Optimal H1 Estimates for two Time-discrete Galerkin Approximations of a Nonlinear Schrodinger Equation

We prove optimal H¹ estimates for the backward Euler and Crank-Nicolsondiscretizations of the Galerkin finite element method appliedto a nonlinear Schrödinger equation. As a by-product, wealso obtain new pointwise error bounds. The analysis relieson a nonlinear stability theory recently developed by López-Marcos& Sanz-Serna 1991.     

Product Approximation for Nonlinear Klein-Gordon Equations

^* Current address: Oxford University Computing Laboratory, Oxford, England OX1 3QD. We present convergence results for the Galerkin method withproduct approximation applied to a family of nonlinear Klein-Gordonequations. In the first part of the paper, we obtain error estimatesunder the assumption that a sufficiently smooth solution exists.This is achieved by adapting the method of Douglas and Dupont[6] to the present case. The second part of the paper dealswith the approximation of ‘weak’ solutions and theanalysis relies on a priori estimates satisfied by the approximatesolution.