In this paper we present a new residual-based reliable a posteriori error estimator for the local discontinuous Galerkin approximations of linear and nonlinear diffusion problems in polygonal regions of R 2. Our analysis, which applies to convex and nonconvex domains, is based on Helmholtz decompositions of the error and a suitable auxiliary polynomial function interpolating the Dirichlet datum. Several examples confirming the reliability of the estimator and providing numerical evidences for its efficiency are given. Furthermore, the associated adaptive method, which considers meshes with and without hanging nodes, is shown to be much more efficient than a uniform refinement to compute the discrete solutions. In particular, the experiments illustrate the ability of the adaptive algorithm to localize the singularities of each problem.
Bibliographical noteFunding Information:
This research was partially supported by CONICYT-Chile through the FONDAP Program in Applied Mathematics, and by the Dirección de Investigación de la Universidad de Concepción, through the Advanced Research Groups Program. Part of this work was completed during the visit of the first author to the School of Mathematics, University of Minnesota, during February–March, 2003, funded by the MECESUP project UCO9907.
- Local discontinuous Galerkin methods
- Residual-based error estimates