Several a posteriori error estimators are introduced and analyzed for a discontinuous Galerkin formulation of a model second-order elliptic problem. In addition to residual-type estimators, we introduce some estimators that are couched in the ideas and techniques of domain decomposition. Results of numerical experiments are presented.

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.

We construct, analyze and numerically validate a class of conservative, discontinuous Galerkin schemes for the Generalized Korteweg-de Vries equation. Up to round-off error, these schemes preserve discrete versions of the first two invariants (the integral of the solution, usually identified with the mass, and the L 2 -norm) of the continuous solution. Numerical evidence is provided indicating that these conservation properties impart the approximations with beneficial attributes, such as more faithful reproduction of the amplitude and phase of traveling-wave solutions. The numerical simulations also indicate that the discretization errors grow only linearly as a function of time.

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 .

No abstract