construction of aharonov berry s superoscillations a simple method is described for constructing functions that superoscillate at an arbitrarily chosen wavelength scale our method is based on the technique of oversampled signal reconstruction this allows us to explicitly demonstrate that the observed fragility of superoscillating wavefunctions is indeed mathematically closely connected to what in the communication theory community is known as the instability of oversampled signal reconstruction confirming a previous conjecture this is of potential interest for example concerning the understanding of the practical difficulties in experimentally producing superoscillatory wavefunctions

a posteriori error estimations of some cell centered finite volume methods for diffusion convection reaction problems this paper presents an a posteriori residual error estimator for diffusion convection reaction problems approximated by some cell centered finite volume methods on isotropic or aniso tropic meshes in rd d 2 or 3 for that purpose we built a reconstructed approximation which is an appropriate interpolant of the finite volume solution the error is then the difference between the exact solution and this interpolant the residual error estimator is based on the jump of the normal derivative of the interpolant we then prove the equivalence between the energy norm of the error and the residual error estimator some numerical tests confirm our theoretical results

a posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions in this paper we shall derive a posteriori error estimates in the l 1 norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions this result is mainly based on some fundamental a priori error estimates published in a recent paper by c chainais hillairet the theoretical results are confirmed by numerical experiments

a posteriori error estimates for combined finite volume finite element discretizations of reactive transport equations on nonmatching grids abstract we derive in this paper guaranteed and fully computable a posteriori error estimates for vertex centered finite volume type discretizations of transient linear convection diffusion reaction equations our estimates enable actual control of the error measured either in the energy norm or in the energy norm augmented by a dual norm of the skew symmetric part of the differential operator lower bounds global in space but local in time are also derived these lower bounds are fully robust with respect to convection or reaction dominance and the final simulation time in the augmented norm setting on the basis of the derived estimates we propose an adaptive algorithm which enables to automatically achieve a user given relative precision this algorithm also leads to efficient calculations as it balances the time and space error contributions as an example we apply our estimates to the combined finite volume finite element scheme including such features as use of mass lumping for the time evolution or reaction terms of upwind weighting for the convection term and discretization on nonmatching meshes possibly containing nonconvex and non star shaped elements a collection of numerical experiments illustrates the efficiency of our estimates and the use of the space time adaptive algorithm

on multivariate lagrange interpolation lagrange interpolation by polynomials in several variables is studied through a finite difference approach we establish an interpolation formula analogous to that of newton and a remainder formula both of them in terms of finite differences we prove that the finite difference admits an integral representation involving simplex spline functions in particular this provides a remainder formula for lagrange interpolation of degree n of a function f which is a sum of integrals of certain n 1 st directional derivatives of f multiplied by simplex spline functions we also provide two algorithms for the computation of lagrange interpolants which use only addition scalar multiplication and point evaluation of polynomials

optimal sampling patterns for zernike polynomials a pattern of interpolation nodes on the disk is studied for which the interpolation problem is theoretically unisolvent and which renders a minimal numerical condition for the collocation matrix when the standard basis of zernike polynomials is used it is shown that these nodes have an excellent performance also from several alternative points of view providing a numerically stable surface reconstruction starting from both the elevation and the slope data sampling at these nodes allows for a more precise recovery of the coefficients in the zernike expansion of a wavefront or of an optical surface

a local problem error estimator for anisotropic tetrahedral finite element meshes the poisson problem is solved by the finite element method on anisotropic tetrahedral or triangular meshes the focus is on adaptive algorithms and in particular on a posteriori error estimators based on the solution of a local problem on anisotropic meshes such estimators cannot be analyzed in the common way known from isotropic meshes the first estimator proposed here is a slight modification of a familiar isotropic counterpart by a rigorous analysis it is proven that this estimator is equivalent to a known anisotropic residual error estimator hence the local problem error estimator also yields reliable upper and lower error bounds on anisotropic meshes the local problems are shown to be well conditioned two further local problem error estimators originally defined for isotropic meshes are investigated in the anisotropic context here the results reveal the significance of the proper choice of local problem furthermore the analysis covers dirichlet neumann and robin boundary conditions particular attention is paid to the robin boundary conditions since they require a different treatment than the neumann boundary conditions a numerical example supports the error analysis

exact nonparaxial transmission of subwavelength detail using superoscillations any object described by a target function that need not be band limited can be sampled at any chosen set of points and then propagated without evanescent waves using a function which is band limited so as to be imaged exactly i e nonparaxially at multiples of a given repetition distance if the samples span a sub wavelength region the repeated images are superoscillatory the number n of samples is equal to the repetition distance measured in wavelengths and also to the number of plane waves in the propagating field if n 1 the waves form a quasi continuum and asymptotics enables an almost explicit description of the superoscillations but the matrix involved is ill conditioned many of its eigenvalues are very small so this method of sub wavelength imaging would be pathologically sensitive to noise and the depth of focus is exponentially small

