# Daniele FUNARO

Department of Chemical and Geological Sciences

Manzini, Gianmarco; Funaro, Daniele; Delzanno, Gian Luca ( 2017 ) - Convergence of spectral discretizations of the Vlasov-Poisson system - SIAM JOURNAL ON NUMERICAL ANALYSIS - n. volume 55 - pp. da 2312 a 2335 ISSN: 0036-1429 [Articolo in rivista (262) - Articolo su rivista]
Abstract

We prove the convergence of a spectral discretization of the Vlasov--Poisson system. The velocity term of the Vlasov equation is discretized using either Hermite functions on the infinite domain or Legendre polynomials on a bounded domain. The spatial term of the Vlasov and Poisson equations is discretized using periodic Fourier expansions. Boundary conditions are treated in weak form through a penalty type term that can be applied also in the Hermite case. As a matter of fact, stability properties of the approximated scheme descend from this added term. The convergence analysis is carried out in detail for the 1D-1V case, but results can be generalized to multidimensional domains, obtained as Cartesian product, in both space and velocity. The error estimates show the spectral convergence under suitable regularity assumptions on the exact solution.

Fatone, Lorella; Funaro, Daniele ( 2017 ) - Isospectral Domains for Discrete Elliptic Operators - JOURNAL OF SCIENTIFIC COMPUTING - pp. da 1 a 22 ISSN: 0885-7474 [Articolo in rivista (262) - Articolo su rivista]
Abstract

Concerning the Laplace operator with homogeneous Dirichlet boundary conditions, the classical notion of isospectrality assumes that two domains are related when they give rise to the same spectrum. In two dimensions, non isometric, isospectral domains exist. It is not known however if all the eigenvalues relative to a specific domain can be preserved under suitable continuous deformation of its geometry. We show that this is possible when the 2D Laplacian is replaced by a finite dimensional version and the geometry is modified by respecting certain constraints. The analysis is carried out in a very small finite dimensional space, but it can be extended to more accurate finite-dimensional representations of the 2D Laplacian, with an increase of computational complexity. The aim of this paper is to introduce the preliminary steps in view of more serious generalizations.

Fatone, Lorella; Funaro, Daniele ( 2015 ) - Optimal collocation nodes for fractional derivative operators - SIAM JOURNAL ON SCIENTIFIC COMPUTING - n. volume 37 [Articolo in rivista (262) - Articolo su rivista]
Abstract

Spectral discretizations of fractional derivative operators are examined, where the approximation basis is related to the set of Jacobi polynomials. The pseudospectral method is implemented by assuming that the grid, used to represent the function to be differentiated, may not be coincident with the collocation grid. The new option opens the way to the analysis of alternative techniques and the search for optimal distributions of collocation nodes, based on the operator to be approximated. Once the initial representation grid has been chosen, indications for how to recover the collocation grid are provided, with the aim of enlarging the dimension of the approximation space. As a result of this process, performances are improved. Applications to fractional type advection-diffusion equation and comparisons in terms of accuracy and efficiency are made. As shown in the analysis, special choices of the nodes can also suggest tricks to speed up computations.

D. Funaro ( 2014 ) - Charging Capacitors According to Maxwell’s Equations: Impossible - ANNALES DE LA FONDATION LOUIS DE BROGLIE - n. volume 39 - pp. da 75 a 93 ISSN: 0182-4295 [Articolo in rivista (262) - Articolo su rivista]
Abstract

The charge of an ideal parallel capacitor leads to the resolution of the wave equation for the electric field with prescribed initial conditions and boundary constraints. Independently of the capacitor’s shape and the applied voltage, none of the corresponding solutions is compatible with the full set of Maxwell’s equations. The paradoxical situation persists even by weakening boundary conditions, resulting in the impossibility to describe a trivial phenomenon such as the capacitor’s charging process, by means of the standard Maxwellian theory.

