Equivalence between the spectral and the finite elements matrices M Ribot M Schatzman

Claude Bernard Lyon

Description

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.

• approximations spectrales de prob- lèmes aux limites elliptiques


• functions theorem


• elements method


• equivalence between


• finite elements


• spectral method


• method

Sujets

• 17764 Schatzman
• Finite element method
• Spectral method
• Method

Informations

• Publié par: pefav
• Nombre de lectures: 11
• Langue: English

Extrait

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 matrixK S for a spectral Legendre–Gauss– Lobatto method on the interval [−with Dirichlet con-1, 1] ditions with the rigidity matrixK F ofP 1 finite elements method. In a second part, we prove the analogous equiva- lence between the mass matrixM S for the spectral method and the mass matrixM F for the finite elements method. −1 We finally establish that the norm ofM M S considered 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 byL N the Legendre polynomial of degreeNand 2 by−1=ξ 0 < ξ 1 <∙ ∙ ∙< ξ N =1 the roots of (1−X)L. N Proposition I.4.5 of Bernardi and Maday [4] gives us the existence of non-negative numbersρ k , 0≤k≤Nsuch that

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2003 Elsevier Science Ltd.All rights reserved. Computational Fluid and Solid Mechanics 2003 K.J. Bathe (Editor)

for all polynomialsof degree at most equal to 2N−1, 1  N  (x) dx=(ξ k )ρ k . k=0 −1 Bernardi and Maday [4] establish an exact formula forρ k which is 2 ρ k =, 0≤k≤N. (1) 2 N(N+1)L(ξ k ) N The collocation method on the knotsξ k is defined by the data of a basis, the Lagrange basis built on those knots and denoted byl k for 0≤k≤N, and the data of an inner product defined by N  (u,v) N =u(ξ k )v(ξ k )ρ k . k=0 Thus, the coefficients of the mass matrixM S for the spec- tral method are given by (M S ) i,j =(l i ,l j ) N =δ i,j ρ j , 1≤i,j≤N−1 and those of the rigidity matrixK S by N     (K)=(l,l)=ρl(ξ). S i,j Nj ik i (ξ k )l j k k=0 Let us now define the matrices for the finite elements method. We denote by k , 1≤k≤N−1 the hat functions centred on the knotsξ k which span the space ofP 1 finite elements. The coefficients of the mass matrixM F are obtained by mass lumping, i.e. by approaching the integral of i  j for 1≤i,j≤N−1 using the trapezoid rule. Thus, M F is diagonal and ξ i+1 −ξ i−1 (M F ) i,i =, 1≤i≤N−1. 2 The rigidity matrixK F is tridiagonal and its coefficient   (K F ) i,j is the integral of . i j