Summary.We approximate the solutions of an initial-and boundary-value problem for nonlinear Schr6dinger equations (with emphasis on the 'cubic' nonlinearity) by two fully discrete finite element schemes based on the standard Galerkin method in space and two implicit, Crank-Nicolson-type second-order accurate temporal discretizations. For both schemes we study the existence and uniqueness of their solutions and prove L 2 error bounds of optimal order of accuracy. For one of the schemes we also analyze one step of Newton's method for solving the nonlinear systems that arise at every time step. We then implement this scheme using an iterative modification of Newton's method that, at each time step t", requires solving a number of sparse complex linear systems with a matrix that does not change with n. The effect of this 'inner' iteration is studied theoretically and numerically.

Abstract. We derive optimal order a posteriori error estimates for time discretizations by both the Crank-Nicolson and the Crank-Nicolson-Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates are second-order Crank-Nicolson reconstructions of the piecewise linear approximate solutions. These functions satisfy two fundamental properties: (i) they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori, and (ii) they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation. The resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem. As a consequence we provide alternative proofs for known a priori rates of convergence for the Crank-Nicolson method.

We consider the initial-value problem for the radially symmetric nonlinear Schrödinger equation with cubic nonlinearity (NLS) in d = 2 and 3 space dimensions. To approximate smooth solutions of this problem, we construct and analyze a numerical method based on a standard Galerkin finite element spatial discretization with piecewise linear, continuous functions and on an implicit Crank-Nicolson type time-stepping procedure. We then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the NLS equation that blow up at the origin as the temporal variable t tends from below to a finite value t. For the blow-up of the amplitude of the solution we recover numerically the well-known rate (t − t) −1/2 for d = 3. For d = 2 our numerical evidence supports the validity of the log log law [ln ln 1 t −t /(t − t)] 1/2 for t extremely close to t. The scheme also approximates well the details of the blow-up of the phase of the solution at the origin as t → t .

Efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in [1]. In this note we present a refined analysis to allow more general nonlinearities. The abstract theory is applied to a quasilinear parabolic equation. Classification (1991): 65M60, 65M12, 65L06 Mathematics Subject

We analyze the discretization of an initial-boundary value problem for the cubic Schrödinger equation in one space dimension by a Crank-Nicolson-type finite difference scheme. We then linearize the corresponding equations at each time level by Newton's method and discuss an iterative modification of the linearized scheme which requires solving linear systems with the same tridiagonal matrix. We prove second-order error estimates.