D. Funaro; E. Kashdan ( 2014 ) - Simulation of Electromagnetic Scattering with Stationary or Accelerating Targets - INTERNATIONAL JOURNAL OF MODERN PHYSICS C - n. volume 26(7) - pp. da 1550075 a 1550075 ISSN: 0129-1831 [Articolo in rivista (262) - Articolo su rivista]
Abstract

The scattering of electromagnetic waves by an obstacle is analyzed through a set of partial differential equations combining the Maxwell's model with the mechanics of fluids. Solitary type EM waves, having compact support, may easily be modeled in this context since they turn out to be explicit solutions. From the numerical viewpoint, the interaction of these waves with a material body is examined. Computations are carried out via a parallel high-order finite-differences code. Due to the presence of a gradient of pressure in the model equations, waves hitting the obstacle may impart acceleration to it. Some explicative 2D dynamical configurations are then studied, enabling the study of photon-particle iterations through classical arguments.

D. Funaro ( 2014 ) - Trapping Electromagnetic Solitons in Cylinders - MATHEMATICAL MODELLING AND ANALYSIS - n. volume 19(1) - pp. da 44 a 51 ISSN: 1392-6292 [Articolo in rivista (262) - Articolo su rivista]
Abstract

Electromagnetic waves in vacuum or dielectrics can be confined in unbounded cylinders in such a way that they turn around the main axis. For particular choices of the cylinder's section, interesting stationary configurations may be assumed. By refining some results obtained in previous papers, additional more complex situations are examined here. For such peculiar guided waves an explicit expression is given in terms of Bessel's functions. Possible applications are in the development of whispering gallery resonators.

Funaro, Daniele ( 2010 ) - Numerical Simulation of Electromagnetic Solitons and their Interaction with Matter - JOURNAL OF SCIENTIFIC COMPUTING - n. volume 45 - pp. da 259 a 271 ISSN: 0885-7474 [Articolo in rivista (262) - Articolo su rivista]
Abstract

A suitable correction of the Maxwell model brings to anenlargement of the space of solutions, allowing for the existenceof solitons in vacuum. We review the basic achievements of thetheory and discuss some approximation results based on an explicitfinite-difference technique. The experiments in two dimensionssimulate travelling solitary electromagnetic waves, and show theirinteraction with conductive walls. In particular, the classicaldispersion, exhibited by the passage of a photon through a smallaperture, is examined.

Chinosi, C.; Della Croce, L.; Funaro, Daniele ( 2010 ) - Rotating Electromagnetic Waves in Toroid-Shaped Regions - INTERNATIONAL JOURNAL OF MODERN PHYSICS C - n. volume 21, n.1 - pp. da 11 a 32 ISSN: 0129-1831 [Articolo in rivista (262) - Articolo su rivista]
Abstract

Electromagnetic waves, solving the full set of Maxwell equationsin vacuum, are numerically computed. These waves occupy a fixedbounded region of the three dimensional space, topologicallyequivalent to a toroid. Thus, their fluid dynamics analogs arevortex rings. An analysis of the shape of the sections of therings, depending on the angular speed of rotation and the majordiameter, is carried out. Successively, spherical electromagneticvortex rings of Hill's type are taken into consideration. For someinteresting peculiar configurations, explicit numerical solutionsare exhibited.

L. Fatone; D. Funaro; G. J. Yoon ( 2008 ) - A Convergence Analysis for the Superconsistent Chebyshev Method - Elsevier BV:PO Box 211, 1000 AE Amsterdam Netherlands:011 31 20 4853757, 011 31 20 4853642, 011 31 20 4853641, EMAIL: nlinfo-f@elsevier.nl, INTERNET: http://www.elsevier.nl, Fax: 011 31 20 4853598 ) - APPLIED NUMERICAL MATHEMATICS - n. volume 58 - pp. da 88 a 100 ISSN: 0168-9274 [Articolo in rivista (262) - Articolo su rivista]
Abstract

The superconsistent collocation method is based on collocation nodes which are different from those used to represent the solution. The two grids are chosen in such a way that the continuous and the discrete operators coincide on a space as larger as possible (superconsistency). There are many documented situations in which this technique provides excellent numerical results. Unfortunately very little theory has been developed. Here, a theoretical convergence analysis for the superconsistent discretization of the second derivative operator, when the representation grid is the set of Chebyshev Gauss–Lobatto nodes is carried out. To this end, a suitable quadrature formula is introduced and studied.

