discontinuous galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity we consider a finite element in space and quadrature in time discretization of a compressible linear quasistatic viscoelasticity problem the spatial discretization uses a discontinous galerkin finite element method based on polynomials of degree r termed dg r and the time discretization uses a trapezoidal rectangle rule approximation to the volterra history integral both semi and fully discrete a priori error estimates are derived without recourse to gronwall s inequality and therefore the error bounds do not show exponential growth in time moreover the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution e g when using the standard k polynomial basis on simplicies or tensor product polynomials k on quadrilaterals when this is not the case e g using k on quadri laterals the convergence rate is suboptimal in r but remains optimal in h we also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the dg r approximation

isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

extreme growth rate of periodic orbits for equivalent differentiable flows abstract we prove that the extreme growth rate of periodic orbits of differentiable flows is preserved by lipschitz equivalence when all the fixed points are hyperbolic and thus it is an invariant in an open dense subset x h r of the set of c r 1 r flows by contrast for any 1 r there exist two infinite dimensional connected subsets of equivalent c r flows disjoint from x h r such that the growth rate of periodic orbits attains infinity and zero respectively

spectral approximation properties of isogeometric analysis with variable continuity abstract we study the spectral approximation properties of isogeometric analysis with local continuity reduction of the basis such continuity reduction results in a decrement in the interconnection between the degrees of freedom of the mesh which allows for large computational savings during the solution of the resulting linear system the continuity reduction results in extra degrees of freedom that modify the approximation properties of the method the convergence rate of such refined isogeometric analysis is equivalent to that of the maximum continuity basis we show how the breaks in continuity and inhomogeneity of the basis lead to artifacts in the frequency spectra such as stopping bands and outliers and present a unified description of these effects in finite element method isogeometric analysis and refined isogeometric analysis accuracy of the refined isogeometric analysis approximations can be improved by using non standard quadrature rules in particular optimal quadrature rules lead to large reductions in the eigenvalue errors and yield two extra orders of convergence as it occurs in standard isogeometric analysis

isogeometric numerical dispersion analysis for two dimensional elastic wave propagation abstract in this paper we carry out a numerical dispersion analysis for the linear two dimensional elastodynamics equations approximated by means of nurbs based isogeometric analysis in the framework of the galerkin method specifically we consider the analysis of harmonic plane waves in an isotropic and homogeneous elastic medium we compare and discuss the errors associated with the compressional and shear wave velocities and we provide the anisotropic curves for numerical approximations obtained by considering b spline and nurbs basis functions of different regularity namely globally c 0 and c p 1 continuous p being the polynomial degree we conclude our analysis by numerically simulating the seismic wave propagation in a sinusoidal shaped valley with discontinuous elastic parameters across an internal interface

a mortar spectral finite element method for complex 2d and 3d elastodynamic problems abstract we present a hybrid spectral element finite element domain decomposition method for solving elastic wave propagation problems aim of the method is to exploit both the enhanced accuracy of spectral elements allowing significant reductions of the computational load and the flexibility of finite elements in treating irregular geometries and non linear media interfaces between finite and spectral elements are managed by the mortar method which enjoys optimal accuracy and allows to discretize the different regions independently several test cases illustrate the robustness of this method and its tolerance to a significant range of geometrical models characterized by different grid refinements

elliptic algebro geometric solutions of the kdv and akns hierarchies an analytic approach we provide an overview of elliptic algebro geometric solutions of the kdv and akns hierarchies with special emphasis on floquet theoretic and spectral theoretic methods our treatment includes an effective characterization of all stationary elliptic kdv and akns solutions based on a theory developed by hermite and picard

new fractional step runge kutta nystrom methods up to order three abstract fractional step runge kutta nystro fsrkn methods have been revealed to be an excellent option to integrate numerically many multidimensional evolution models governed by second order in time partial differential equations these methods combined with suitable spatial discretizations lead to strong computational cost reductions respect to many classical implicit time integrators in this paper we present the construction process of several implicit fsrkn methods of two and three levels which attain orders up to three and satisfy adequate stability properties we have also performed some numerical experiments in order to show the unconditionally convergent behavior of these schemes as well as their computational advantages

