Equivalence between the spectral and the finite elements matrices M Ribot M Schatzman
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2110 Equivalence between the spectral and the finite elements matrices M. Ribot ?, M. Schatzman MAPLY, CNRS and Université Claude Bernard-Lyon 1, 69622 Villeurbanne Cedex, France Abstract In this paper, we prove the spectral equivalence of the mass matrix for a Legendre–Gauss–Lobatto method with the mass matrix of finite elements method; we also prove analogous results on rigidity matrices. For this purpose, we establish some asymptotic formulae for Legendre polynomials and for the roots of their derivatives. Keywords: Preconditioner; Spectral method; Finite elements method; Mass matrix; Rigidity matrix; Legendre polynomials 1. Introduction A well-known preconditioner for spectral methods uses finite differences or low degree finite elements on the nodes of the spectral method. This is a very efficient process, which has been validated numerically by Orszag [1], Deville and Mund [2], Canuto and Quarteroni [3] and others . . . but for which the theory was lacking. In this paper, we first show the spectral equivalence of the rigidity matrix KS for a spectral Legendre–Gauss– Lobatto method on the interval [?1,1] with Dirichlet con- ditions with the rigidity matrix KF of P1 finite elements method. In a second part, we prove the analogous equiva- lence between the mass matrix MS for the spectral method and the mass matrix MF for the finite elements method.
Voir
- approximations spectrales de prob- lèmes aux limites elliptiques
- functions theorem
- elements method
- equivalence between
- finite elements
- spectral method
- method
2110
Equivalence between the spectral and the finite elements matrices
∗ M. Ribot, M. Schatzman MAPLY, CNRS and Université Claude BernardLyon 1, 69622 Villeurbanne Cedex, France
Abstract In this paper, we prove the spectral equivalence of the mass matrix for a Legendre–Gauss–Lobatto method with the mass matrix of finite elements method; we also prove analogous results on rigidity matrices. For this purpose, we establish some asymptotic formulae for Legendre polynomials and for the roots of their derivatives. Keywords:Preconditioner; Spectral method; Finite elements method; Mass matrix; Rigidity matrix; Legendre polynomials
1. Introduction
A well-known preconditioner for spectral methods uses finite differences or low degree finite elements on the nodes of the spectral method. This is a very efficient process, which has been validated numerically by Orszag [1], Deville and Mund [2], Canuto and Quarteroni [3] and others. . .but for which the theory was lacking. In this paper, we first show the spectral equivalence of the rigidity matrixKSfor a spectral Legendre–Gauss– Lobatto method on the interval [−with Dirichlet con-1, 1] ditions with the rigidity matrixKFofP1finite elements method. In a second part, we prove the analogous equiva-lence between the mass matrixMSfor the spectral method and the mass matrixMFfor the finite elements method. −1 We finally establish that the norm ofM MSconsidered as F 1 an operator from the discrete Sobolev spaceHto itself is N bounded from below and from above independently onN. All these results can be generalised to a square with Dirich-let conditions with the tensored matrices 1⊗K+K⊗1 and 1⊗M+M⊗1.
2. Definitions and expressions of the mass and rigidity matrices
Denote byLNthe Legendre polynomial of degreeNand 2 by−1=ξ0< ξ1<∙ ∙ ∙< ξN=1 the roots of (1−X)L. N Proposition I.4.5 of Bernardi and Maday [4] gives us the existence of non-negative numbersρk, 0≤k≤Nsuch that
∗ Corresponding author. Tel.:+33 4 72 44 79 49; Fax:+33 4 72 44 80 53; E-mail: ribot@maply.univ-lyon1.fr
2003 Elsevier Science Ltd.All rights reserved. Computational Fluid and Solid Mechanics 2003 K.J. Bathe (Editor)
for all polynomialsof degree at most equal to 2N−1, 1 N (x) dx=(ξk)ρk. k=0 −1 Bernardi and Maday [4] establish an exact formula forρk which is 2 ρk=, 0≤k≤N. (1) 2 N(N+1)L(ξk) N The collocation method on the knotsξkis defined by the data of a basis, the Lagrange basis built on those knots and denoted bylkfor 0≤k≤N, and the data of an inner product defined by N (u,v)N=u(ξk)v(ξk)ρk. k=0 Thus, the coefficients of the mass matrixMSfor the spec-tral method are given by (MS)i,j=(li,lj)N=δi,jρj, 1≤i,j≤N−1 and those of the rigidity matrixKSby N (K)=(l,l)=ρl(ξ). S i,j Nj ik i(ξk)lj k k=0 Let us now define the matrices for the finite elements method. We denote byk, 1≤k≤N−1 the hat functions centred on the knotsξkwhich span the space ofP1finite elements. The coefficients of the mass matrixMFare obtained by mass lumping, i.e. by approaching the integral ofijfor 1≤i,j≤N−1 using the trapezoid rule. Thus, MFis diagonal and ξi+1−ξi−1 (MF)i,i=, 1≤i≤N−1. 2 The rigidity matrixKFis tridiagonal and its coefficient (KF)i,jis the integral of . i j
