preconditioning the poincare steklov operator by using green s function this paper is concerned with the poincare steklov operator that is widely used in domain decomposition methods it is proved that the inverse of the poincare steklov operator can be expressed explicitly by an integral operator with a kernel being the green s function restricted to the interface as an application for the discrete poincare steklov operator with respect to either a line edge or a star shaped web associated with a single vertex point a preconditioner can be constructed by first imbedding the line as the diameter of a disk or the web as a union of radii of a disk and then using the green s function on the disk the proposed technique can be effectively used in conjunction with various existing domain decomposition techniques especially with the methods based on vertex spaces from multi subdomain decomposition some numerical results are reported

an energy space finite element approach for elliptic dirichlet boundary control problems in this paper we present a finite element analysis for a dirichlet boundary control problem where the dirichlet control is considered in a convex closed subspace of the energy space h 1 2 gamma h 1 2 as an equivalent norm in h 1 2 gamma h 1 2 we use the energy norm induced by the so called steklov poincare operator which realizes the dirichlet to neumann map and which can be implemented by using standard finite element methods the presented stability and error analysis of the discretization of the resulting variational inequality is based on the mapping properties of the solution operators related to the primal and adjoint boundary value problems and their finite element approximations some numerical results are given which confirm on one hand the theoretical estimates but on the other hand indicate the differences when modelling the control in l 2 gamma l 2

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

quasi optimal approximation of surface based lagrange multipliers in finite element methods we show quasi optimal a priori convergence results in the l 2 and h 1 2 norm for the approximation of surface based lagrange multipliers such as those employed in the mortar finite element method we improve on the estimates obtained in the standard saddle point theory where error estimates for both the primal and dual variables are obtained simultaneously and thus only suboptimal a priori estimates for the dual variable are reached for the lowest order case i e k 1 an additional factor of sqrt h ln h and for higher order cases i e k 1 an additional factor of sqrt h in the a priori bound for the dual variable can be recovered the proof is based on the use of new estimates for the primal variable in strips of width o h near these surfaces

error estimates for variational normal derivatives and dirichlet control problems with energy regularization this article deals with error estimates for the finite element approximation of variational normal derivatives and as a consequence error estimates for the finite element approximation of dirichlet boundary control problems with energy regularization the regularity of the solution is carefully carved out exploiting weighted sobolev and holder spaces this allows to derive a sharp relation between the convergence rates for the approximation and the structure of the geometry more precisely the largest opening angle at the vertices of polygonal domains numerical experiments confirm that the derived convergence rates are sharp

error analysis for a finite element approximation of elliptic dirichlet boundary control problems we consider the galerkin finite element approximation of an elliptic dirichlet boundary control model problem governed by the laplacian operator the analytical setting of this problem uses l 2 controls and a very weak formulation of the state equation however the corresponding finite element approximation uses standard continuous trial and test functions for this approximation we derive a priori error estimates of optimal order which are confirmed by numerical experiments the proofs employ duality arguments and known results from the l p error analysis for the finite element dirichlet projection

on the regularity of the solutions of dirichlet optimal control problems in polygonal domains a linear quadratic dirichlet control problem governed by an elliptic equation posed on a possibly nonconvex polygonal domain is analyzed detailed regularity results are provided in classical sobolev slobodetski spaces in particular it is proved that in the presence of pointwise control constraints the optimal control is continuous despite the nonconvexity of the domain

on kicked systems modulated along the thue morse sequence we give necessary conditions for the fourier transform of the time evolution operator of kicked systems modulated along the thue morse sequence to present dirac s delta functions the case of two level systems is analysed in detail and it is found based on the trace map and numerical investigations that generically its time evolution has no quasi periodic component

energy corrected fem for optimal dirichlet boundary control problems in the presence of re entrant corners the solution of pde constrained optimization problems in general has singular components even when the given data are smooth consequently the use of standard finite element discretization gives rise to the so called pollution effect this means that only a reduced convergence rate compared to the best approximation error is obtained we discuss how optimal convergence rates in weighted norms can be achieved using the idea of energy corrected finite element methods applied to optimal dirichlet boundary control problem in the energy space we present optimal error estimates in weighted norms for the state variable and for the control several numerical examples illustrate the obtained theoretical results