isogeometric analysis of the advective cahn hilliard equation spinodal decomposition under shear flow we present a numerical study of the spinodal decomposition of a binary fluid undergoing shear flow using the advective cahn hilliard equation a stiff nonlinear parabolic equation characterized by the presence of fourth order spatial derivatives our numerical solution procedure is based on isogeometric analysis an approximation technique for which basis functions of high order continuity are employed these basis functions allow us to directly discretize the advective cahn hilliard equation without resorting to a mixed formulation we present steady state solutions for rectangular domains in two dimensions and for the first time in three dimensions we also present steady state solutions for the two dimensional taylor couette cell to enforce periodic boundary conditions in this curved domain we derive and utilize a new periodic bezier extraction operator we present an extensive numerical study showing the effects of shear rate surface tension and the geometry of the domain on the phase evolution of the binary fluid theoretical and experimental results are compared with our simulations

discontinuous galerkin finite element method for the wave equation the symmetric interior penalty discontinuous galerkin finite element method is presented for the numerical discretization of the second order wave equation the resulting stiffness matrix is symmetric positive definite and the mass matrix is essentially diagonal hence the method is inherently parallel and leads to fully explicit time integration when coupled with an explicit time stepping scheme optimal a priori error bounds are derived in the energy norm and the l 2 norm for the semidiscrete formulation in particular the error in the energy norm is shown to converge with the optimal order cal o h min s ell with respect to the mesh size h the polynomial degree ell and the regularity exponent s of the continuous solution under additional regularity assumptions the l 2 error is shown to converge with the optimal order cal o h min s ell numerical results confirm the expected convergence rates and illustrate the versatility of the method

a quadrature based superconvergent isogeometric frequency analysis with macro integration cells and quadratic splines abstract a quadrature based superconvergent isogeometric method is presented for efficient frequency analysis with macro integration cells and quadratic splines the present development particularly refers to the isogeometric analysis of scalar wave equations the single element two element and three element macro integration cells are proposed to develop a set of explicit superconvergent quadrature rules with a 6th order frequency accuracy in contrast to the 4th order frequency accuracy associated with the standard isogeometric formulation with consistent mass matrix the 1d quadrature rules with various macro integration cells are derived from the condition of exact integration of the higher order mass matrix as well as the stiffness matrix consequently 3 point 5 point and 7 point quadrature rules with identical precision are established for the single element two element and three element macro integration cells respectively by construction these quadrature rules exactly recover the isogeometric higher order mass matrix formulation with frequency superconvergence in 1d case the 2d and 3d superconvergent quadrature rules with versatile integration cells are directly constructed through the tensor product formulation of the 1d integration algorithms it is shown that the multidimensional isogeometric analysis employing the proposed quadrature rules for both mass and stiffness matrices does produce the frequency superconvergence simultaneously without the wave propagation direction dependence problem which needs special treatments for the multidimensional higher order mass matrix formulation the proposed approach is featured by its simplicity for numerical implementation and efficiency using macro integration cells numerical examples confirm the efficacy of the present methodology

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

stability properties of one parameter flows this thesis is in two parts the first part consists of chapters 1 4 and the second part is chapter 5 in the first part we consider the idea of approximating pieces of orbits by a single orbit there are many examples of such properties approximation property a p see chapter one definitions 1 2 1 4 specification property s p 24 pseudo orbit tracing property p o t p 25 and definition 4 1 in chapter one we show that a p for a homeomorphism flow is equivalent to topological transitivity and density of periodic points and how this property a p is invariant under topological conjugacy in theorem 1 we prove that an expansive homeomorphism which is topologically mixing and has p o t p also has the s p in chapter two theorem 2 we prove that the p o t p for flows is invariant under topological conjugacy with preserved orientation velocity changes also we prove in theorem 3 that the suspension flow 2 for a homeomorphism t x x has p o t p if and only if t has the p o t p in chapter three we prove that an expansive flow which has the p o t p is topologically stable theorem 4 in chapter four the last theorem in this part is that every flow without fixed points on a compact manifold m which is topologically stable has the p o t p then some important corollaries are deduced in the second part of this thesis please see full thesis for rest of abstract