on hyperinterpolation on the unit ball abstract we prove estimates on the lebesgue constants of the hyperinterpolation operator for functions on the unit ball b d r d with respect to gegenbauer weight functions 1 x 2 1 2 the relationship between orthogonal polynomials on the sphere and ball is exploited to achieve this result which provides an improvement on known estimates of the lebesgue constant for hyperinterpolation operators on b 2

a posteriori error analysis of nonconforming finite element methods for convection diffusion problems abstract a unified framework is established for the a posteriori error analysis of nonconforming finite element approximations to convection diffusion problems under some certain conditions the theory assures the semi robustness of residual error estimates in the usual energy norm and the robustness in a modified norm and applies to several nonconforming finite elements such as the crouzeix raviart triangular element the nonconforming rotated nr parallelogram element of rannacher and turek the constrained nr parallelogram element etc based on the general error decomposition in different norms we show that the key ingredients of error estimation are the existence of a bounded linear operator v h c v h nc with some elementary properties and the estimation on the consistency error related to the particular numerical scheme the numerical results are presented to illustrate the practical behavior of the error estimator and check the theoretical predictions

anisotropic mesh refinement in stabilized galerkin methods the numerical solution of a convection diffusion reaction model problem is considered in two and three dimensions a stabilized finite element method of galerkin least square type accomodates diffusion dominated as well as convection and or reaction dominated situations the resolution of boundary layers occuring in the singularly perturbed case is achieved using anisotropic mesh refinement in boundary layer regions in this paper the standard analysis of the stabilized galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement simplicial lagrangian elements of arbitrary order are used

stabilized galerkin methods and layer adapted grids for elliptic problems abstract global error estimates are considered for stabilized galerkin finite element methods to second order elliptic boundary value problems with emphasis on the singularly perturbed case the critical point is the appropriate choice of numerical damping parameters furthermore we address the resolution of exponentially decaying boundary layers with anisotropic finite elements on layer adapted grids the numerical analysis is based on both anisotropic interpolation estimates and sharp estimates of derivatives

a posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection diffusion equations this paper is devoted to the study of a posteriori and a priori error estimates for the scalar nonlinear convection diffusion equation c t nabla cdot u f c varepsilon delta c 0 the estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the l 1 norm in the situation where the diffusion parameter varepsilon is smaller or comparable to the mesh size numerical experiments underline the theoretical results

estimations a posteriori d un schema de volumes finis pour un probleme non lineaire resumedans cet article nous developpons des estimations a posteriori de l erreur commise en approchant un probleme elliptique non lineaire par une methode de volumes finis mvf de type vertex centered les estimateurs obtenus mesurent le residu de l equation forte et l irregularite de la solution discrete qui se traduit par des sauts a travers les inter elelments une condition sur la fonction flux numerique sera imposee pour traiter la non conformite du probleme au cours de l evaluation de l estimation summary we derive a posteriori error estimates for vertex centered finite volume discretizations of a class of non linear elliptic pdes the error estimates are of residual type and incorporate residuals of the pde in its strong form on the control volumes and jumps of the pde s prinicipal part across the element boundaries the non conformity of the discretization is taken into account by a condition on the numerical flux function

constructing nested nodal sets for multivariate polynomial interpolation we present a robust method for choosing multivariate polynomial interpolation nodes our algorithm is an optimization method to greedily minimize a measure of interpolant sensitivity a variant of a weighted lebesgue function nodes are therefore chosen that tend to control oscillations in the resulting interpolant this method can produce an arbitrary number of nodes and is not constrained by the dimension of a complete polynomial space our method is therefore flexible nested nodal sets are produced in spaces of arbitrary dimensions and the number of nodes added at each stage can be arbitrary the algorithm produces a nodal set given a probability measure on the input space thus parameterizing interpolants with respect to finite measures we present examples to show that the method yields nodal sets that behave well with respect to standard interpolation diagnostics the lebesgue constant the vandermonde determinant and the vandermonde condition number we also show that a nongreedy version of the

computing fekete and lebesgue points simplex square disk we have computed point sets with maximal absolute value of the vandermonde determinant fekete points or minimal lebesgue constant lebesgue points on three basic bidimensional compact sets the simplex the square and the disk using routines of the matlab optimization toolbox we have obtained some of the best bivariate interpolation sets known so far

