quantum monodromy and nonconcentration near a closed semi hyperbolic orbit for a large class of semiclassical operators p h z which includes schrodinger operators on manifolds with boundary we construct the quantum monodromy operator m z associated to a periodic orbit of the classical flow using estimates relating m z and p h z we prove semiclassical estimates for small complex perturbations of p h z in the case is semi hyperbolic as our main application we give logarithmic lower bounds on the mass of eigenfunctions away from semi hyperbolic orbits of the associated classical flow as a second application of the monodromy operator construction we prove if is an elliptic orbit then p h admits quasimodes which are well localized near

a recursive approach to local mesh refinement in two and three dimensions abstract the newest vertex strategy for local refinement in two dimensions is reviewed and discussed a recursive algorithm for local refinement of tetrahedral meshes in three dimensions based on similar bisection strategy is described the recursive approach requires certain restrictions on the initial mesh on the other hand under this condition the refinement process can be kept as local as needed and it can be fully inverted simple data structures and derefinement algorithms are also outlined

selective refinement derefinement algorithms for sequences of nested triangulations the construction of changing sequences of irregular and nested triangulations based on the use of conforming refinement derefinement algorithms for triangulations is presented and discussed this strategy is particularly appropriate to combine adaptivity and full multigrid algorithms for dealing with moving fronts or fluid dynamics problems it is shown that the quality of all the triangulations iteratively generated depends only on the geometric characteristics of the initial grid a data structure suitable to create manage and modify series of nested triangulations as well as the main features of the deref prototype package are described and numerical examples are given

homogenization limits and wigner transforms we present a theory for carrying out homogenization limits for quadratic functions called energy densities of solutions of initial value problems ivps with anti self adjoint spatial pseudo differential operators pdos the approach is based on the introduction of phase space wigner matrix measures that are calculated by solving kinetic equations involving the spectral properties of the pdo the weak limits of the energy densities are then obtained by taking moments of the wigner measure the very general theory is illustrated by typical examples like semi classical limits of schrodinger equations with or without a periodic potential the homogenization limit of the acoustic equation in a periodic medium and the classical limit of the dirac equation 1997 john wiley sons inc

an error estimate free and remapping free variational mesh refinement and coarsening method for dissipative solids at finite strains a variational h adaptive finite element formulation is proposed the distinguishing feature of this method is that mesh refinement and coarsening are governed by the same minimization principle characterizing the underlying physical problem hence no error estimates are invoked at any stage of the adaption procedure as a consequence linearity of the problem and a corresponding hilbert space functional framework are not required and the proposed formulation can be applied to highly non linear phenomena the basic strategy is to refine respectively unrefine the spatial discretization locally if such refinement respectively unrefinement results in a sufficiently large reduction respectively sufficiently small increase in the energy this strategy leads to an adaption algorithm having o n complexity local refinement is effected by edge bisection and local unrefinement by the deletion of terminal vertices dissipation is accounted for within a time discretized variational framework resulting in an incremental potential energy in addition the entire hierarchy of successive refinements is stored and the internal state of parent elements is updated so that no mesh transfer operator is required upon unrefinement the versatility and robustness of the resulting variational adaptive finite element formulation is illustrated by means of selected numerical examples

decay of correlations for normally hyperbolic trapping we prove that for evolution problems with normally hyperbolic trapping in phase space correlations decay exponentially in time normally hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it flows with this structure include contact anosov flows classical flows in molecular dynamics and null geodesic flows for black holes metrics the decay of correlations is a consequence of the existence of resonance free strips for green s functions cut off resolvents and polynomial bounds on the growth of those functions in the semiclassical parameter

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

local smoothing for scattering manifolds with hyperbolic trapped sets we prove a resolvent estimate for the laplace beltrami operator on a scattering manifold with a hyperbolic trapped set and as a corollary deduce local smoothing we use a result of nonnenmacher zworski to provide an estimate near the trapped region a result of burq and cardoso vodev to provide an estimate near infinity and the microlocal calculus on scattering manifolds to combine the two

goal oriented adaptive finite element methods for elliptic problems revisited a goal oriented a posteriori error estimation of an output functional for elliptic problems is presented continuous finite element approximations are used in quadrilateral and triangular meshes the algorithm is similar to the classical dual weighted error estimation however the dual weight contains solutions of the proposed patch problems the patch problems are introduced to apply clement and scott zhang type interpolation operators to estimate point values with the finite element polynomials the algorithm is shown to be reliable efficient and convergent

