Gaussian quadrature rules and \(A\)-stability of Galerkin schemes for ODE. (English) Zbl 1024.65068

Bensebah, Ali; Dubeau, François; Gélinas, Jacques

Int. J. Math. Math. Sci. 2003, No. 31, 1947-1959 (2003).

This paper deals with the A-stability properties of some Galerkin methods for the numerical solution of ordinary differential equations (ODEs). Firstly, a general variational formulation of the original initial value problem which allows to include both continuous and discontinuous Galerkin formulations is given. Then, by using the standard scalar product for the original problem and a discrete semi scalar product with respect to the \(m\) roots of the polynomial \( \pi_m(x) - \gamma \pi_{m-1}(x)\) where \( \pi_m\) is the \(m\)th Legendre polynomial in \( [-1,1]\) with the normalization \( \pi_m(1)=1\) and \( \gamma \in [-1,1] \) a parameter, together with some properties of orthogonal polynomials, the A-stability properties of several continuous and discontinuous Galerkin schemes for some values of \( \gamma \) are derived.
It must be noticed that this paper presents an alternative proof of A-stability of some well known collocation methods.

Reviewer: Manuel Calvo (Zaragoza)

MSC:

65L20	Stability and convergence of numerical methods for ordinary differential equations
65L60	Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L05	Numerical methods for initial value problems involving ordinary differential equations
41A55	Approximate quadratures

Keywords:

initial value problems; Gaussian quadrature rules; Galerkin schemes; A-stability; collocation methods