a posteriori error estimations of some cell centered finite volume methods this paper presents the natural framework to residual based a posteriori error estimation of some cell centered finite volume methods for the laplace equation in r d d 2 or 3 for that purpose we associate with the finite volume solution a reconstructed approximation which is a kind of morley interpolant the error is then the difference between the exact solution and this morley interpolant the residual error estimator is based on the jump of normal and tangential derivatives of the morley interpolant we then prove the equivalence between the discrete h1 seminorm of the error and the residual error estimator numerical tests confirm our theoretical results

a posteriori error estimators for convection diffusion equations we derive a posteriori error estimators for convection diffusion equations with dominant convection the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh peclet number the estimators are either based on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

a pde approach to fractional diffusion we derive a computable a posteriori error estimator for the harmonic extension problem which localizes the fractional powers of elliptic operators supplemented with dirichlet boundary conditions our a posteriori error estimator relies on the solution of small discrete problems on anisotropic cylindrical stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation under suitable assumptions we design a simple adaptive algorithm and present numerical experiments which reveal a competitive performance

an a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes a new a posteriori residual error estimator is defined and rigorously analysed for anisotropic tetrahedral finite element meshes all considerations carry over to anisotropic triangular meshes with minor changes only the lower error bound is obtained by means of bubble functions and the corresponding anisotropic inverse inequalities in order to prove the upper error bound it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration to measure this correspondence a so called matching function is defined and its discussion shows it to be a useful tool with its help anisotropic interpolation estimates and subsequently the upper error bound are proven additionally it is pointed out how to treat robin boundary conditions in a posteriori error analysis on isotropic and anisotropic meshes a numerical example supports the anisotropic error analysis

a posteriori error estimator for finite volume methods in this paper first we discuss a technique to compare finite volume method and some well known finite element methods namely the dual mixed methods and nonconforming primal methods for elliptic equations these both equivalences are exploited to give us a posteriori error estimator for finite volume methods this estimator is explicitly given easy to compute and asymptotically exact without any regularity of the solution in unstructured grids

goal oriented a posteriori error estimation for finite volume methods a general framework for goal oriented a posteriori error estimation for finite volume methods is developed the framework does not rely on recasting finite volume methods as special cases of finite element methods but instead directly determines error estimators from the discretized finite volume equations thus the framework can be applied to arbitrary finite volume methods it also provides the proper functional settings to address well posedness issues for the primal and adjoint problems numerical results are presented to illustrate the validity and effectiveness of the a posteriori error estimates and their applicability to adaptive mesh refinement

residual flux based a posteriori error estimates for finite volume and related locally conservative methods we derive in this paper a posteriori error estimates for discretizations of convection diffusion reaction equations in two or three space dimensions our estimates are valid for any cell centered finite volume scheme and in a larger sense for any locally conservative method such as the mimetic finite difference covolume and other we consider meshes consisting of simplices or rectangular parallelepipeds and also provide extensions to nonconvex cells and nonmatching interfaces we allow for the cases of inhomogeneous and anisotropic diffusion dispersion tensors and of convection dominance the estimates are established in the energy semi norm for a locally postprocessed approximate solution preserving the conservative fluxes and are of residual type they are fully computable all occurring constants are evaluated explicitly and locally efficient give a local lower bound on the error times an efficiency constant so that they can serve both as indicators for adaptive refinement and for the actual control of the error they are semi robust in the sense that the local efficiency constant only depends on local variations in the coefficients and becomes optimal as the local peclet number gets sufficiently small numerical experiments confirm their accuracy

a posteriori error estimation for the discrete duality finite volume discretization of the laplace equation an efficient and fully computable a posteriori error bound is derived for the discrete duality finite volume discretization of the laplace equation on very general two dimensional meshes the main ingredients are the equivalence of this method with a finite element like scheme and tools from the finite element framework numerical tests are performed with a stiff solution on highly nonconforming locally refined meshes and with a singular solution on triangular meshes

a dual graph norm refinement indicator for finite volume approximations of the euler equations we generalise and apply a refinement indicator of the type originally designed by mackenzie suli and warnecke in 15 and 16 for linear friedrichs systems to the euler equations of inviscid compressible fluid flow the euler equations are symmetrized by means of entropy variables and locally linearized about a constant state to obtain a symmetric hyperbolic system to which an a posteriori error analysis of the type introduced in 15 can be applied we discuss the details of the implementation of the refinement indicator into the dlr tau code which is based on a finite volume method of box type on an unstructured grid and present numerical results