duality and unified analysis of discrete approximations in structural dynamics and wave propagation comparison of p method finite elements with k method nurbs we study the discretization behavior of classical finite element and nurbs approximations on problems of structural vibrations and wave propagation we find that on the basis of equal numbers of degrees of freedom and bandwidth nurbs have superior approximation properties in fact we observe that the high mode behavior of classical finite elements is divergent with the order of approximation a surprisingly negative result on the other hand nurbs offer almost spectral approximation properties and all modes converge with increasing order of approximation

topological entropy of fixed point free flows topological entropy was introduced as an invariant of topological conjugacy and also as an analogue of measure theoretic entropy topological entropy for one parameter flows on a compact metric spaces is defined by bowen general statements are proved about this entropy but it is not easy to calculate the topological entropy and to show it is invariant under conjugacy for all this i would like to try to pose a new direction and study a definition for the topological entropy that involves handling the technical difficulties that arise from allowing reparametrizations of orbits some well known results are proved as well using this definition these results enable us to prove some results which seem difficult to prove using bowen s definition also we show here that this definition is equivalent to bowen s definition for any flow without fixed points on a compact metric space finally it is shown that the topological entropy of an expansive flow can be defined globally on a local cross sections

on the stability of the rayleigh ritz method for eigenvalues this paper studies global stability properties of the rayleigh ritz approximation of eigenvalues of the laplace operator the focus lies on the ratios hat lambda k lambda k of the kth numerical eigenvalue hat lambda k and the kth exact eigenvalue lambda k in the context of classical finite elements the maximal ratio blows up with the polynomial degree for b splines of maximum smoothness the ratios are uniformly bounded with respect to the degree except for a few instable numerical eigenvalues which are related to the presence of essential boundary conditions these phenomena are linked to the inverse inequalities in the respective approximation spaces

isogeometric approximation of cardiac electrophysiology models on surfaces an accuracy study with application to the human left atrium abstract we consider isogeometric analysis in the framework of the galerkin method for the spatial approximation of cardiac electrophysiology models defined on nurbs surfaces specifically we perform a numerical comparison between basis functions of degree p 1 and globally c k continuous with k 0 or p 1 to find the most accurate approximation of a propagating front with the minimal number of degrees of freedom we show that b spline basis functions of degree p 1 which are c p 1 continuous capture accurately the front velocity of the transmembrane potential even with moderately refined meshes similarly we show that for accurate tracking of curved fronts high order continuous b spline basis functions should be used finally we apply isogeometric analysis to an idealized human left atrial geometry described by nurbs with physiologically sound fiber directions and anisotropic conductivity tensor to demonstrate that the numerical scheme retains its favorable approximation properties also in a more realistic setting

optimal discontinuous galerkin methods for wave propagation we have developed and analyzed a new class of discontinuous galerkin methods dg which can be seen as a compromise between standard dg and the finite element fe method in the way that it is explicit like standard dg and energy conserving like fe in the literature there are many methods that achieve some of the goals of explicit time marching unstructured grid energy conservation and optimal higher order accuracy but as far as we know only our new algorithms satisfy all the conditions we propose a new stability requirement for our dg the stability analysis is based on the careful selection of the two fe spaces which verify the new stability condition the convergence rate is optimal with respect to the order of the polynomials in the fe spaces moreover the convergence is described by a series of numerical experiments

