scaling and eigenstates for a class of one dimensional quasiperiodic lattices numerical and analytical calculations of the electronic properties of a one dimensional quasiperiodic lattice are presented the lattice consists of two types of bonds a and b whose distribution follows the recursion formula sl 1 sml 1snl where m n 1 2 3 and sl is a building block sequence for l or 2 an exact evaluation of the wavefunctions yields the scaling for several pairs of values of m and n numerical calculations of the wavefunction and the resistance reveal a rich self similar structure as well as localisation for one of the quasiperiodic sequences studied

general relaxation schemes in multigrid algorithms for higher order singularity methods abstract relaxation schemes based on an approximate and incomplete factorization technique af are described these af schemes allow one to construct a fast multigrid method for solving integral equations of the second as well as of the first kind novel items are the smoothing factors found for integral equations of the first kind and the comparison with similar results for equations of the second kind application of the mg algorithm shows convergence of a second order accurate panel method to the level of the truncation error within two multigrid cycles

an a posteriori error estimate for finite element approximations of the navier stokes equations abstract in this paper an a posteriori error estimate for the steady state navier stokes equations based on element residual methods is developed here a special norm is designed to measure the error in velocity and pressure for incompressible flow problems we prove that under mild assumptions the new norm is asymptotically equivalent to the product space norm for the navier stokes equations moreover by summing the solutions of local elliptic residual problems the computed global error is shown to be always bounded below by the true global error when measured in the new norm a numerical experiment on the backstep channel problem is given to demonstrate the effectiveness of the error estimate

reduced integration at superconvergent points in isogeometric analysis abstract we propose a new reduced integration rule for isogeometric analysis iga based on the concept of variational collocation it has been recently shown that if a discrete space is constructed by smooth and pointwise non negative basis functions there exists a set of points named cauchy galerkin cg points such that collocation performed at these points can reproduce the galerkin solution of various boundary value problems exactly since cg points are not known a priori estimates are necessary in practice and can be found based on superconvergence theory in this contribution we explore the use of estimated cg points i e superconvergent points as numerical quadrature points to obtain an efficient and stable reduced quadrature rule in iga we use the weighted residual formulation as basis for our new quadrature rule so that the proposed approach can be considered intermediate between the standard accurately integrated galerkin variational formulation and the direct evaluation of the strong form in collocation approaches the performance of the method is demonstrated by several examples for odd degrees of discretization we obtain spatial convergence rates and accuracy very close to those of accurately integrated standard galerkin with a quadrature rule of two points per parametric direction independently of the degree

overlapping and non overlapping domain decomposition methods for large scale meshless efg simulations abstract meshless methods have a number of virtues in problems concerning crack growth and propagation strain localization dynamic shear band propagation projectile penetration among others the main drawback of such methods is that the resulting matrices are more densely populated and the computational cost for the formulation and solution of the problem is much higher than the conventional fem this is the reason for their limited application to academic and small scale problems until now in this paper we introduce a novel approach for reducing the computational cost of meshless element free galerkin efg methods by employing domain decomposition techniques on the physical as well as on the algebraic domains specifically we propose the implementation of the dual domain decomposition feti family of methods that have been successfully used with fem in many problems in computational mechanics we address the issues of dividing an overlapping meshfree efg domain to several overlapping subdomains following the algebraic decomposition of the feti method and alternatively on non overlapping subdomains by modifying the required displacement compatibility conditions and by improving the accuracy of the solution through a stress update procedure

isogeometric collocation for elastostatics and explicit dynamics we extend the development of collocation methods within the framework of isogeometric analysis iga to multi patch nurbs configurations various boundary and patch interface conditions and explicit dynamic analysis the methods developed are higher order accurate stable with no hourglass modes and efficient in that they require a minimum number of quadrature evaluations the combination of these attributes has not been obtained previously within standard finite element analysis

commuting finite blaschke products with no fixed points in the unit disk abstract in this paper we study when two finite blaschke products commute we complete previous results by chalendar and mortini when they have a fixed point in the unit disk and by arteaga when they do not have a fixed point in the unit disk

parallel computation of meshless methods for explicit dynamic analysis a parallel computational implementation of modern meshless methods is presented for explicit dynamic analysis the procedures are demonstrated by application of the reproducing kernel particle method rkpm aspects of a coarse grain parallel paradigm are detailed for a lagrangian formulation using model partitioning integration points are uniquely defined on separate processors and particle definitions are duplicated as necessary so that all support particles for each point are defined locally on the corresponding processor several partitioning schemes are considered and a reduced graph based procedure is presented partitioning issues are discussed and procedures to accommodate essential boundary conditions in parallel are presented explicit mpi message passing statements are used for all communications among partitions on different processors the effectiveness of the procedure is demonstrated by highly deformable inelastic example problems

