on the finite volume element method for general self adjoint elliptic problems the finite volume element method fve is a discretization technique for partial differential equations this paper develops discretization energy error estimates for general self adjoint elliptic boundary value problems with fve based on triangulations on which there exist linear finite element spaces and a very general type of control volumes covolumes the energy error estimates of this paper are also optimal but the restriction conditions for the covolumes given in r e bank and d j rose siam j numer anal 24 1987 pp 777 787 z q cai numer math 58 1991 pp 713 735 are removed the authors finally provide a counterexample to show that an expected l2 error estimate does not exist in the usual sense it is conjectured that the optimal order of u u h 0 omega should be o h for the general case

a hybrid multilevel schwarz method for the bidomain model abstract a hybrid multilevel schwarz method is studied numerically for the anisotropic bidomain model in both two and three dimensions this multiscale system models the electrical activity of the heart and it consists of two degenerate parabolic non linear reaction diffusion equations coupled with a stiff system of ordinary differential equations the numerical discretization of the whole system by finite elements in space and semi implicit methods in time generates ill conditioned linear systems that must be solved at each time step the multilevel algorithm studied employs a hierarchy of nested meshes with overlapping schwarz preconditioners on each level and is additive within the levels and multiplicative among the levels we perform several parallel tests on two linux clusters showing that the convergence of the method is independent of the number of subdomains scalability the discretization parameters and the number of levels optimality moreover the comparison with the traditional block jacobi ilu parallel preconditioner and the v cycle multigrid parallel preconditioner shows that on a whole heart cycle simulation the proposed method attains the best performances in terms of cpu times

multigrid algorithms for a vertex centered covolume method for elliptic problems we analyze v cycle multigrid algorithms for a class of perturbed problems whose perturbation in the bilinear form preserves the convergence properties of the multigrid algorithm of the original problem as an application we study the convergence of multigrid algorithms for a covolume method or a vertex centered finite volume element method for variable coefficient elliptic problems on polygonal domains as in standard finite element methods the v cycle algorithm with one pre smoothing converges with a rate independent of the number of levels various types of smoothers including point or line jacobi and gauss seidel relaxation are considered

error estimates in l 2 h 1 and l in covolume methods for elliptic and parabolic p roblems a unified approach in this paper we consider covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane we introduce a general approach for connecting these methods with finite element method analysis this unified approach is used to prove known convergence results in the h 1 l 2 norms and new results in the max norm for the elliptic problems we demonstrate that the error u u h between the exact solution u and the approximate solution u h in the maximum norm is o h 2 ln h in the linear element case furthermore the maximur norm error in the gradient is shown to be of first order similar results hold for the parabolic problems

simply connected k contact and sasakian manifolds of dimension 7 we construct a compact simply connected 7 dimensional manifold admitting a k contact structure but not a sasakian structure we also study rational homotopy properties of such manifolds proving in particular that a simply connected 7 dimensional sasakian manifold has vanishing cup product h 2 times h 2 rightarrow h 4 and that it is formal if and only if all its triple massey products vanish

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

on the arithmetic of shalika models and the critical values of l functions for gl2n let pi be a cohomological cuspidal automorphic representation of rm gl 2n bbb a over a totally real number field f suppose that pi has a shalika model we define a rational structure on the shalika model of pi f comparing it with a rational structure on a realization of pi f in cuspidal cohomology in top degree we define certain periods omega epsilon pi f we describe the behavior of such top degree periods upon twisting pi by algebraic hecke characters chi of f then we prove an algebraicity result for all the critical values of the standard l functions l s pi otimes chi here we use the recent work of b sun on the non vanishing of a certain quantity attached to pi infty as applications we obtain algebraicity results in the following cases firstly for the symmetric cube l functions attached to holomorphic hilbert modular cusp forms we also discuss the situation for higher symmetric powers secondly for certain self dual of symplectic type rankin selberg l functions for rm gl 3 times rm gl 2 assuming langlands functoriality this generalizes to certain rankin selberg l functions of rm gl n times rm gl n 1 thirdly for the degree four l functions attached to siegel modular forms of genus 2 and full level moreover we compare our top degree periods with periods defined by other authors we also show that our main theorem is compatible with conjectures of deligne and gross