robustness of isogeometric structural discretizations under severe mesh distortion abstract this paper investigates higher order and higher continuity functions in isogeometric structural analysis under distortion of the control and physical meshes first the concepts behind isogeometric analysis are briefly reviewed the effect of perturbing control points upon the geometrical mapping is observed in the one dimensional setting the concept is extended to multiple dimensions with a series of cases where the interior of a cube is distorted in different ways and patch tests are performed to assess accuracy a circular fillet patch is introduced through the appropriate degeneration of a control mesh in order to address the problem of modeling small features in finite element models the fillet patch is tested with two problems of linear elasticity and is shown to be accurate the investigation is further extended to the setting of large deformation nearly incompressible nonlinear elasticity with two examples which make use of the recently proposed f projection method

rational solutions of nonlinear evolution equations vertex operators and bispectrality abstract we prove that if q x t 2 t m and r x t 2 t m are certain rational solutions of the akns hierarchy then there are eigenfunctions x k of the akns zs operator l x q r x satisfying a differential equation in the spectral parameter k of the form b k k x where b k k is a matrix differential operator independent of x and is a nonconstant function of x we also discuss the relation between this result and a similar one for the rational solutions of the schrodinger operator with potentials in the manifold of rational solutions of the kdv hierarchy

liquid vapor phase transition thermomechanical theory entropy stable numerical formulation and boiling simulations abstract we develop a new continuum mechanics modeling framework for liquid vapor flows with particular focus on the van der waals fluid by invoking microforce theory the coleman noll procedure is generalized to derive consistent constitutive relations in the presence of non local effects a new thermodynamically consistent algorithm for the van der waals model is designed using functional entropy variables and a new temporal scheme employing a family of new quadrature rules we show that the resulting fully discrete scheme is unconditionally stable in entropy and second order time accurate isogeometric analysis is utilized for spatial discretization the analytical properties of the formulation are corroborated by benchmark problems three sets of application problems are simulated to demonstrate the capability of the model and the algorithm our methodology provides a particularly useful predictive tool for boiling flows

isogeometric analysis of geometric partial differential equations abstract we consider the numerical approximation of geometric partial differential equations pdes defined on surfaces in the 3d space in particular we focus on the geometric pdes deriving from the minimization of an energy functional by l 2 gradient flow we analyze two energy functionals the area which leads to the mean curvature flow a nonlinear second order pde and the willmore energy leading to the willmore flow a nonlinear fourth order pde we consider surfaces represented by single patch tensor product nurbs and discretize the pdes by means of nurbs based isogeometric analysis in the framework of the galerkin method to approximate the high order geometric pdes we use high order continuous nurbs basis functions for the time discretization of the nonlinear geometric pdes we use backward differentiation formulas bdf with extrapolation of the geometric quantities involved in the weak formulation of the problem in this manner we solve a linear problem at each time step we report numerical results concerning the mean curvature and willmore flows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme

topological entropies of equivalent smooth flows we construct two equivalent smooth flows one of which has positive topological entropy and the other has zero topological entropy this provides a negative answer to a problem posed by ohno

non conforming high order approximations of the elastodynamics equation in this paper we formulate and analyze two non conforming high order strategies for the approximation of elastic wave problems in heterogeneous media namely the mortar spectral element method and the discontinuous galerkin spectral element method starting from a common variational formulation we make a full comparison of the two techniques from the points of view of accuracy convergence grid dispersion and stability

isogeometric analysis of the electrophysiology in the human heart numerical simulation of the bidomain equations on the atria abstract we consider isogeometric analysis iga for the numerical solution of the electrophysiology of the atria which in this work is modeled by means of the bidomain equations on thin surfaces first we consider the bidomain equations coupled with the roger mcculloch ionic model on simple slabs here our goal is to evaluate the effects of the spatial discretization by iga and the use of different b spline basis functions on the accuracy of the approximation in particular regarding the accuracy of the front velocity and the dispersion error specifically we consider basis functions with high polynomial degree p and global high order continuity c p 1 in the computational domain our results show that the use of such basis functions is beneficial to the accurate approximation of the solution then we consider a realistic application of the bidomain equations coupled with the courtemanche ramirez nattel ionic model on the two human atria which are represented by means of two nurbs surfaces

topological pressure of continuous flows without fixed points in this paper we give the equivalent definitions of topological pressure for flows by using spanning sets weakly spanning sets strongly separated sets and tracing sets respectively we get an inequality between the topological pressures of lipschitz conjugate flows and prove that the topological pressure of expansive flows with tracing property can be described by its periodic orbits