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis abstract we continue the study initiated in hughes et al 2010 in search of optimal quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis these rules are optimal in the sense that there exists no other quadrature rule that can exactly integrate the elements of the given spline space with fewer quadrature points we extend the algorithm presented in hughes et al 2010 with an improved starting guess which combined with arbitrary precision arithmetic results in the practical computation of quadrature rules for univariate non uniform splines up to any precision explicit constructions are provided in sixteen digits of accuracy for some of the most commonly used uniform spline spaces defined by open knot vectors we study the efficacy of the proposed rules in the context of full and reduced quadrature applied to two and three dimensional diffusion reaction problems using tensor product and hierarchically refined splines and prove a theorem rigorously establishing the stability and accuracy of the reduced rules

integration by interpolation and look up for galerkin based isogeometric analysis abstract even though isogeometric analysis has a clear advantage regarding the number of degrees of freedom needed to achieve a certain level of accuracy the time needed for matrix assembly by means of numerical integration of stiffness or mass matrix entries constitutes a severe bottleneck in the process in order to overcome this difficulty we propose the new approach of integration by interpolation and lookup iil firstly applying spline interpolation to the common factors in the occurring integrals approximately transforms them into integrals of piecewise polynomial functions whose integrands are expressed in tensor product b spline form the common factors represent the influence of the geometry mapping i e the nurbs domain parameterization and the contributions of possibly non constant material coefficients secondly these integrals are evaluated exactly using pre computed look up tables for integrals of tri products of univariate b splines and their derivatives for the model case of elliptic problems we perform a theoretical analysis to demonstrate that the iil method maintains the overall approximation order of the galerkin discretization provided that the spline interpolation is sufficiently accurate moreover we provide a comparison of the computational complexity of our method with that of a standard gauss quadrature method finally we present experimental results to illustrate the performance of the iil method and to support our theoretical results

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows ii efficient flux quadrature a new and efficient quadrature rule for the flux integrals arising in the space time discontinuous galerkin discretization of the euler equations in a moving and deforming space time domain is presented and analyzed the quadrature rule is a factor three more efficient than the commonly applied quadrature rule and does not affect the local truncation error and stability of the numerical scheme the local truncation error of the resulting numerical discretization is determined and is shown to be the same as when product gauss quadrature rules are used details of the approximation of the dissipation in the numerical flux are presented which render the scheme consistent and stable the method is successfully applied to the simulation of a three dimensional transonic flow over a deforming wing

a coupled element free galerkin boundary element method for stress analysis of two dimensional solids element free galerkin efg method is a newly developed meshless method for solving partial differential equations using moving least squares interpolants it is however computationally expensive for many problems a coupled efg boundary element be method is proposed in this paper to improve the solution efficiency a procedure is developed for the coupled efg be method so that the continuity and compatibility are preserved on the interface of the two domains where the efg and be methods are applied the present coupled efg be method has been coded in fortran the validity and efficiency of the efg be method are demonstrated through a number of examples it is found that the present method can take the full advantages of both efg and be methods it is very easy to implement and very flexible for computing displacements and stresses of desired accuracy in solids with or without infinite domains

orbits of non elliptic disc automorphisms on h p motivated by the invariant subspace problem we describe explicitly the closed subspace h 2 generated by the limit points in the h 2 norm of the orbit of a thin blaschke product b under composition operators c phi induced by non elliptic automorphisms this description exhibits a surprising connection to model spaces finally we give a constructive characterization of the c phi eigenfunctions in h p for 1 p infinity c 2011 elsevier inc all rights reserved

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

fast formation and assembly of finite element matrices with application to isogeometric linear elasticity abstract recently a new formation and assembly strategy was proposed in 1 which resulted in significant speedups in the formation and assembly time of the galerkin mass matrix in isogeometric analysis the strategy relies on three key ingredients 1 assembly row by row instead of element by element and an efficient formation strategy based on 2 sum factorization and 3 weighted quadrature that is applied to each specific row of the matrix compared to traditional element procedures applied to three dimensional problems the computational complexity is lowered from o p 9 per degree of freedom to o p 4 where p is the order of polynomials this is close to the theoretical minimum of o p 3 attained by for example collocation consequently this type of formation and assembly scales favorably with polynomial degree which opens the way for high order isogeometric analysis employing k refinement that is use of maximally smooth higher order splines in this work we discuss various important details for the practical implementation of the weighted quadrature formation strategy proposed in 1 specifically we extend the weighted quadrature scheme to accurately integrate the elements of the stiffness matrix in linear elasticity and propose a means of distributing quadrature points for non uniform mixed continuity spline spaces furthermore we discuss efficient access and assignment into the prevalent sparse matrix data structures namely compressed sparse row csr and compressed sparse column csc in particular row by row or column by column assembly allows matrix rows or columns respectively to be formed contiguously in the storage order of the sparse matrix thereby minimizing the memory overhead and eliminating the addition assignment operation on sparse matrices several three dimensional benchmark problems illustrate the efficiency and efficacy of the proposed formation and assembly technique applied to isogeometric linear elasticity we show that the accuracy of full gauss quadrature is maintained while the computational burden of forming the matrix equations is significantly reduced

fixed points of commuting holomorphic mappings other than the wolff point let be the unit disc of c and let f g hol be such that f o g g o f for a 1 let fix a f p l lim r 1 f rp p lim r 1 f rp a we study the behavior of g on fix a f in particular we prove that g fix a f fix a f as a consequence besides conditions for fix a f fix a g 0 we prove a conjecture of c cowen in case f and g are univalent mappings

multigrid methods for space fractional partial differential equations we propose some multigrid methods for solving the algebraic systems resulting from finite element approximations of space fractional partial differential equations sfpdes it is shown that our multigrid methods are optimal which means the convergence rates of the methods are independent of the mesh size and mesh level moreover our theoretical analysis and convergence results do not require regularity assumptions of the model problems numerical results are given to support our theoretical findings