a variationally consistent mesh adaptation method for triangular elements in explicit lagrangian dynamics in this paper a variational formulation for mesh adaptation procedures involving local mesh changes for triangular meshes is presented such local adaptive changes are very well suited for explicit methods as they do not involve significant computational expense they also greatly simplify the projection of field variables from the old to the new meshes crucially the variational nature of the formulation used to derive the equilibrium equations at steps where adaptation takes place ensures that conservation of linear and angular momentum is obtained int j numer meth engng 2000 49 1295 1325 several examples in 2 d showing the application of the proposed adaptive algorithms are used to demonstrate the validity of the methodology proposed copyright 2009 john wiley sons ltd

parametrices and dispersive estimates for schrodinger operators with variable coefficients in this article we consider variable coefficient time dependent schr odinger evolutions in bbb r n using phase space methods we construct outgoing parametrices and prove strichartz type estimates globally in time this is done in the context of c 2 metrics which satisfy a weak asymptotic flatness condition at infinity

boundary stabilization of transmission problems we study the transmission problem in bounded domains with dissipative boundary conditions under some natural assumptions we prove uniform bounds of the corresponding resolvents on the real axis at high frequency and as a consequence we obtain regions free of eigenvalue to this end we extend the result of cardoso et al distribution of resonances and local energy decay in the transmission problem ii math res lett 6 377 1999 under more general assumptions as an application we get exponential decay of the energy of the solutions of the corresponding mixed boundary value problems

near sharp strichartz estimates with loss in the presence of degenerate hyperbolic trapping we consider an n dimensional spherically symmetric asymptotically euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic by separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond ehrenfest time we show that solutions to the linear schrodinger equation with initial conditions localized on a spherical harmonic satisfy strichartz estimates with a loss depending only on the dimension n and independent of the degeneracy the strichartz estimates are sharp up to an arbitrary 0 loss this is in contrast to christianson and wunsch amer j math 2013 where it is shown that solutions satisfy a sharp local smoothing estimate with loss depending only on the degeneracy of the trapped set independent of the dimension

an algorithm for a special case of a generalization of the richardson extrapolation process a special case of a generalization of the richardson extrapolation process is considered and its complete solution is given in closed form using this an algorithm for implementing the extrapolation is devised it is shown that this algorithm needs a very small amount of arithmetic operations and very little storage convergence and stability properties for some cases are also considered

decay and non decay of the local energy for the wave equation on the de sitter schwarzschild metric we describe an expansion of the solution of the wave equation on the de sitter schwarzschild metric in terms of resonances the principal term in the expansion is due to a resonance at 0 the error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an varepsilon derivative loss in the angular directions

smoothing and dispersive estimates for 1d schrodinger equations with bv coefficients and applications we prove smoothing estimates for schrodinger equations i t x a x x 0 with a x bv real and bounded from below we then bootstrap these estimates to obtain optimal strichartz and maximal function estimates all of which turn out to be identical to the constant coefficient case we also provide counterexamples showing a bv to be in a sense a minimal requirement finally we provide an application to sharp well posedness for a generalized benjamin ono equation

convergence acceleration methods the past decade abstract the aim of this paper is to present a review of the most significant results obtained the past ten years on convergence acceleration methods it is divided into four sections dealing respectively with the theory of sequence transformations the algorithms for such transformations the practical problems related to their implementation and their applications to various subjects of numerical analysis

an adaptive finite element splitting method for the incompressible navier stokes equations we present an adaptive finite element method for the incompressible navier stokes equations based on a standard splitting scheme the incremental pressure correction scheme the presented method combines the efficiency and simplicity of a splitting method with the powerful framework offered by the finite element method for error analysis and adaptivity an a posteriori error estimate is derived which expresses the error in a goal functional of interest as a sum of contributions from spatial discretization time discretization and a term that measures the deviation of the splitting scheme from a pure galerkin scheme the computational error numerical examples are presented which demonstrate the performance of the adaptive algorithm and high quality efficiency indices it is further demonstrated that the computational error of the navier stokes momentum equation is linear in the size of the time step while the computational error of the continuity equation is quadratic in the size of the time step