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

discrete dispersion relation for hp version finite element approximation at high wave number the dispersive properties of high order finite element schemes are analyzed in the setting of the helmholtz equation and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order it is shown that the numerical dispersion displays three different types of behavior depending on the size of the order of the method relative to the mesh size and the wave number quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes all estimates are fully explicit and are shown to be sharp limits are obtained where transitions between the different regimes occur and used to provide guidelines for the selection of the mesh size and the polynomial order in terms of the wave number so that the dispersion error is controlled

the use of the local truncation error to improve arbitrary order finite elements for the linear wave and heat equations abstract the local truncation error in space and time can be efficiently used for the analysis and the increase in accuracy of the linear and high order finite elements in the 1 d 2 d and 3 d cases on uniform and non uniform meshes several applications of the local truncation error are considered in the paper it is proven that for the 1 d wave equation with a piece wise constant wave velocity the local truncation error is zero if the linear finite elements with the element size proportional to the wave velocity the lumped mass matrix and the central difference method with the time increments equal to the stability limit are used it is shown in the 1 d and multidimensional cases that the optimal lumped mass matrix can be calculated by the minimization of the order of the local truncation error and yields the maximum possible order of accuracy the minimization of the order of the local truncation error allows us to develop the linear finite elements and the isogeometric high order elements with improved accuracy i e accuracy is improved from order 2 p the conventional elements to order 4 p the new elements where p is the order of the polynomial approximations new high order boundary conditions are developed in order to keep a high order accuracy of the developed technique the new elements can be equally applied to linear wave propagation and heat transfer problems it is also shown that non uniform meshes may lead to inaccurate results due to the increase in the local truncation error the difference in accuracy between the quadrilateral and triangular linear elements is analyzed with the suggested approach the presented numerical examples are in good agreement with the theoretical results the approach considered in the paper can be easily applied to the analysis of different aspects of finite elements techniques as well as other numerical approaches

studies of refinement and continuity in isogeometric structural analysis we investigate the effects of smoothness of basis functions on solution accuracy within the isogeometric analysis framework we consider two simple one dimensional structural eigenvalue problems and two static shell boundary value problems modeled with trivariate nurbs solids we also develop a local refinement strategy that we utilize in one of the shell analyses we find that increased smoothness that is the k method leads to a significant increase in accuracy for the problems of structural vibrations over the classical c0c0 continuous p method whereas a judicious insertion of c0c0 continuous surfaces about singularities in a mesh otherwise generated by the k method usually outperforms a mesh in which all basis functions attain their maximum level of smoothness we conclude that the potential for the k method is high but smoothness is an issue that is not well understood due to the historical dominance of c0c0 continuous finite elements and therefore further studies are warranted

on the integrable variant of the boussinesq system painleve property rational solutions a related many body system and equivalence with the akns hierarchy abstract an infinite family of rational solutions of the integrable version of the boussinesq system u t w x uu x 0 w t u xxx uw x 0 and the associated higher order flows is constructed the differential equations governing the motion of the poles are derived and their complete integrability demonstrated this system embeds into two disjoint calogero moser systems coupled only through a constraint a novel treatment of this constraint is given using the equivalence of to the real version of the nonlinear schrodinger system it is seen that rational solutions of the akns hierarchy have been constructed as well finally it is shown that all solutions of these hierarchies including the rational ones also solve the first modified kp equation

coupling finite element and spectral methods first results a poisson equation on a rectangular domain is solved by coupling two methods the domain is divided in two squares a finite element approximation is used on the first square and a spectral discretization is used on the second one two kinds of matching conditions on the interface are presented and compared in both cases error estimates are proved

