Gaussian quadrature rules and \(A\)-stability of Galerkin schemes for ODE (Q1811957)

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.