on the connectivity of the julia sets of meromorphic functions we prove that every transcendental meromorphic map f with disconnected julia set has a weakly repelling fixed point this implies that the julia set of newton s method for finding zeroes of an entire map is connected moreover extending a result of cowen for holomorphic self maps of the disc we show the existence of absorbing domains for holomorphic self maps of hyperbolic regions whose iterates tend to a boundary point in particular the results imply that periodic baker domains of newton s method for entire maps are simply connected which solves a well known open question

an efficient nesting sub domain gradient smoothing integration algorithm with quadratic exactness for galerkin meshfree methods abstract an efficient nesting sub domain gradient smoothing integration algorithm is proposed for galerkin meshfree methods with particular reference to the quadratic exactness this approach is consistently built upon the smoothed gradients of meshfree shape functions defined on two level nesting triangular sub domains where each integration cell consists of four equal area nesting sub domains first a rational measure is designed to evaluate the error of the gradient smoothing integration for the stiffness matrix thereafter through a detailed analysis of the gradient smoothing integration errors associated with the two level nesting triangular sub domains a quadratically exact algorithm for the stiffness matrix integration is established through optimally combining the contributions from the two level nesting sub domains meanwhile the integration of force terms consistent with the stiffness integration is presented in order to ensure exact quadratic solutions within the galerkin formulation it is noted that the proposed approach with quadratic exactness shares the same foundation as the well established stabilized conforming nodal integration method with linear exactness i e the smoothed derivatives of meshfree shape functions are directly built upon the values of meshfree shape functions on the boundary of the integration cells and the time consuming derivative computations are completely avoided moreover the present formulation has even less integration sampling points than the four point gauss integration numerical examples show very favorable performance regarding the accuracy and efficiency for the proposed approach

dynamics of transcendental henon maps the dynamics of transcendental functions in the complex plane has received a significant amount of attention in particular much is known about the description of fatou components besides the types of periodic fatou components that can occur for polynomials there also exist so called baker domains periodic components where all orbits converge to infinity as well as wandering domains in trying to find analogues of these one dimensional results it is not clear which higher dimensional transcendental maps to consider in this paper we find inspiration from the extensive work on the dynamics of complex henon maps we introduce the family of transcendental henon maps and study their dynamics emphasizing the description of fatou components we prove that the classification of the recurrent invariant fatou components is similar to that of polynomial henon maps and we give examples of baker domains and wandering domains

renormalization group approach to the local green functions of a family of generalized fibonacci lattices we study the local electronic properties of a family of generalized fibonacci lattices associated with the sequences which are given by the inflation rule a b ab n a where n is an arbitrary positive integer greater than one a unified real space renormalization group approach is presented to calculated the local green function and the local density of states at any given site

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

discontinuous approximation of viscous two phase flow in heterogeneous porous media runge kutta discontinuous galerkin rkdg and discontinuous finite volume element dfve methods are applied to a coupled flow transport problem describing the immiscible displacement of a viscous incompressible fluid in a non homogeneous porous medium the model problem consists of nonlinear pressure velocity equations assuming brinkman flow coupled to a nonlinear hyperbolic equation governing the mass balance saturation equation the mass conservation properties inherent to finite volume based methods motivate a dfve scheme for the approximation of the brinkman flow in combination with a rkdg method for the spatio temporal discretization of the saturation equation the stability of the uncoupled schemes for the flow and for the saturation equations is analyzed and several numerical experiments illustrate the robustness of the numerical method

subnormality and composition operators on h2 if is a non constant analytic function defined on the unit disk d such that d d the composition operator c is the operator on the hardy space h2 of the unit disk defined by c f f in this paper we investigate the relationship between properties of the symbol and the subnormality of the operators c and c the main theorem shows under a regularity hypothesis that c is subnormal if and only if is in a restricted class of linear fractional transformations the two previously known special cases the affine transformations associated with the cesaro operator and the automorphisms are included as the extreme examples the proof of the main theorem consists of verifying a moment condition and yields in addition the construction of a measure that exhibits cgj as multiplication by an analytic function on p2 and the identification of the minimal normal extension as a weighted sum of shifts

classification of semigroups of linear fractional maps in the unit ball we give a complete classiflcation up to conjugation of continuous semigroups of linear fractional self maps of the unit ball

iteration and the solution of functional equations for functions analytic in the unit disk this paper considers the classical functional equations of schroeder f o p xf and abel f o p f 1 and related problems of fractional iteration where 9p is an analytic mapping of the open unit disk into itself the main theorem states that under very general conditions there is a linear fractional transformation 4 and a function a analytic in the disk such that a oa p and that with suitable normalization 4 and a are unique in particular the hypotheses are satisfied if 4p is a probability generating function that does not have a double zero at 0 this intertwining relates solutions of functional equations for t to solutions of the corresponding equations for d for example it follows that if 4p has no fixed points in the open disk then the solution space of f o p xf is infinite dimensional for every nonzero x although the discrete semigroup of iterates of t usually cannot be embedded in a continuous semigroup of analytic functions mapping the disk into itself we find that for each z in the disk all sufficiently large fractional iterates of p can be defined at z this enables us to find a function meromorphic in the disk that deserves to be called the infinitesimal generator of the semigroup of iterates of p if the iterates of p can be embedded in a continuous semigroup we show that the semigroup must come from the corresponding semigroup for 4 and thus be real analytic in t the proof of the main theorem is not based on the well known limit technique introduced by koenigs 1884 but rather on the construction of a riemann surface on which an extension of 9p is a bijection much work is devoted to relating characteristics of p to the particular linear fractional transformation constructed in the theorem

