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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)

- Xiaobing Feng, Ohannes A. Karakashian
- SIAM J. Numerical Analysis
- 2001

The convergence of the discontinuous Galerkin method for the nonlinear (cubic) Schrödinger equation is analyzed in this paper. We show the existence of the resulting approximations and prove optimal order error estimates in L ∞ (L 2). These estimates are valid under weak restrictions on the space-time mesh.

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)

- Xiaobing Feng, Ohannes A. Karakashian
- Math. Comput.
- 2007

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 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)

- Xiaobing Feng, Ohannes A. Karakashian
- J. Sci. Comput.
- 2005

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)

We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear evolution equations and extend thus recent results concerning the discretization of nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit–explicit multistep schemes as well as the combination… (More)