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Several a posteriori error estimators are introduced and analyzed for a discontin-uous 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. 1. Introduction. One(More)
A residual type a posteriori error estimator is introduced and analyzed for a discontinuous Galerkin formulation of a model second-order elliptic problem with Dirichlet-Neumann type boundary conditions. An adaptive algorithm using this estimator together with specific marking and refinement strategies is constructed and shown to achieve any specified error(More)
Fully discrete discontinuous Galerkin methods with variable meshes in time are developed for the fourth order Cahn-Hilliard equation arising from phase transition in materials science. The methods are formulated and analyzed in both two and three dimensions, and are proved to give optimal order error bounds. This coupled with the flexibility of the methods(More)
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.(More)
We approximate the solutions of an initial-and boundary-value problem for nonlinear Schrödinger 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 discretiza-tions. For both schemes we study the(More)
Implicit Runge–Kutta methods in time are used in conjunction with the Galerkin method in space to generate stable and accurate approximations to solutions of the nonlinear (cubic) Schrödinger equation. The temporal component of the discretization error is shown to decrease at the classical rates in some important special cases.
We consider space-time continuous Galerkin methods with mesh modification in time for semilinear second order hyperbolic equations. We show a priori estimates in the energy norm without mesh conditions. Under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates. Estimates of the jump in the Riesz(More)
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(More)