on fractional iterates of a brouwer homeomorphism embeddable in a flow we present a method for finding continuous and consequently homeomorphic orientation preserving iterative roots of a brouwer homeomorphism which is embeddable in a flow to obtain the roots we use a countable family of maximal parallelizable regions of the flow which is a cover of the plane the maximal parallelizable regions are unions of equivalence classes of an appropriate equivalence relation we show that if an equivalence class is invariant under the nth iterate of a brouwer homeomorphism g then it is invariant under g we use this fact to prove that each maximal parallelizable region of the flow must be invariant under all homeomorphic orientation preserving iterative roots of the given brouwer homeomorphism

an implicit high order hybridizable discontinuous galerkin method for nonlinear convection diffusion equations in this paper we present hybridizable discontinuous galerkin methods for the numerical solution of steady and time dependent nonlinear convection diffusion equations the methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary applying the newton raphson procedure and the hybridization technique we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every newton iteration the high number of globally coupled degrees of freedom in the discontinuous galerkin approximation is therefore significantly reduced we then extend the method to time dependent problems by approximating the time derivative by means of backward difference formulae when the time marching method is p 1 th order accurate and when polynomials of degree p 0 are used to represent the scalar variable each component of the flux and the approximate trace we observe that the approximations for the scalar variable and the flux converge with the optimal order of p 1 in the l 2 norm finally we apply element by element postprocessing schemes to obtain new approximations of the flux and the scalar variable the new approximate flux which has a continuous interelement normal component is shown to converge with order p 1 in the l 2 norm the new approximate scalar variable is shown to converge with order p 2 in the l 2 norm the postprocessing is performed at the element level and is thus much less expensive than the solution procedure for the time dependent case the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required extensive numerical results are provided to demonstrate the performance of the present method

mixed stress displacement isogeometric collocation for nearly incompressible elasticity and elastoplasticity abstract we propose a mixed stress displacement isogeometric collocation method for nearly incompressible elastic materials and for materials exhibiting von mises plasticity the discretization is based on isogeometric analysis iga with non uniform rational b splines nurbs as basis functions as compared to conventional iga galerkin formulations isogeometric collocation methods offer a high potential of computational cost reduction for higher order discretizations as they eliminate the need for quadrature in the proposed mixed formulation both stress and displacement fields are approximated as primary variables with the aim of treating volumetric locking and instability issues which occur in displacement based isogeometric collocation for nearly incompressible elasticity and von mises plasticity the performance of the proposed approach is demonstrated by several numerical examples

a nonlinear filter for high order discontinuous galerkin discretizations with discontinuity resolution within the cell the nonlinear filter introduced by yee et al 1999 27 and extensively used in the development of low dissipative well balanced high order accurate finite difference schemes is adapted to the finite element context of discontinuous galerkin dg discretizations the filter operator is constructed in the canonical computational domain for the standard cubical element where it is applied to the computed conservative variables in a direction per direction basis filtering becomes possible for all element types in unstructured meshes using collapsed coordinate transformations the performance of the proposed nonlinear filter for dg discretizations is demonstrated and evaluated for different orders of expansions for one dimensional and multidimensional problems with exact solutions it is shown that for higher order discretizations discontinuity resolution within the cell is achieved and the design order of accuracy is preserved the filter is applied for a number of standard inviscid flow test problems including strong shocks interactions to demonstrate that the proposed dissipative mechanism for dg discretizations yields superior results compared to the results obtained with the total variation bounded tvb limiter and high order hierarchical limiting the proposed approach is suitable for p adaptivity in order to locally enhance resolution of three dimensional flow simulations that include discontinuities and complex flow features

finite difference ghost point multigrid methods on cartesian grids for elliptic problems in arbitrary domains in this paper we present a numerical method for solving elliptic equations in an arbitrary domain described by a level set function with general boundary conditions dirichlet neumann robin etc on cartesian grids using finite difference discretization and non eliminated ghost values a system of n i n g equations in n i n g unknowns is obtained by finite difference discretization on the n i internal grid points and second order interpolation to define the conditions for the n g ghost values the resulting large sparse linear system is then solved by a multigrid technique the novelty of the papers can be summarized as follows general strategy to discretize the boundary condition to second order both in the solution and its gradient a relaxation of inner equations and boundary conditions by a fictitious time method inspired by the stability conditions related to the associated time dependent problem with a convergence proof for the first order scheme an effective geometric multigrid which maintains the structure of the discrete system at all grid levels it is shown that by increasing the relaxation step of the equations associated to the boundary conditions a convergence factor close to the optimal one is obtained several numerical tests including variable coefficients anisotropic elliptic equations and domains with kinks show the robustness efficiency and accuracy of the approach

gpu acceleration of stiffness matrix calculation and efficient initialization of efg meshless methods abstract meshless methods have a number of virtues in problems concerning crack growth and propagation large displacements strain localization and complex geometries among other despite the fact that they do not rely on a mesh meshless methods require a preliminary step for the identification of the correlation between nodes and gauss points before building the stiffness matrix this is implicitly performed with the mesh generation in fem but must be explicitly done in efg methods and can be time consuming furthermore the resulting matrices are more densely populated and the computational cost for the formulation and solution of the problem is much higher than the conventional fem this is mainly attributed to the vast increase in interactions between nodes and integration points due to their extended domains of influence for these reasons computing the stiffness matrix in efg meshless methods is a very computationally demanding task which needs special attention in order to be affordable in real world applications in this paper we address the pre processing phase dealing with the problem of defining the necessary correlations between nodes and gauss points and between interacting nodes as well as the computation of the stiffness matrix a novel approach is proposed for the formulation of the stiffness matrix which exhibits several computational merits one of which is its amenability to parallelization which allows the utilization of graphics processing units gpus to accelerate computations