D. Funaro ( 2008 ) - Electromagnetism and the Structure of Matter - World Scientific Singapore SGP) [Monografia o trattato scientifico (276) - Monografia/Trattato scientifico]
Abstract

The classical theory of electromagnetism is entirely revised in this book by proposing a variant of Maxwell equations that allows solitonic solutions (photons). The Lagrangian is the standard one, but it is minimized on a constrained space that enforces the wave packets to follow the rules of geometrical optics. Exact solutions are explicitly shown; this opens a completely new perspective for the study of light wave phenomena. In the framework of general relativity, the equations are written in covariant form. A coupling with the metric is obtained through the Einstein equation, whose solutions are computed exactly in a lot of original situations. Finally, the explicit construction of elementary particles, consisting of rotating photons, is indicated. The results agree qualitatively and quantitatively with what it is actually observed. This opens the path to an understanding of the structure of matter and its properties, also aimed to provide a causal explanation to quantum phenomena.

L. Fatone; D. Funaro; V. Scannavini ( 2007 ) - Finite-Differences Preconditioners for Superconsistent Pseudospectral Approximations - MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE - n. volume v. 41, n. 6 - pp. da 1021 a 1039 ISSN: 0764-583X [Articolo in rivista (262) - Articolo su rivista]
Abstract

The superconsistent collocation method, which is based on a collocation grid different from the one used to represent the solution, has proven to be very accurate in the resolution of various functional equations. Excellent results can be also obtained for what concerns preconditioning. Some analysis and numerous experiments, regarding the use of finite-differences preconditioners, for matrices arising from pseudospectral approximations of advection-diffusion boundary value problems, are presented and discussed, both in the case of Legendre and Chebyshev representation nodes.

D. Funaro ( 2006 ) - Analisi Numerica - - - Ist. Enciclopedia Italiana (Treccani) Roma ITA) [Voce (in dizionario o enciclopedia) (271) - Voce in Dizionario o Enciclopedia]
Abstract

D. Funaro; G. Pontrelli ( 2005 ) - A general class of finite-difference methods for the linear transport equation - COMMUNICATIONS IN MATHEMATICAL SCIENCES - n. volume 3 - pp. da 403 a 423 ISSN: 1539-6746 [Articolo in rivista (262) - Articolo su rivista]
Abstract

A wide family of finite-differences methods for the linear advection equation, based on a six-point stencil, is presented. The family depends on three parameters and includes most of the classical linear schemes. A stability and consistency analysis is carried out of the parameters. The problem of the determination of the parameters providing the best approximation is also addressed.

L. Fatone; D. Funaro; R. Giova ( 2005 ) - Finite-difference schemes for transport-dominated equations using special collocation nodes - NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS - n. volume 21(4) - pp. da 649 a 671 ISSN: 0749-159X [Articolo in rivista (262) - Articolo su rivista]
Abstract

We introduce finite-difference schemes based on a special upwind-type collocation grid, in order to obtain approximations of the solution of linear transport-dominated advection-diffusion problems. The method is well suited when the diffusion parameter is very small compared to the discretization parameter. A theory is developed and many numerical experiments are shown.

D. Funaro ( 2004 ) - Superconsistent discretizations of integral type equations - APPLIED NUMERICAL MATHEMATICS - n. volume 48 - pp. da 1 a 11 ISSN: 0168-9274 [Articolo in rivista (262) - Articolo su rivista]
Abstract

We recall some of the results, obtained by using the collocation method at special nodes, in the approximation of boundary value problems for partial differential equations. Similar techniques are applied for discretizing, by the Gaussian collocation method, a singular integral equation of the Carleman type. The idea, documented by some comparative tests, can be generalized to other approximation methods and/or integral equations of a different kind. It can also provide a link between different numerical approaches. (C) 2003 IMACS. Published by Elsevier B.V. All rights reserved.

D. Funaro ( 2003 ) - Superconsistent Discretizations with Application to Hyperbolic Equation - SIBIRSKII ZHURNAL VYCHISLITEL'NOI MATEMATIKI - n. volume v. 6, n.1 - pp. da 89 a 99 ISSN: 1560-7526 [Articolo in rivista (262) - Articolo su rivista]
Abstract

A family of finite difference methods for the linear hyperbolic equations, constructed on a six-point stencil, is presented. The family depends on 3 parameters and includes many of the classical linear schemes. The approximation method is based on the use of two different grids. One grid is used to represent the approximated solution, the other (the collocation grid) is where the equation is to be satisfied. The two grids are related in such a way that the exact and the discrete operators have a common space which is as large as possible.

