Convergence Estimates for Galerkin Methods for Variable Coefficient Initial Value Problems

The use of Galerkin’s method for the approximate solution of the initial value problem for certain simple equations ${{\partial u} / {\partial t = Pu}}$, where P is a differential operator of order m with respect to x, is analyzed when the approximate solution is sought in the space of smooth splines of order $\mu $ based on a uniform mesh with mesh-width h. It is proved that at the mesh-points the error can be made to be $O(h^\nu )$, where $v = 2\mu - m$ for m even, $v = 2\nu - m + 1$ for m odd.