additive average schwarz method for a crouzeix raviart finite volume element discretization of elliptic problems with heterogeneous coefficients in this paper we introduce an additive schwarz method for a crouzeix raviart finite volume element discretization of a second order elliptic problem with discontinuous coefficients where the discontinuities are both inside the subdomains and across and along the subdomain boundaries we show that depending on the distribution of the coefficient in the model problem the parameters describing the generalized minimal residual method gmres convergence rate of the proposed method depend linearly on the mesh parameters inlineequation not available see fulltext also under certain restrictions on the distribution of the coefficient the convergence of the gmres method is independent of jumps in the coefficient

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

special values of adjoint l functions and congruences for automorphic forms on gl n over a number field we prove an integrality result for the value at s 1 of the adjoint l function associated to a cohomological cuspidal automorphic representation on rm gl n over any number field we then show that primes outside an exceptional set dividing this special value give rise to congruences between automorphic forms we also prove a non vanishing property at infinity for the relevant rankin selberg l functions on rm gl n times rm gl n

a finite volume method based on the crouzeix raviart element for elliptic pde s in two dimensions we introduce and analyse a finite volume method for the discretization of elliptic boundary value problems in mathbb r 2 the method is based on nonuniform triangulations with piecewise linear nonconforming spaces we prove optimal order error estimates in the l 2 norm and a mesh dependent h 1 norm

rationality results for the exterior and the symmetric square l function with an appendix by nadir matringe let g mathrm gl 2n over a totally real number field f and n ge 2 let pi be a cuspidal automorphic representation of g mathbb a which is cohomological and a functorial lift from so 2n 1 the latter condition can be equivalently reformulated that the exterior square l function of pi has a pole at s 1 in this paper we prove a rationality result for the residue of the exterior square l function at s 1 and also for the holomorphic value of the symmetric square l function at s 1 attached to pi as an application of the latter we also obtain a period free relation between certain quotients of twisted symmetric square l functions and a product of gaus sums of hecke characters

the lefschetz property formality and blowing up in symplectic geometry in this paper we study the behaviour of the lefschetz property under the blow up construction we show that it is possible to reduce the dimension of the kernel of the lefschetz map if we blow up along a suitable submanifold satisfying the lefschetz property we use this together with results about massey products to construct compact nonformal symplectic manifolds satisfying the lefschetz property

cohomologically induced distinguished representations and cohomological test vectors let g be a real reductive group and let be a character of a reductive subgroup h of g we construct invariant linear functionals on certain cohomologically induced representations of g and we show that these linear functionals do not vanish on the bottom layers applying this construction we prove two archimedean nonvanishing hypotheses which are vital to the arithmetic study of special values of certain l functions via modular symbols

on domain decomposition algorithms for covolume methods for elliptic problems in this paper some new two level additive schwarz domain decomposition methods are constructed and used as preconditioners for gmres for the nonsymmetric systems arising from the covolume methods for elliptic problems without any restriction on the coarse mesh size the convergence of the preconditioned gmres is obtained for both circumcentric and barycentric covolume methods with inexact solvers for subproblems the rates of convergence are shown to be independent of the number of subregions and these methods are optimal for generous overlap

the finite volume element method for diffusion equations on general triangulations this paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in mathcal r 2 using a special type of control volume the theory applies to diffusion equations of the form begin gathered nabla a nabla u f quad text in omega hfill u 0 quad text on partial omega hfill end gathered under fairly general conditions the theory establishes o h estimates of the error in a discrete mathcal h 1 seminorm under an additional assumption concerning local uniformity of the triangulation the estimate is improved to o h 2