D. Funaro ( 2002 ) - Superconsistent Discretizations - JOURNAL OF SCIENTIFIC COMPUTING - n. volume v. 17, n. 1-4 - pp. da 67 a 79 ISSN: 0885-7474 [Articolo in rivista (262) - Articolo su rivista]
Abstract

We say that the approximation of a linear operator issuperconsistent when the exact and the discrete operatorscoincide on a functional space whose dimension is bigger thanthe number of degrees of freedom needed in the constructionof the discretization. By providing several examples we show how to build superconsitent schemes.

D. Funaro ( 2001 ) - A superconsistent Chebyshev collocation method for second-order differential operators - NUMERICAL ALGORITHMS - n. volume 28 - pp. da 151 a 157 ISSN: 1017-1398 [Articolo in rivista (262) - Articolo su rivista]
Abstract

A standard way to approximate the model problem -u = f, with u(+/-1) = 0, is to collocate the differential equation at the zeros of T-n': x(i), i = 1,..., n - 1, having denoted by T,, the nth Chebyshev polynomial. We introduce an alternative set of collocation nodes z(i), i = 1,..., n - 1, which will provide better numerical performances. The approximated solution is still computed at the nodes {x(i)}, but the equation is required to be satisfied at the new nodes {z(i)}, which are determined by asking an extra degree of consistency in the discretization of the differential operator.

D. Funaro ( 2000 ) - About 3-D Spectral Element Computations - Recent Trends in Numerical Analysis, Advances in the Theory of Computational Mathematics - Nova Science Pub. Huntington NY USA) - n. volume 3 - pp. da 163 a 173 ISBN: 978 [Contributo in Atti di convegno (273) - Relazione in Atti di Convegno]
Abstract

D. Funaro ( 1999 ) - A note on second-order finite-difference schemes on uniform meshes for advection-diffusion equations - NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS - n. volume 15 - pp. da 581 a 588 ISSN: 0749-159X [Articolo in rivista (262) - Articolo su rivista]
Abstract

An artificial-viscosity finite-difference scheme is introduced for stabilizing the solutions of advection-diffusion equations. Although only the linear one-dimensional case is discussed, the method is easily susceptible to generalization. Some theory and comparisons with other well-known schemes are carried out. The aim is, however, to explain the construction of the method, rather than considering sophisticated applications.

D. Funaro; G. Pontrelli ( 1999 ) - Spline approximation of advection-diffusion problems using upwind type collocation nodes - JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS - n. volume 110 - pp. da 141 a 153 ISSN: 0377-0427 [Articolo in rivista (262) - Articolo su rivista]
Abstract

A spline collocation method for linear advection-diffusion equations is proposed. The method is based on an operator-dependent collocation grid, and provides stabilized approximated solutions, with respect to the coefficient of the diffusive term, when problems are advection dominated. (C) 1999 Elsevier Science B.V. All rights reserved.

D. Funaro ( 1997 ) - Some remarks about the collocation method on a modified Legendre grid - COMPUTERS & MATHEMATICS WITH APPLICATIONS - n. volume 33 - pp. da 95 a 103 ISSN: 0898-1221 [Articolo in rivista (262) - Articolo su rivista]
Abstract

We compare the results obtained by applying the standard collocation method at the Legendre Gauss-Lobatto nodes, for a model problem simulating a steady advection-diffusion equation, with those obtained by collocating at a new set of nodes. These nodes are derived from a suitable modification of the Legendre grid, according to the magnitude of the ratio between the advective and the diffusive parts of the differential operator.

Daniele Funaro ( 1997 ) - Spectral Elements for Transport-Dominated Equations - Springer Verlag Heidelberg DEU) - pp. da 0 a 222 ISBN: 9783540626497 [Monografia o trattato scientifico (276) - Monografia/Trattato scientifico]
Abstract

The book deals with the numerical approximation of various PDEs using the spectral element method, with particular emphasis for elliptic equations dominated by first-order terms. It provides a simple introduction to spectral elements with additional new tools (upwind grids and preconditioners). Applications to fluid dynamics and semiconductor devices are considered, as well as in other models were transport-diffusion equations arise. The aim is to provide the reader with both introductive and more advanced material on spectral Legendre collocation methods. The book however does not cover all the aspects of spectral methods. Engineers, physicists and applied mathematicians may study how to implement the collocation method and use the results to improve their computational codes.

D. FUNARO ( 1993 ) - A NEW SCHEME FOR THE APPROXIMATION OF ADVECTION-DIFFUSION EQUATIONS BY COLLOCATION - SIAM JOURNAL ON NUMERICAL ANALYSIS - n. volume 30 - pp. da 1664 a 1676 ISSN: 0036-1429 [Articolo in rivista (262) - Articolo su rivista]
Abstract

A competitive algorithm, which allows the computation of approximated polynomial solutions of advection-diffusion equations in the square, is presented. The equation is collocated at a special grid and the corresponding system is solved by a low-cost preconditioned iterative procedure. The method provides accurate results even when the solution presents sharp boundary layers.

Daniele Funaro ( 1992 ) - Polynomial Approximation of Differential Equations - Springer Verlag Heidelberg DEU) - pp. da 0 a 333 ISBN: 0387552308 [Monografia o trattato scientifico (276) - Monografia/Trattato scientifico]
Abstract

This book is a basic and comprehensive introduction to the use of spectral methods for the approximation of the solution to ordinary differential equations and time-dependent boundary-value problems. The algorithms are presented and studied both from the point of view of the theoreticalanalysis of convergence and the numerical implementation. Unlike other texts devoted to the subject this is a concise introduction that is ideally suited to the novice and practitioner alike, enabling them to assimilate themethods quickly and efficiently.

D. FUNARO; O. KAVIAN ( 1991 ) - APPROXIMATION OF SOME DIFFUSION EVOLUTION-EQUATIONS IN UNBOUNDED-DOMAINS BY HERMITE FUNCTIONS - MATHEMATICS OF COMPUTATION - n. volume 57 - pp. da 597 a 619 ISSN: 0025-5718 [Articolo in rivista (262) - Articolo su rivista]
Abstract

Spectral and pseudospectral approximations of the heat equation are analyzed. The solution is represented in a suitable basis constructed with Hermite polynomials. Stability and convergence estimates are given and numerical tests are discussed.

D. FUNARO; D. GOTTLIEB ( 1991 ) - CONVERGENCE RESULTS FOR PSEUDOSPECTRAL APPROXIMATIONS OF HYPERBOLIC SYSTEMS BY A PENALTY-TYPE BOUNDARY TREATMENT - MATHEMATICS OF COMPUTATION - n. volume 57 - pp. da 585 a 596 ISSN: 0025-5718 [Articolo in rivista (262) - Articolo su rivista]
Abstract

In a previous paper we have presented a new method of imposing boundary conditions in the pseudospectral Chebyshev approximation of a scalar hyperbolic equation. The novel idea of the new method is to collocate the equation at the boundary points as well as in the inner grid points, using the boundary conditions as penalty terms. In this paper we extend the above boundary treatment to the case of pseudospectral approximations to general constant-coefficient hyperbolic systems of equations, and we provide error estimates for the pseudospectral Legendre method. The same scheme can be implemented also in the general (even nonlinear) case.

D. FUNARO ( 1990 ) - COMPUTATIONAL ASPECTS OF PSEUDOSPECTRAL LAGUERRE APPROXIMATIONS - APPLIED NUMERICAL MATHEMATICS - n. volume 6 - pp. da 447 a 457 ISSN: 0168-9274 [Articolo in rivista (262) - Articolo su rivista]
Abstract

Pseudospectral approximations in unbounded domains by Laguerre polynomials lead to ill-conditioned algorithms. We introduce a scaling function and appropriate numerical procedures to limit these phenomena.

O. COULAUD; D. FUNARO; O. KAVIAN ( 1990 ) - LAGUERRE SPECTRAL APPROXIMATION OF ELLIPTIC PROBLEMS IN EXTERIOR DOMAINS - COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING - n. volume 80 - pp. da 451 a 458 ISSN: 0045-7825 [Articolo in rivista (262) - Articolo su rivista]
Abstract

D. FUNARO; W. HEINRICHS ( 1990 ) - SOME RESULTS ABOUT THE PSEUDOSPECTRAL APPROXIMATION OF ONE-DIMENSIONAL 4TH-ORDER PROBLEMS - NUMERISCHE MATHEMATIK - n. volume 58 - pp. da 399 a 418 ISSN: 0029-599X [Articolo in rivista (262) - Articolo su rivista]
Abstract

We analyze the pseudospectral approximation of fourth order problems. We give convergence results in the one dimensional case. Numerical experiments are shown in two dimensions for the approximation of the rhombic plate bending problem. Eigenvalues and preconditioning are also investigated.

D. FUNARO; D. GOTTLIEB ( 1988 ) - A NEW METHOD OF IMPOSING BOUNDARY-CONDITIONS IN PSEUDOSPECTRAL APPROXIMATIONS OF HYPERBOLIC-EQUATIONS - MATHEMATICS OF COMPUTATION - n. volume 51 - pp. da 599 a 613 ISSN: 0025-5718 [Articolo in rivista (262) - Articolo su rivista]
Abstract

A new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested. This method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions. Stability and convergence results are proven for the Chebyshev approximation of linear scalar hyperbolic equations. The eigenvalues of this method applied to parabolic equations are shown to be real and negative.

Daniele Funaro;Alfio Quarteroni;Paola Zanolli ( 1988 ) - An Iterative Procedure with Interface Relaxation for Domain Decomposition Methods - SIAM JOURNAL ON NUMERICAL ANALYSIS - n. volume 25 - pp. da 1213 a 1236 ISSN: 0036-1429 [Articolo in rivista (262) - Articolo su rivista]
Abstract

A domain decomposition method for second-order elliptic problems is considered. An iterative procedure that reduces the problem to a sequence of mixed boundary value problems on each subdomain is proposed. At each iteration, a relaxation is accomplished at the subdomain interfaces. In several circumstances, a value of the relaxation parameter that yields exact convergence in a finite number of iterations is explicitly found. Moreover, when such a value is not available, an appropriate strategy for the automatic selection of the relaxation parameter at each iteration is indicated and analyzed.This iterative method is then applied to the spectral collocation approximation of the differential problem. The same kind of convergence results are proven. Many numerical experiments show the effectiveness of the method proposed here.

D. FUNARO ( 1988 ) - DOMAIN DECOMPOSITION METHODS FOR PSEUDO SPECTRAL APPROXIMATIONS .1. 2ND ORDER EQUATIONS IN ONE DIMENSION - NUMERISCHE MATHEMATIK - n. volume 52 - pp. da 329 a 344 ISSN: 0029-599X [Articolo in rivista (262) - Articolo su rivista]
Abstract

Multidomain pseudo spectral approximations of second order boundary value problems in one dimension are considered. The equation is collocated at the Chebyshev nodes inside each subinterval. Different patching conditions at the interfaces are analyzed. Results of stability and convergence are given.

C. CANUTO; D. FUNARO ( 1988 ) - THE SCHWARZ ALGORITHM FOR SPECTRAL METHODS - SIAM JOURNAL ON NUMERICAL ANALYSIS - n. volume 25 - pp. da 24 a 40 ISSN: 0036-1429 [Articolo in rivista (262) - Articolo su rivista]
Abstract

Recently, the Schwarz alternating method has been successfully coupled to spatial discretizations of spectral type, in order to solve boundary value, problems in complex, geometries with infinite order accuracy. In this paper, a simple version of the method is considered. A proof of its convergence is given in the energy norm, exploiting the properties of discrete-harmonic polynomials and a discrete maximum principle for spectral methods. More general situations can be handled theoretically in one space dimension.

D. FUNARO ( 1987 ) - A PRECONDITIONING MATRIX FOR THE CHEBYSHEV DIFFERENCING OPERATOR - SIAM JOURNAL ON NUMERICAL ANALYSIS - n. volume 24 - pp. da 1024 a 1031 ISSN: 0036-1429 [Articolo in rivista (262) - Articolo su rivista]
Abstract

An efficient preconditioner for the Chebyshev differencing operator is considered. The corresponding preconditioned eigenvalues are real and positive and lie between 1 and ${\pi / 2}$. An eixpelicit formula for these eigenvalues and the corresponding eigenfunctions is given. The results are generalized to the case of operators related to Chebyshev discretizations of systems of linear differential equations.