gpu accelerated boundary element method for helmholtz equation in three dimensions recently the application of graphics processing units gpus to scientific computations is attracting a great deal of attention because gpus are getting faster and more programmable in particular nvidia s gpus called compute unified device architecture enable highly mutlithreaded parallel computing for non graphic applications this paper proposes a novel way to accelerate the boundary element method bem for three dimensional helmholtz equation using cuda adopting the techniques for the data caching and the double single precision floating point arithmetic we implemented a gpu accelerated bem program for geforce 8 series gpus the program performed 6 23 times faster than a normal bem program which was optimized for an intel s quad core cpu for a series of boundary value problems with 8000 128000 unknowns and it sustained a performance of 167 gflop s for the largest problem 1 058 000 unknowns the accuracy of our bem program was almost the same as that of the regular bem program using the double precision floating point arithmetic in addition our bem was applicable to solve realistic problems in conclusion the present gpu accelerated bem works rapidly and precisely for solving large scale boundary value problems for helmholtz equation copyright 2009 john wiley sons ltd

geometric models and compactness of composition operators abstract this work explores some of the terrain between functional equations geometric function theory and operator theory it is inspired by the fact that whenever a composition operator or one of its powers is compact on the hardy space h 2 then its eigenfunctions cannot grow too quickly on the unit disc the goal is to show that under certain natural and necessary additional conditions there is a converse slow growth of eigenfunctions implies compactness we interpret the slow growth condition in terms of the geometry of the principal eigenfunction of the composition operator the konigs function of the inducing map we emphasize throughout the importance of this eigenfunction in providing a simple geometric model for the operator s inducing map

portrait of the four regular super exponentials to base sqrt 2 we introduce the concept of regular super functions at a fixed point it is derived from the concept of regular iteration a super function f of h is a solution of f z 1 h f z we provide a condition for f being entire we also give two uniqueness criteria for regular super functions in the particular case h x b x we call f super exponential h has two real fixed points for b between 1 and e 1 e exemplary we choose the base b sqrt 2 and portray the four classes of real regular super exponentials in the complex plane there are two at fixed point 2 and two at fixed point 4 each class is given by the translations along the x axis of a suitable representative both super exponentials at fixed point 4 one strictly increasing and one strictly decreasing are entire both super exponentials at fixed point 2 one strictly increasing and one strictly decreasing are holomorphic on a right half plane all four super exponentials are periodic along the imaginary axis only the strictly increasing super exponential at 2 can satisfy f 0 1 and can hence be called tetrational we develop numerical algorithms for the precise evaluation of these functions and their inverses in the complex plane we graph the two corresponding different half iterates of h z sqrt 2 z an apparent symmetry of the tetrational to base sqrt 2 disproved

development of an adaptive discontinuous galerkin finite element method for advection reaction equations abstract this paper presents an adaptive discontinuous galerkin finite element method to solve linear advection reaction equations the method was developed for the prediction of hemolysis in prosthetic devices where the flow may include recirculation zones to ensure the well posedness of the problem we solve the transient form of the pde time discretization is achieved with an implicit euler scheme in order to obtain the steady state solution rapidly linear interpolation functions are used to approximate the hemolysis field the a posteriori estimation of the error is obtained by solving an advection reaction equation for the error which is approximated with quadratic interpolation functions then elementary estimations of the error are computed using a semi norm developed to achieve the best possible results even in the presence of recirculation code verification to assess the convergence and conservation properties of the method is performed with the method of manufactured solutions and with a conservation test using a sharp gaussian function as a reaction term excellent results are obtained in both cases predictions of hemolysis in a dialysis cannula are presented as an engineering application of our method

angular and unrestricted limits of one parameter semigroups in the unit disk abstract we study local boundary behaviour of one parameter semigroups of holomorphic functions in the unit disk earlier under some additional condition the position of the denjoy wolff point it was shown in 13 that elements of one parameter semigroups have angular limits everywhere on the unit circle and unrestricted limits at all boundary fixed points we prove stronger versions of these statements with no assumption on the position of the denjoy wolff point in contrast to many other problems in the question of existence for unrestricted limits it appears to be more complicated to deal with the boundary denjoy wolff point the case not covered in 13 than with all the other boundary fixed points of the semigroup

a gpu based multi level algorithm for boundary value problems abstract a novel geometric single grid multi level gsgml algorithm is presented for the numerical solution of boundary value problems bvps the algorithm is specifically designed for parallel scalability with high occupancy of streaming processors inside many core architectures like the graphics processing unit gpu the algorithm consists of iterative superposed operations on a single grid and is composed of two full grid routines a restriction and a coarsened interpolation relaxation the restriction is a local average of scalar or vector fields and the interpolation relaxation is a simultaneous application of coarsened finite difference operators and interpolations in complementary subsets of the grid the routines are scheduled in a saw like refining cycle convergence to machine precision is achieved repeating the full cycle using residuals and superposition the algorithm is attractive for its computational simplicity and provides a competitive gpu based fast solver for elliptic partial differential equations like the poisson helmholtz equation applications shown in this work include the deformation of two dimensional grids the computation of three dimensional streamlines for a singular trifoil knot vortex and the calculation of three dimensional electric potentials in heterogeneous dielectric media