sharp low frequency resolvent estimates on asymptotically conical manifolds on a class of asymptotically conical manifolds we prove two types of low frequency estimates for the resolvent of the laplace beltrami operator the first result is a uniform l 2 rightarrow l 2 bound for scal r 1 delta g z 1 scal r 1 when mbox re z is small with the optimal weight scal r 1 the second one is about powers of the resolvent for any integer n we prove uniform l 2 rightarrow l 2 bounds for scal epsilon r n epsilon 2 delta g z n scal epsilon r n when mbox re z belongs to a compact subset of 0 infty and 0 epsilon ll 1 these results are obtained by proving similar estimates on a pure cone with a long range perturbation of the metric at infinity

local bisection refinement for n simplicial grids generated by reflection a simple local bisection refinement algorithm for the adaptive refinement of n simplicial grids is presented the algorithm requires that the vertices of each simplex be ordered in a special way re

variational h adaption in finite deformation elasticity and plasticity we ropose a variational h adaption strategy in which the evolution of the mesh is driven directly by the governing minimum principle this minimum principle is the principle of minimum potential energy in the case of elastostatics and a minimum principle for the incremental static problem of elasto viscoplasticity in particular the mesh is refined locally when the resulting energy or incremental pseudo energy released exceeds a certain threshold value in order to avoid global recomputes we estimate the local energy released by mesh refinement by means of a lower bound obtained by relaxing a local patch of elements this bound can be computed locally which reduces the complexity of the refinement algorithm to o n we also demonstrate how variational h refinement can be combined with variational r refinement to obtain a variational hr refinement algorithm because of the strict variational nature of the h refinement algorithm the resulting meshes are anisotropic and outperform other refinement strategies based on aspect ratio or other purely geometrical measures of mesh quality the versatility and rate of convergence of the resulting approach are illustrated by means of selected numerical examples copyright 2007 john wiley sons ltd

semiclassical non concentration near hyperbolic orbits for a large class of semiclassical pseudodifferential operators including schrodinger operators p h h2 g v x on compact riemannian manifolds we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits more precisely we show that if a is a pseudodifferential operator which is microlocally equal to the identity near the hyperbolic orbit and microlocally zero away from the orbit then u c log 1 h h p h u clog 1 h i a u this generalizes earlier estimates of colin de verdiere and parisse y colin de verdiere b parisse equilibre instable en regime semi classique i concentration microlocale comm partial differential equations 19 1994 1535 1563 equilibre instable en regime semi classique ii conditions de bohr sommerfeld ann inst h poincare phys theor 61 1994 347 367 obtained for a special case and of burq and zworski n burq m zworski geometric control in the presence of a black box j amer math soc 17 2004 443 471 for real hyperbolic orbits

a new adaptive mesh refinement strategy for numerically solving evolutionary pde s a graph based implementation of quadtree meshes for dealing with adaptive mesh refinement amr in the numerical solution of evolutionary partial differential equations is discussed using finite volume methods the technique displays a plug in feature that allows replacement of a group of cells in any region of interest for another one with arbitrary refinement and with only local changes occurring in the data structure the data structure is also specially designed to minimize the number of operations needed in the amr implementation of the new scheme allows flexibility in the levels of refinement of adjacent regions moreover storage requirements and computational cost compare competitively with mesh refinement schemes based on hierarchical trees low storage is achieved for only the children nodes are stored when a refinement takes place these nodes become part of a graph structure thus motivating the denomination autonomous leaves graph alg for the new scheme neighbors can then be reached without accessing their parent nodes additionally linear system solvers based on the minimization of functionals can be easily employed alg was not conceived with any particular problem or geometry in mind and can thus be applied to the study of several phenomena some test problems are used to illustrate the effectiveness of the technique

a novel variational algorithmic formulation for wrinkling at finite strains based on energy minimization application to mesh adaption this paper is concerned with an efficient novel algorithmic formulation for wrinkling at finite strains in contrast to previously published numerical implementations the advocated method is fully variational more precisely the parameters describing wrinkles or slacks together with the unknown deformation mapping are computed jointly by minimizing the potential energy of the considered mechanical system furthermore the wrinkling criteria are naturally included within the presented variational framework the proposed method allows to employ three dimensional constitutive models without any additional modification i e a projection in plane stress space is not required analogously to the wrinkling parameters the non vanishing out of plane component of the strain tensor results conveniently from relaxing the respective helmholtz energy of the membrane the proposed framework is very general and does not rely on any assumption regarding the symmetry of the material i e arbitrary anisotropic hyperelastic models can be consistently taken into account the advantages associated with such a variational method are manifold for instance it opens up the possibility of applying standard optimization algorithms to the numerical implementation this is especially important for highly non linear or singular problems such as wrinkling on the other hand minimization principles provide a suitable basis for a posteriori error estimation and thus for adaptive finite element formulations as a prototype a variational error indicator leading to an efficient h adaption for wrinkling is briefly discussed the performance of the wrinkling approach is demonstrated by selected finite element analyses