optimal reduction of numerical dispersion for wave propagation problems part 1 application to 1 d isogeometric elements abstract a numerical technique with the optimal coefficients of the stencil equation has been suggested based on this approach new high order isogeometric elements with the reduced dispersion error have been developed for wave propagation problems in the 1 d case by the modification of the mass and stiffness matrices the order of the dispersion error is improved from order 2 p the conventional elements to order 4 p the new elements where p is the order of the polynomial approximations it was shown that the new approach yields the maximum order of the dispersion error for the stencil equations related to the high order isogeometric elements the analysis of the dispersion error of the high order isogeometric elements with the lumped mass matrix has also shown that independent of the procedures for the calculation of the lumped mass matrix the second order of the dispersion error cannot be improved with the conventional stiffness matrix however the dispersion error for the lumped mass matrix can be improved from the second order to order 2 p by the modification of the stiffness matrix the numerical examples confirm the computational efficiency of the new high order isogeometric elements with reduced dispersion we have also showed that numerical results obtained by the new and conventional high order isogeometric elements may include spurious oscillations due to the dispersion error these oscillations can be quantified and filtered by the two stage time integration technique recently developed in our papers the approach developed in the paper can be directly applied to other space discretization techniques with similar stencil equations

isogeometric analysis and proper orthogonal decomposition for parabolic problems we investigate the combination of isogeometric analysis iga and proper orthogonal decomposition pod based on the galerkin method for model order reduction of linear parabolic partial differential equations for the proposed fully discrete scheme the associated numerical error features three components due to spatial discretization by iga time discretization with the theta scheme and eigenvalue truncation by pod first we prove a priori error estimates of the spatial iga semi discrete scheme then we show stability and prove a priori error estimates of the space time discrete scheme and the fully discrete iga theta pod galerkin scheme numerical tests are provided to show efficiency and accuracy of nurbs based iga for model order reduction in comparison with standard finite element based pod techniques

explicit second order isogeometric discretizations for acoustic wave problems abstract this paper presents a stability analysis of isogeometric iga and explicit newmark discretizations for acoustic wave problems with absorbing boundary conditions in spite of the very ill conditioned iga mass and stiffness matrices especially with respect to the polynomial degree p of the b splines and non uniform rational b splines nurbs basis functions employed the stability bounds of the proposed newmark iga method depend linearly on the meshsize h of the iga mesh and inversely on the iga polynomial degree p several numerical tests in the plane confirm these stability bounds and additionally explore the newmark iga order of convergence with respect to h p t domain deformation and the absorbing boundary conditions performance in the presence of a ricker wavelet in a nurbs domain

bowen entropy of sets of generic points for fixed point free flows abstract let x be a compact metric flow without fixed points we will be concerned with the entropy of flows which takes into consideration all possible reparametrizations of flows in this paper by establishing the brin katok s entropy formula for flows without fixed points in the non ergodic case we prove the following result for an ergodic invariant measure h t o p b g h 1 where g is the set of generic points for and h t o p b g is the bowen entropy on g this extends the classical result of bowen in 1973 to fixed point free flows moreover it is shown that the bowen entropy can be determined via the local entropies of measures

finite element and nurbs approximations of eigenvalue boundary value and initial value problems abstract we study the spectral approximation properties of finite element and nurbs spaces from a global perspective we focus on eigenfunction approximations and discover that the l 2 norm errors for finite element eigenfunctions exhibit pronounced spikes about the transition points between branches of the eigenvalue spectrum this pathology is absent in nurbs approximations by way of the pythagorean eigenvalue error theorem we determine that the squares of the energy norm errors of the eigenfunctions are the sums of the eigenvalue errors and the squares of the l 2 norm eigenfunction errors the spurious behavior of the higher eigenvalues for standard finite elements is well known and therefore inherited by the energy norm errors along with the spikes in the l 2 norm of the eigenfunction errors the eigenvalue pathology is absent for nurbs the implications of these results to the corresponding elliptic boundary value problem and parabolic and hyperbolic initial value problems are discussed