large calculation of the flow over a hypersonic vehicle using a gpu graphics processing units are capable of impressive computing performance up to 518gflops peak performance various groups have been using these processors for general purpose computing most efforts have focussed on demonstrating relatively basic calculations e g numerical linear algebra or physical simulations for visualization purposes with limited accuracy this paper describes the simulation of a hypersonic vehicle configuration with detailed geometry and accurate boundary conditions using the compressible euler equations to the authors knowledge this is the most sophisticated calculation of this kind in terms of complexity of the geometry the physical model the numerical methods employed and the accuracy of the solution the navier stokes stanford university solver nssus was used for this purpose nssus is a multi block structured code with a provably stable and accurate numerical discretization which uses a vertex based finite difference method a multi grid scheme is used to accelerate the solution of the system based on a comparison of the intel core 2 duo and nvidia 8800gtx speed ups of over 40x were demonstrated for simple test geometries and 20x for complex geometries

identity principles for commuting holomorphic self maps of the unit disc abstract let f g be two commuting holomorphic self maps of the unit disc if f and g agree at the common wolff point up to a certain order of derivatives no more than 3 if the wolff point is on the unit circle then f g

gpu accelerated computation of the isogeometric analysis stiffness matrix abstract due to high regularity across mesh elements isogeometric analysis achieves higher accuracy per degree of freedom and improved spectrum properties among others compared with finite element analysis however this inherent feature of isogeometric analysis increases the density of the stiffness matrix and requires more elaborate numerical integration schemes for its computation for these reasons the assembly of the stiffness matrix in isogeometric analysis is a computationally demanding task which needs special attention in order to be affordable for real world applications in this paper we address the computational efficiency of assembling the stiffness matrix using the standard element wise gaussian quadrature a novel approach is proposed for the formulation of the stiffness matrix which exhibits several computational merits among them its amenability to parallelization and the efficient utilization of the graphics processing units to drastically accelerate computations

a spacetime discontinuous galerkin method for scalar conservation laws abstract we present a spacetime discontinuous galerkin sdg finite element method for scalar hyperbolic conservation laws the method is consistent with the integral forms of the conservation law and its associated entropy condition written in terms of the physical godunov flux on each spacetime element the discrete basis functions are piecewise continuous in spacetime admitting discontinuities across element boundaries the resulting bubnov galerkin method is high order stable for both convex and non convex flux functions it does not require stabilization beyond the basic galerkin projection the sdg method is applicable to both layered and unstructured spacetime grids an element by element solution scheme delivers o n computational complexity where n is the number of elements when applied to spacetime meshes that conform to a special causality constraint we employ two different limiters to control the local overshoot and undershoot that the basic sdg projection generates near shocks

a new hierarchical partition of unity formulation of efg meshless methods abstract meshless methods have shown increased accuracy and better convergence rates compared to other well known simulation methods in a variety of computational mechanics problems their computational cost is relatively higher especially in adaptivity analysis when new nodes are inserted at each step where the construction of updated shape functions requires a significant amount of computational effort in this paper two hierarchical formulations are proposed in the context of an h type refinement scheme the addition of new nodes and subsequently the re calculation of the influenced moment matrices that are necessary for obtaining the shape functions and their derivatives and subsequently for the construction of the stiffness matrix are properly addressed both hierarchical schemes do not need the recalculation of the initial stiffness matrix but only the additional node contributions to each shape function field while the second scheme produces purely hierarchically refined stiffness matrices leaving the initial stiffness matrix unmodified

weighted composition semigroups on hardy spaces abstract the operator semigroups of the title are studied on hardy spaces conditions are found for strong continuity and the infinitesimal generator and its point spectrum are identified a particular semigroup is used to study an averaging operator

the stable subset of a univalent self map we give a complete description of the stable subset the union of all backward orbit with bounded step and of the pre models of a univalent self map f x rightarrow x where x is a kobayashi hyperbolic cocompact complex manifold such as the ball or the polydisc in mathbb c q the result is obtained studying the complex structure of a decreasing intersection of complex manifolds all biholomorphic to x

a priori mesh quality estimation via direct relation between truncation error and mesh distortion the purpose of the present work is the derivation and evaluation of a priori mesh quality indicators for structured unstructured as well as hybrid grids emphasis is placed on deriving direct relations between the indicators and mesh distortion the work is based on use of the finite volume discretization for evaluation of first order spatial derivatives the analytic form of the truncation error is derived and applied to elementary types of mesh distortion including typical hybrid grid interfaces the corresponding analytic expressions provide direct relations between computational accuracy and the degree of stretching skewness shearing and non alignment of the mesh

phase transition in aperiodic quantum ising models phase transition in the aperiodic quantum ising models were studied in which the transverse magnetic fields are uniform and the couplings are arranged according to the aperiodic sequences that are generated by two types of two tile inflation rules with initial condition a p b q a a a m11 b m12 b b a m21 b m22 m12 m21 not equal 0 and a a a m11 b m12 b b b m21 a m22 m12 m22 not equal 0 when the inflation rules have the pisot vijayaraghavan pv property in number theory the generated aperiodic systems undergo phase transition at the critical points when the inflation rules do not have the pv property it is found that only by changing the initial conditions the inflation rules can produce aperiodic systems exhibiting phase transition as well as aperiodic systems which do not exhibit any phase transition