regularity and decay of solutions to the stark evolution equation long time behavior and regularity are studied for solutions of the stark equationut i x1 v x u u 0 x l2 rn it is shown that for a class of short range potentialsv x the gain of local smoothness and the decay as t are close to those of the corresponding schrodinger equationut i v x u

proving the non degeneracy of the longest edge trisection by a space of triangular shapes with hyperbolic metric from an initial triangle three triangles are obtained joining the two equally spaced points of the longest edge with the opposite vertex this construction is the base of the longest edge trisection method let be an arbitrary triangle with minimum angle let be any triangle generated in the iterated application of the longest edge trisection let be the minimum angle of thus c with c 3 arctan 3 11 is proved in this paper a region of the complex half plane endowed with the poincare hyperbolic metric is used as the space of triangular shapes the metric properties of the piecewise smooth complex dynamic defined by the longest edge trisection are studied this allows us to obtain the value c

algorithms for refining triangular grids suitable for adaptive and multigrid techniques two general algorithms for refining triangular computational meshes based on the bisection of triangles by the longest side are presented and discussed the algorithms can be applied globally or locally for selective refinement of any conforming triangulation and always generate a new conforming triangulation after a finite number of interactions even when locally used the algorithms also ensure that all angles in subsequent refined triangulations are greater than or equal to half the smallest angle in the original triangulation the shape regularity of all triangles is maintained and the transition between small and large triangles is smooth in a natural way proofs of the above properties are presented the second algorithm is a simpler improved version of the first which retains most of the properties of the latter the algorithms can be used either for constructing irregular computational meshes or for locally refining any given triangulation in this sense they can be adequately combined with adaptive and or multigrid techniques for solving finite element systems examples of the application of the algorithms are given and two possible generalizations are pointed out

a grid generator based on 4 triangles conforming mesh refinement algorithms a procedure to generate 2d meshes of triangles in a quite general fashion allowing for local and selective mesh refinement is presented and discussed the grid generator is based on the iterative application of 4 triangles conforming mesh refinement algorithms for triangulations which are also introduced in this paper these algorithms are modified versions of those proposed in int j numer methods eng 20 745 756 1984 and they can be used for global refinement of a grid as well as for local refinement the grid generator works in the following way given any initial coarse triangulation which properly defines the geometry of the problem a set of user defined refinement subregions ri i 1 2 n and an associated set of tolerance parameters hi an irregular and conforming final triangulation is automatically constructed in such a way that the diameter of all the triangles contained in ri is smaller than hi i 1 2 n moreover all angles in the final triangulation are greater than or equal to half the smallest angle in the initial coarse one the refinement is propagated only to assure the conformity and smoothness of the grid and consequently the number of involved nodes will be minimized the refinement of the final mesh will be determined by the subregions ri and the parameters ni and will be essentially independent of the initial coarse grid the 4 triangles conforming mesh refinement algorithms are presented and their properties discussed the implementation of these techniques is discussed and examples of the application of the grid generator are given

local smoothing for the schrodinger equation with a prescribed loss we consider a family of surfaces of revolution each with a single periodic geodesic which is degenerately unstable we prove a local smoothing estimate for solutions to the linear schrodinger equation with a loss that depends on the degeneracy and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale as a byproduct of our proof we obtain a cutoff resolvent estimate with a sharp polynomial loss

local modification of meshes for adaptive and or multigrid finite element methods abstract a mesh generation approach that allows the local modification of meshes of simplices is reviewed and discussed the generation and modification of sequences of nested irregular grids as needed for multigrid algorithms in the adaptive finite element setting can also be managed in 2 dimensions the meshes constructed are guaranteed to be nondegenerate and smooth empirical evidence has shown the 3d algorithm is in practice a powerful and realiable tool for mesh refinement

multilevel solvers for unstructured surface meshes parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing such parameterization approaches give rise to large and exceedingly ill conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies since the underlying meshes are very fine to begin with such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy in this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature we introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner