Approximation results for orthogonal polynomials in Sobolev spaces

We analyze the approximation properties of some interpolation operators and some L2-orthogonal projection operators related to systems of polynomials which are orthonormal with respect to a weight function o(x1, . . ., Xd), d > 1. The error estimates for the Legendre system and the Chebyshev system of the first kind are given in the norms of the Sobolev spaces H'. These results are useful in the numerical analysis of the approximation of partial differential equations by spectral methods. 0. Introduction. Spectral methods are a classical and largely used technique to solve differential equations, both theoretically and numerically. During the years they have gained new popularity in automatic computations for a wide class of physical problems (for instance in the fields of fluid and gas dynamics), due to the use of the Fast Fourier Transform algorithm. These methods appear to be competitive with finite difference and finite element methods and they must be decisively preferred to the last ones whenever the solution is highly regular and the geometric dimension of the domain becomes large. Moreover, by these methods it is possible to control easily the solution (filtering) of those numerical problems affected by oscillation and instability phenomena. The use of spectral and pseudo-spectral methods in computations in many fields of engineering has been matched by deeper theoretical studies; let us recall here the pioneering works by Orszag [25], [26], Kreiss and Oliger [14] and the monograph by Gottlieb and Orszag [13]. The theoretical results of such works are mainly concerned with the study of the stability of approximation of parabolic and hyperbolic equations; the solution is assumed to be infinitely differentiable, so that by an analysis of the Fourier coefficients an infinite order of convergence can be achieved. More recently (see Pasciak [27], Canuto and Quarteroni [10], [11], Maday and Quarteroni [20], [211, [22], Mercier [23]), the spectral methods have been studied by the variational techniques typical of functional analysis, to point out the dependence of the approximation error (for instance in the L2-norm, or in the energy norm) on the regularity of the solution of continuous problems and on the discretization parameter (the dimension of the space in which the approximate solution is sought). Indeed, often the solution is not infinitely differentiable; on the other hand, sometimes even if the solution is smooth, its derivatives may have very Received August 9, 1980; revised June 12, 1981. 1980 Mathematics Subject Classification. Primary 41A25; Secondary 41A 10, 41A05. 1982 American Mathematical Society 0025-571 8/82/0000-0470/$06.00 (67 This content downloaded from 207.46.13.111 on Tue, 09 Aug 2016 06:29:39 UTC All use subject to http://about.jstor.org/terms 68 C. CANUTO AND A. QUARTERONI large norms which affect negatively the rate of convergence (for instance in problems with boundary layers). Both spectral and pseudo-spectral methods are essentially Ritz-Galerkin methods (combined with some integration formulae in the pseudo-spectral case). It is well known that when Galerkin methods are used the distance between the exact and the discrete solution (approximation error) is bounded by the distance between the exact solution and its orthogonal projection upon the subspace (projection error), or by the distance between the exact solution and its interpolated polynomial at some suitable points (interpolation error). This upper bound is often realistic, in the sense that the asymptotic behavior of the approximation error is not better than the one of the projection (or even the interpolation) error. Even more, in some cases the approximate solution coincides with the projection of the true solution upon the subspace (for instance when linear problems with constant coefficients are approximated by spectral methods). This motivates the interest in evaluating the projection and the interpolation errors in differently weighted Sobolev norms. So we must face a situation different from the one of the classical approximation theory where the properties of approximation of orthogonal function systems, polynomial and trigonometric, are studied in the LP-norms, and mostly in the maximum norm (see, e.g., Butzer and Berens [6], Butzer and Nessel [7], Nikol'skiT [24], Sansone [291, Szego [30], Triebel [31], Zygmund [32]; see also Bube [5]). Approximation results in Sobolev norms for the trigonometric system have been obtained by Kreiss and Oliger [15]. In this paper we consider the systems of Legendre orthogonal polynomials, and of Chebyshev orthogonal polynomials of the first kind in dimension d > 1. The reason for this interest must be sought in the applications to spectral approximations of boundary value problems. Indeed, if the boundary conditions are not periodic, Legendre approximation seems to be the easiest to be investigated (the weight w is equal to 1). On the other hand, the Chebyshev approximation is the most effective for practical computations since it allows the use of the Fast Fourier Transform algorithm. The techniques used to obtain our results are based on the representation of a function in the terms of a series of orthogonal polynomials, on the use of the so-called inverse inequality, and finally on the operator interpolation theory in Banach spaces. For the theory of interpolation we refer for instance to Calderon [8], Lions [17], Lions and Peetre [19], Peetre [28]; a recent survey is given, e.g., by Bergh and Lofstrom [4]. An outline of the paper is as follows. In Section 1 some approximation results for the trigonometric system are recalled; the presentation of the results to the interpolation is made in the spirit of what will be its application to Chebyshev polynomials. In Section 2 we consider the La-projection operator upon the space of polynomials of degree at most N in any variable (w denotes the Chebyshev or Legendre weight). In Section 3 a general interpolation operator, built up starting by integration formulas which are not necessarily the same in different spatial dimensions, is considered, and its approximation properties are studied. In [22] Maday and Quarteroni use the results of Section 2 to study the approximation properties of some projection operators in higher order Sobolev norms. Recently, an interesting method which lies inbetween finite elements and This content downloaded from 207.46. 13.111 on Tue, 09 Aug 2016 06:29:39 UTC All use subject to http://about.jstor. org/terms ORTHOGONAL POLYNOMIALS IN SOBOLEV SPACES 69 spectral methods has been investigated from the theoretical point of view by Babuska, Szabo and Katz [3]. In particular they obtain approximation properties of polynomials in the norms of the usual Sobolev spaces. Acknowledgements. Some of the results of this paper were announced in [9]; we thank Professor J. L. Lions for the presentation to the C. R. Acad. Sci. of Paris. We also wish to express our gratitude to Professors F. Brezzi and P. A. Raviart for helpful suggestions and continuous encouragement. Notations. Throughout this paper we shall use the following notations: I will be an open bounded interval c R, whose variable is denoted by x; Q the product Id C Rd (d integer > 1) whose variable is denoted by x = (x(.')I_ d; for a multi-integer k E Zd, we set ikV = jd X I'12 and IkloK = m x 1, Dj = a/ax@). The symbol X'J=p (q eventually + oo) will denote the summation over all integral k such that p 0 in U. Set L2(Q) = ({: Q -C I 0 is measurable and ( 0, set Hs ( () = C E L(Q) I 1111ksI, < +? }, where /d 2 11I412I= kENd f DI L/)4 D w dx.