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 du/dt equals Pu, where P is a differential operator of order m with respect to x, and du/dt indicates a partial derivative, 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 nu equals 2 mu minus m for m even, nu equals 2 mu minus m plus 1 for m odd.