One-step piecewise polynomial Galerkin methods for initial value problems

Abstract:

A new approach to the numerical solution of systems of first-order ordinary differential equations is given by finding local Galerkin approximations on each subinterval of a given mesh of size $h$. One step at a time, a piecewise polynomial, of degree $n$ and class ${C^0}$, is constructed, which yields an approximation of order $O({h^{2n}})$ at the mesh points and $O({h^{n + 1}})$ between mesh points. In addition, the $j$th derivatives of the approximation on each subinterval have errors of order $O({h^{n - j + 1}}),1 \leqq j \leqq n$. The methods are related to collocation schemes and to implicit Runge-Kutta schemes based on Gauss-Legendre quadrature, from which it follows that the Galerkin methods are $A$-stable.