covariant information density cutoff in curved space time in information theory the link between continuous information and discrete information is established through well known sampling theorems sampling theory explains for example how frequency filtered music signals are reconstructible perfectly from discrete samples in this letter sampling theory is generalized to pseudo riemannian manifolds this provides a new set of mathematical tools for the study of space time at the planck scale theories formulated on a differentiable space time manifold can be equivalent to lattice theories there is a close connection to generalized uncertainty relations which have appeared in string theory and other studies of quantum gravity

a posteriori error estimates and adaptive mesh refinement for the coupling of the finite volume method and the boundary element method we consider the coupling of the finite volume method and the boundary element method of erath siam j numer anal 50 2012 pp 574 594 in two and three dimensions this method can be used for example to approximate a solution of the transport of a concentration in a fluid where no boundary conditions are available we derive residual based a posteriori estimates also for an upwind version these upper and lower bounds measure the error in an energy semi norm between the exact solution and the numerical solution the upper bound is robust in the sense that it does not depend on the variation of the model data the lower bound however depends additionally on the local peclet number the local contributions of the a posteriori estimates are used to steer an adaptive mesh refining algorithm this strategy turns out to be very suitable for the numerical treatment of transmission problems which have singularities or boundary internal layers several numerical examples illustrate the effectivenes

analysis of superoscillatory wave functions surprisingly differentiable functions are able to oscillate arbitrarily faster than their highest fourier component would suggest the phenomenon is called superoscillation recently a practical method for calculating superoscillatory functions was presented and it was shown that superoscillatory quantum mechanical wave functions should exhibit a number of counter intuitive physical effects following up on this work we here present more general methods which allow the calculation of superoscillatory wave functions with custom designed physical properties we give concrete examples and we prove results about the limits to superoscillatory behavior we also give a simple and intuitive new explanation for the exponential computational cost of superoscillations

a new conservative numerical scheme for flow problems on unstructured grids and unbounded domains abstract we consider the prototype of a flow and transport problem in two dimensions in particular in many fluid dynamics problems the boundary conditions may be unknown or the domain may be unbounded furthermore only numerical schemes which conserve mass and are stable with respect to dominating convection are appropriate to solve these problems therefore we develop a new discrete scheme to address these issues more precisely we couple the cell centered finite volume method and the boundary element method and introduce suitable transmission conditions also semi robust a posteriori estimates of residual type are developed which allow us to use an adaptive mesh refinement algorithm to treat efficiently problems which have singularities or boundary internal layers several examples illustrate the effectiveness of our new adaptive coupling method

on a hermite interpolation by polynomials of two variables a problem of hermite interpolation by polynomials of two variables is studied the interpolation matches preassigned data of function values and consecutive normal derivatives on a set of points on several circles centered at the origin it includes lagrange interpolation as a special case the uniqueness of the interpolation is established when the points are equidistant on the circles while the points on different circles may differ by arbitrary rotations this leads to a cubature formula on the unit disc which can be given explicitly without knowing the explicit formula of the interpolation polynomial

black holes bandwidths and beethoven it is usually believed that a function t whose fourier spectrum is bounded can vary at most as fast as its highest frequency component max this is in fact not the case as aharonov berry and others drastically demonstrated with explicit counterexamples so called superoscillations it has been claimed that even the recording of an entire beethoven symphony can occur as part of a signal with a 1 hz bandwidth bandlimited functions also occur as ultraviolet regularized fields their superoscillations have been suggested for example to resolve the trans planckian frequencies problem of black hole radiation here we give an exact proof for generic superoscillations namely we show that for every fixed bandwidth there exist functions that pass through any finite number of arbitrarily prespecified points further we show that in spite of the presence of superoscillations the behavior of bandlimited functions can be characterized reliably namely through an uncertainty relation the standard devi

a robust nonconforming h 2 element finite element methods for some elliptic fourth order singular perturbation problems are discussed we show that if such problems are discretized by the nonconforming morley method in a regime close to second order elliptic equations then the error deteriorates in fact a counterexample is given to show that the morley method diverges for the reduced second order equation as an alternative to the morley element we propose to use a nonconforming h 2 element which is h 1 conforming we show that the new finite element method converges in the energy norm uniformly in the perturbation parameter

a posteriori error analysis of a cell centered finite volume method for semilinear elliptic problems in this paper we conduct a goal oriented a posteriori analysis for the error in a quantity of interest computed from a cell centered finite volume scheme for a semilinear elliptic problem the a posteriori error analysis is based on variational analysis residual errors and the adjoint problem to carry out the analysis we use an equivalence between the cell centered finite volume scheme and a mixed finite element method with special choice of quadrature