dynamic p adaptive runge kutta discontinuous galerkin methods for the shallow water equations abstract in this paper dynamic p adaptive runge kutta discontinuous galerkin rkdg methods for the two dimensional shallow water equations swe are investigated the p adaptive algorithm that is implemented dynamically adjusts the order of the elements of an unstructured triangular grid based on a simple measure of the local flow properties of the numerical solution time discretization is accomplished using optimal strong stability preserving ssp rk methods the methods are tested on two idealized problems of coastal ocean modeling interest with complex bathymetry namely the idealization of a continental shelf break and a coastal inlet numerical results indicate the stability robustness and accuracy of the algorithm and it is shown that the use of dynamic p adaptive grids offers savings in cpu time relative to grids with elements of a fixed order p that use either local h refinement or global p refinement to adequately resolve the solution while offering comparable accuracy

a point interpolation meshless method based on radial basis functions a point interpolation meshless method is proposed based on combining radial and polynomial basis functions involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis this non singularity is useful in constructing well performed shape functions furthermore the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property this makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least squares approximation in addition the partial derivatives of shape functions are easily obtained thus improving computational efficiency examples on curve surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high copyright 2002 john wiley sons ltd

mean growth of koenigs eigenfunctions in 1884 g koenigs solved schroeder s functional equation f o a af in the following context is a given holomorphic function mapping the open unit disk u into itself and fixing a point a e u f is holomorphic on u and a is a complex scalar koenigs showed that if 0 equation for p has a unique holomorphic solution cr satisfying a co p a a and o 0 1 moreover he showed that the only other solutions are the obvious ones given by constant multiples of powers of a we call a the koenigs eigenfunction of motivated by fundamental issues in operator theory and function theory we seek to understand the growth of integral means of koenigs eigenfunctions for 0 of to belong to the hardy space hp and show that the condition is necessary when is analytic on the closed disk for many mappings the condition may be expressed as a relationship between p a and derivatives of at points on ou that are fixed by some iterate of our work depends upon a formula we establish for the essential spectral radius of any composition operator on the hardy space hp department of mathematics washington and lee university lexington virginia 24450 e mail address pbourdonfwlu edu department of mathematics michigan state university east lansing michigan 48824 e mail address shapirofmath msu edu this content downloaded from 157 55 39 211 on sat 25 jun 2016 06 33 26 utc all use subject to http about jstor org terms

a quadtree adaptive multigrid solver for the serre green naghdi equations the serre green naghdi sgn equations also known as the fully nonlinear boussinesq wave equations accurately describe the behaviour of dispersive shoaling water waves this article presents and validates a novel combination of methods for the numerical approximation of solutions to the sgn equations the approach preserves the robustness of the original finite volume saint venant solver in particular for the treatment of wetting drying and equilibrium states the linear system of coupled vector equations governing the dispersive sgn momentum sources is solved simply and efficiently using a generic multigrid solver this approach generalises automatically to adaptive quadtree meshes adaptive mesh refinement is shown to provide orders of magnitude gains in speed and memory when applied to the dispersive propagation of waves during the tohoku tsunami the source code test cases and examples are freely available

on a class of composition semigroups in hardy spaces the purpose of this note is to show that r 1 r is not always compact and then to give conditions for com pactness the separation of the semigroups giving rise to compact resolvent from those which do not seems to be very delicate our conditions are in terms of a certain associated univalent function to be introduced below

fast formation of isogeometric galerkin matrices by weighted quadrature abstract in this paper we propose an algorithm for the formation of matrices of isogeometric galerkin methods the algorithm is based on three ideas the first is that we perform the external loop over the rows of the matrix the second is that we calculate the row entries by weighted quadrature the third is that we exploit the local tensor product structure of the basis functions while all ingredients have a fundamental role for computational efficiency the major conceptual change of paradigm with respect to the standard implementation is the idea of using weighted quadrature the test function is incorporated in the integration weight while the trial function the geometry parametrization and the pdes coefficients form the integrand function this approach is very effective in reducing the computational cost while maintaining the optimal order of approximation of the method analysis of the cost is confirmed by numerical testing where we show that for p large enough the time required by the floating point operations is less than the time spent in unavoidable memory operations the sparse matrix allocation and memory write the proposed algorithm allows significant time saving when assembling isogeometric galerkin matrices for all the degrees of the test spline space and paves the way for a use of high degree k refinement in isogeometric analysis

invertible composition operators on hp abstract the weakly closed algebras generated by certain sets of composition operators are shown to be reflexive a structure theorem for invertible composition operators on h2 is obtained and used to show that such operators are reflexive the structure theorem shows that invertible hyperbolic composition operators are similar to cosubnormal operators built up from bilateral weighted shifts another consequence of the structure theorem is that the composition operators induced by hyperbolic disc automorphisms are universal thus the general invariant subspace problem for hilbert space operators is contained in the problem of determining the invariant subspace lattices of these operators

the h p adaptive finite element method for the numerical simulation of compressible flow an h p adaptive finite element method is presented for the numerical simulation of compressible fluid flow in addition to refining the mesh in regions where the estimated error is large the polynomial degree of the elements is increased it is shown that by using high order polynomials highly accurate approximations are obtained to a regularized form of the equations governing viscous compressible flow