singular continuous spectrum on a cantor set of zero lebesgue measure for the fibonacci hamiltonian it is rigorously proven that the spectrum of the tight binding fibonacci hamiltonian h mn m n 1 m n 1 m n n 1 n where 5 1 2 and means integer part is a cantor set of zero lebesgue measure for all real nonzero and the spectral measures are purely singular continuous this follows from a recent result by kotani coupled with the vanishing of the lyapunov exponent in the spectrum

recurrent versus diffusive dynamics for a kicked quantum system we study the dynamics of a two level quantum system subject to a time dependent kicking perturbation modulated along the thue morse sequence for a nontrivial set of the parameters the quantum autocorrelation function is explicitly calculated and splits into a purely recurrent and a purely diffusive part furthermore the diffusive part is directly related to the singular continuous correlation measure of the thue morse sequence

spectrum and extended states in a harmonic chain with controlled disorder effects of the thue morse symmetry along the lines of previous work we give the general framework together with a detailed and rigorous study of the spectrum and born von karman eigenstates of a 1d harmonic chain with controlled disorder determined by the thue morse sequence the spectrum is a cantor like set we prove numerically that its measure is zero and calculate its bouligand minkowski dimension box dimension we prove that the value of the ids on each of the gaps is 2k 1 3 2 p withk andp integers finally we also prove that points in a dense subset of the spectrum give rise to extended states an exceptional property due to the symmetry of the thue morse substitution which can have important applications to multilayered structures and we illustrate this situation

a short remark regarding pohozaev type results on general domains assuming finite morse index we are interested in nonnegative nontrivial solutions of u up in u 0 on 1 where 1 p and a bounded smooth domain in rn with 3 n 9 we show that given a nonnegative integer m there is some large p m such that the only nonnegative solution u of morse index at most m is u 0 2010 mathematics subject classification

on the stability of the l_2 projection in fractional sobolev spaces in this paper we prove the stability of the l 2 projection onto the finite element trial space of piecewise polynomial in particular piecewise linear basis functions in h s omega for s in 0 1 we formulate explicit and computable local mesh conditions to be satisfied which depend on the sobolev index s in conclusion we prove a stability condition needed in the numerical analysis of mixed and hybrid boundary element methods as well as in the construction of efficient preconditioners in adaptive boundary and finite element methods

traces extensions and co normal derivatives for elliptic systems on lipschitz domains this is the post print version of the article the official published version can be accessed from the link below copyright 2011 elsevier

multifractal analysis in reciprocal space and the nature of the fourier transform of self similar structures the authors propose to use multifractal analysis in reciprocal space as a tool to characterise in a statistical and global sense the nature of the fourier transform of geometrical models for atomic structures this approach is especially adequate for shedding some new light on a class of structures introduced recently which exhibit singular scattering using the language of measure theory the fourier intensity of these models is presumably singular continuous and therefore represents an intermediate type of order between periodic or quasiperiodic structures characterised by bragg peaks atomic fourier transform and amorphous structures which exhibit diffuse scattering absolutely continuous fourier transform this general approach is illustrated in several examples of self similar one dimensional sequences and structures generated by substitutions a special emphasis is put on the relationship between the nature of the fourier intensity of these models and the f alpha spectrum obtained by multifractal analysis in reciprocal space

solution regularity and co normal derivatives for elliptic systems with non smooth coefficients on lipschitz domains abstract elliptic pde systems of the second order with coefficients from l or holder lipschitz spaces are considered in the paper continuity of the operators in corresponding sobolev spaces is stated and the internal local solution regularity theorems are generalized to the non smooth coefficient case for functions from the sobolev space h s 1 2 s 3 2 definitions of non unique generalized and unique canonical co normal derivatives are considered which are related to possible extensions of a partial differential operator and the pde right hand side from the domain to its boundary it is proved that the canonical co normal derivatives coincide with the classical ones when both exist a generalization of the boundary value problem settings which makes them insensitive to the co normal derivative inherent non uniqueness is given

