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 ∂ u / ∂ 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 μ 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^ν ), where v = 2μ - m for m even, v = 2ν - m + 1 for m odd.