huygens principle for hyperbolic operators and integrable hierarchies abstract we show that the stationary solutions of the canonical akns hierarchy of nonlinear evolution equations yield perturbations of dirac operators that satisfy a strict form of huygens principle namely the domain of dependence of such dirac operators at any given point y is contained in the light cone s hypersurface issued from y by the canonical akns hierarchy we mean that the differential polynomials defining the flows are isobaric with respect to certain weights the method we employ is of interest by itself indeed we consider the riesz kernels associated to a given hyperbolic differential operator and expand the fundamental solution of perturbations of this operator in a series in such riesz kernels using the coefficients of this hadamard type expansion we introduce a family of vector fields for the d alembertian such vector field family corresponds to the kdv hierarchy and for the dirac operators they include the akns one

differential equations in the spectral parameter for matrix differential operators we use matrix darboux transformation to generate a class of matrix differential operators l that have the following property there exist a family x k of eigenfunctions of l also satisfying a differential equation in the spectral parameter k of the form b k k x whereb k k is a diffrential and is a non constant function of x we also obtain nth order scalar operators which have this property

iterative solution of helmholtz problem with high order isogeometric analysis and finite element method at mid range frequencies abstract solving wave problems with isogeometric analysis has attracted a significant attention in the past few years it is well known that keeping a fixed number of degrees of freedom per wavelength leads to an increased error as higher wavenumbers are considered this behaviour often cited as the pollution error improves significantly with isogeometric analysis when compared to the conventional finite element method the improvement in handling pollution along with the ability to represent exact geometries has been the main reasons behind the attention that isogeometric analysis has received furthermore using high order elements also presents major advantages over low order elements for this range of frequencies however it remains to be studied how iterative linear solvers often necessary for solving high frequency wave problems perform when using isogeometric analysis compared to the finite element method especially at high polynomial orders this paper is one of the first studies in this direction in this work we investigate the generalised minimal residual method a standard krylov subspace iterative technique for solving the linear system resulting form isogeometric analysis furthermore we look into the use of some recently proposed preconditioners for helmholtz problem such as shifted laplace or ilu with a complex shift preconditioners and how they perform with high order isogeometric analysis and finite element method in general the results show improvement when using isogeometric analysis in terms of the number of iterations required for convergence compared to the finite element method for both preconditioned and non preconditioned linear systems we use eigenvalue spectra to understand this improvement

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

the bounded complexity function versus the unbounded complexity function abstract we construct two point wise periodic flows which are equivalent such that all the complexity functions of one flow are bounded while the other flow has an unbounded complexity function

quasi shadowing and limit quasi shadowing for quasi partially hyperbolic strings of flows abstract in this paper we study the quasi shadowing property for quasi partially hyperbolic pseudo orbit made by quasi partially hyperbolic strings we also investigate the corresponding l p limit and asymptotic quasi shadowing properties as an application we prove that the set of points with periodic center leaves is dense in the non wondering set for some partially hyperbolic flows

heat kernel coefficients for the matrix schrodinger operator the heat kernel coefficients hk to the schrodinger operator with a matrix potential are investigated we present algorithms and explicit expressions for the taylor coefficients of the hk special terms are discussed and for the one dimensional case some improved algorithms are derived

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

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

pressures for flows on arbitrary subsets abstract with the help of the theory of caratheodory structure the topological pressure for flows on non compact sets is introduced and the measure theoretic pressure for flows is defined as well it is shown that the topological pressure for flows on arbitrary sets is equal to the one of its time one map moreover given an ergodic measure for a flow there exists a certain set on which the topological pressure is equal to the measure theoretic one a variational principle for the topological pressure defined above is also established at the end the pressures for a frame flow are studied

non uniformly hyperbolic flows and shadowing abstract we consider a hyperbolic ergodic measure of a c 1 flow on a compact manifold under the hypothesis that there are no fixed points and that the oseledec splitting of the normal bundle satisfies a limit domination property we prove that the measure has a shadowing property as an application of this result we prove that the measure can be approached on the weak topology by measures supported on hyperbolic periodic orbits

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