efficient quadrature for nurbs based isogeometric analysis abstract we initiate the study of efficient quadrature rules for nurbs based isogeometric analysis a rule of thumb emerges the half point rule indicating that optimal rules involve a number of points roughly equal to half the number of degrees of freedom or equivalently half the number of basis functions of the space under consideration the half point rule is independent of the polynomial order of the basis efficient rules require taking into account the precise smoothness of basis functions across element boundaries several rules of practical interest are obtained and a numerical procedure for determining efficient rules is presented we compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics the new rules represent improvements over those used previously in isogeometric analysis

commuting analytic functions let f and g not conformal automorphisms of the unit disk be analytic mappings of the unit disk into itself we say f and g commute if f o g g o f this paper characterizes those functions g that commute with a given function f several corollaries of this characterization give qualitative information about g given similar information about f and examples are given in each case to show the limitations of the conclusions some of the qualitative properties considered are univalence fixed point sets and whether two such g must commute with each other

an exact systematized renormalization group approach for periodic and aperiodic systems a systematized renormalization group approach is present to study systematically the physical properties of the one dimensional systems arranged in the 1d sequence which are constructed by the inflation rule a b to am11bm12 bm21am22 a basic decimation rule is introduced which is intuitive simplified and exact to investigate the electronic properties of the systems which contain crystals and quasicrystals as typical examples the spectra of global electronic bands of the 1d systems with m11 m12 m21 m22 1 1 0 1 1 1 1 1 and 1 1 2 1 are numerically calculated

megapixel topology optimization on a graphics processing unit we show how the computational power and programmability of modern graphics processing units gpus can be used to efficiently solve large scale pixel based material distribution problems using a gradient based optimality criterion method to illustrate the principle a so called topology optimization problem that results in a constrained nonlinear programming problem with over 4 million decision variables is solved on a commodity gpu

spectral multigrid methods for elliptic equations ii a detailed description of spectral multigrid methods is provided this includes the interpolation and coarse grid operators for both periodic and dirichlet problems the spectral methods for periodic problems use fourier series and those for dirichlet problems are based upon chebyshev polynomials an improved preconditioning for dirichlet problems is given numerical examples and practical advice are included

second angular derivatives and parabolic iteration in the unit disk in this paper we deal with second angular derivatives at denjoy wolff points for parabolic functions in the unit disc namely we study and analyze the existence and the dynamical meaning of this second angular derivative for instance we provide several characterizations of that existence in terms of the so called koenigs function it is worth pointing out that there are two quite different classes of parabolic iteration those with positive hyperbolic step and those with zero hyperbolic step in the first case the koenigs function is in the caratheodory class but in the second case it is even unknown if it is normal therefore the ideas and techniques to approach these two cases are really different in the end we also present several rigidity results related to the second angular derivatives at denjoy wolff points

a limiting approach for dg discretizations on mixed type meshes abstract a limiting procedure of discontinuous galerkin dg discretizations that is suitable for computations of high speed flows with strong shocks around complex geometries using a p adaptive procedure on mixed type quadrilateral triangular meshes is developed a total variation bounded tvb approach is the building block of the limiting procedure the main difference of this work with other recent limiting approaches for dg discretizations that were based on tvb limiting is that a unified approach for limiting on mixed type meshes is obtained this is achieved by applying the tvb limiter on the transformed canonical elements where appropriate estimates of the variation of the approximate solution derivatives are feasible applications for mixed type meshes and p adaptive calculations with the dg method are performed for a number of standard test cases for shock capturing schemes to illustrate the potential of the proposed limiting approach it is demonstrated in the computed results that good resolution of discontinuities and embedded smooth flow features is obtained with quadrilateral triangular and mixed type meshes furthermore the resolution of the present approach on arbitrary type meshes is equally good with the resolution obtained with other limiting approaches of equivalent order obtained on rectangular meshes

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

simultaneous models for commuting holomorphic self maps of the ball abstract we prove that a finite family of commuting holomorphic self maps of the unit ball b q c q admits a simultaneous holomorphic conjugacy to a family of commuting automorphisms of a possibly lower dimensional ball and that such conjugacy satisfies a universal property as an application we describe when a hyperbolic and a parabolic holomorphic self map of b q can commute

a simple algorithm for obtaining nearly optimal quadrature rules for nurbs based isogeometric analysis we develop new quadrature rules for isogeometric analysis based on the solution of a local nonlinear problem a simple and robust algorithm is developed to determine the rules which are exact for important b spline spaces of uniform and geometrically stretched knot spacings we consider both periodic and open knot vector configurations and illustrate the efficiency of the rules on selected boundary value problems we find that the rules are almost optimally efficient but much easier to obtain than optimal rules which require the solution of global nonlinear problems that are often ill posed

application of discontinuous galerkin method in supersonic and hypersonic gas flows abstract most high order methods often suffer from two and three dimensional unstructured grids particularly for supersonic and hypersonic gas flows we applied a high order mixed modal discontinuous galerkin method with a coupled slope limiter to investigate multidimensional cases of supersonic and hypersonic gas flows furthermore we provide detailed implementations and validations of two and three dimensional problems examples include double mach reflection problems and supersonic and hypersonic gas flows around a cylinder in addition we provide case studies of a 3d hypersonic vehicle to evaluate its computational performance on complex hypersonic gas flows this study can be regarded as an attempt to apply the discontinuous galerkin method to engineering problems