algorithm which greatly reduces both the memory requirement and cpu time the finite element equations are solved efficiently using the multifrontal algorithm a mathematical analysis is conducted to study the convergence of the method finally a number of numerical examples are given to illustrate its accuracy and efficiency

analysis of a coupled finite infinite element method for exterior helmholtz problems this analysis of convergence of a coupled fem iem is based on our previous work on the fem and the iem for exterior helmholtz problems the key idea is to represent both the exact and the numerical solution by the dirichlet to neumann operators that they induce on the coupling hypersurface in the exterior of an obstacle the investigation of convergence can then be related to a spectral analysis of these dtn operators we give a general outline of our method and then proceed to a detailed investigation of the case that the coupling surface is a sphere our main goal is to explore the convergence mechanism in this context we show well posedness of both the continuous and the discrete models we further show that the discrete inf sup constants have a positive lower bound that does not depend on the number of dof of the iem the proofs are based on lemmas on the spectra of the continuous and the discrete dtn operators where the spectral characterization of the discrete dtn operator is given as a conjecture from numerical experiments in our convergence analysis we show algebraic in terms of n convergence of arbitrary order and generalize this result to exponential convergence