Nonlinear discontinuous Petrov–Galerkin methods

Discontinuous Galerkin methods for nonlinear elasticity

Superconvergent discontinuous Galerkin methods for nonlinear elliptic equations

Based on the analysis of Cockburn et. al. [Math. Comp. 78 (2009), pp. 1-24] for a selfadjoint linear elliptic equation, we first discuss superconvergence results for nonselfadjoint linear elliptic problems using discontinuous Galerkin methods. Further, we have extended our analysis to derive superconvergence results for quasilinear elliptic problems. When piecewise polynomials of degree k ≥ 1 a...

Local discontinuous Galerkin methods for nonlinear dispersive equations

We develop local discontinuous Galerkin (DG) methods for solving nonlinear dispersive partial differential equations that have compactly supported traveling waves solutions, the so-called ‘‘compactons’’. The schemes we present extend the previous works of Yan and Shu on approximating solutions for linear dispersive equations and for certain KdV-type equations. We present two classes of DG metho...

Adaptive Discontinuous Galerkin Methods for Nonlinear Diffusion-Convection-Reaction Equations

In this work, we apply the adaptive discontinuous Galerkin (DGAFEM) method to the convection dominated non-linear, quasi-stationary diffusion convection reaction equations. We propose an efficient preconditioner using a matrix reordering scheme to solve the sparse linear systems iteratively arising from the discretized non-linear equations. Numerical examples demonstrate effectiveness of the DG...

Local discontinuous Galerkin methods for nonlinear Schrödinger equations

In this paper we develop a local discontinuous Galerkin method to solve the generalized nonlinear Schrödinger equation and the coupled nonlinear Schrödinger equation. L stability of the schemes are obtained for both of these nonlinear equations. Numerical examples are shown to demonstrate the accuracy and capability of these methods. 2004 Elsevier Inc. All rights reserved. MSC: 65M60; 35Q55