spectral properties of one dimensional quasi crystals in this paper we prove that the one dimensional schrodinger operator onl 2 with potential given by upsilon n lambda chi 1 alpha 1 x n alpha alpha notin mathbb q has a cantor spectrum of zero lebesgue measure for any irrational and any 0 we can thus extend the kotani result on the absence of absolutely continuous spectrum for this model to all mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpepge9fr xfr x frpewzqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiaadihaiiaacq wfiiiztuudjxwak1uy0hmmaehbfv3yslgzg0uy0hgiud3bagqbbiab nj8ubaa 4628 x in mathbb t

local energy estimates for the finite element method on sharply varying grids local energy error estimates for the nite element method for el liptic problems were originally proved in 1974 by nitsche and schatz these estimates show that the local energy error may be bounded by a local ap proximation term plus a global pollution term that measures the inuence of solution quality from outside the domain of interest and is heuristically of higher order however the original analysis of nitsche and schatz is restricted to quasi uniform grids we present local a priori energy estimates that are valid on shape regular grids an assumption which allows for highly graded meshes and which much more closely matches the typical practical situation our chief technical innovation is an improved superapproximation result

spectral properties of one dimensional schrodinger operators with potentials generated by substitutions we investigate one dimensional discrete schrodinger operators whose potentials are invariant under a substitution rule the spectral properties of these operators can be obtained from the analysis of a dynamical system called the trace map we give a careful derivation of these maps in the general case and exhibit some specific properties under an additional easily verifiable ypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying schrodinger operator is singular and supported on a set of zero lebesgue measure a condition allowing to exclude point spectrum is also given the application of our theorems is explained on a series of examples

error estimates for the numerical approximation of dirichlet boundary control for semilinear elliptic equations we study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control the control is the trace of the state on the boundary of the domain which is assumed to be a convex polygonal open set in mathbb r 2 piecewise linear finite elements are used to approximate the control as well as the state we prove that the error estimates are of order o h 1 1 p for some p 2 which is consistent with the w 1 1 p p gamma regularity of the optimal control

structure factor of 1d systems superlattices based on two letter substitution rules i delta bragg peaks the recent generalization to the case of arbitrary tile lengths and arbitrary scattering factors of the calculation of the structure factor of 1d substitutional systems is studied in detail this method makes a easy to find all the peaks in the diffraction spectrum of a system the well known periodic and quasiperiodic spectra with delta peaks at integer multiples of a single number and integer linear combinations of two incommensurate frequencies respectively were found to be the l 0 subsets of two more general types of spectra infinite periodic or limit periodic and infinite quasiperiodic or limit quasiperiodic characterized by rational numbers of the type m nl l 0 infinity in place of the above integers substitution rules that produce quasicrystalline quasiperiodic and infinite quasiperiodic spectra give the same type of spectrum for all values of the ratio rho rho a rho b of the two tile lengths rho a and rho b this is not the case for the other rules thus the same substitution rule such as the copper mean rule can give an infinite periodic spectrum for a single rational ratio rho rho a rho b of the two tile lengths rho a and rho b a periodic like spectrum for other rational rho and a spectrum in some aspects similar to that of a random system when rho is an irrational number on the other hand a thue morse system diffracts as a periodic crystal when p not 1 but has no non trivial delta peaks when rho 1 other thue morse like systems can have infinite periodic spectra for all rho

inhomogeneous dirichlet conditions in a priori and a posteriori finite element error analysis the numerical solution of elliptic boundary value problems with finite element methods requires the approximation of given dirichlet data ud by functions ud h in the trace space of a finite element space on d in this paper quantitative a priori and a posteriori estimates are presented for two choices of ud h namely the nodal interpolation and the orthogonal projection in l2 d onto the trace space two corresponding extension operators allow for an estimate of the boundary data approximation in global h1 and l2 a priori and a posteriori error estimates the results imply that the orthogonal projection leads to better estimates in the sense that the influence of the approximation error on the estimates is of higher order than for the nodal interpolation