Adaptivity and Error Estimation for Discontinuous Galerkin Methods
We test the a posteriori error estimates of discontinuous Galerkin (DG) discretization errors (Adjerid and Baccouch, J. Sci. Comput. 33(1):75–113, 2007; Adjerid and Baccouch, J. Sci. Comput. 38(1):15–49, 2008; Adjerid and Baccouch Comput. Methods Appl. Mech. Eng. 200:162–177, 2011) for hyperbolic problems on adaptively refined unstructured triangular meshes. A local error analysis allows us to construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems on each element. The Taylor-expansion-based error analysis (Adjerid and Baccouch, J. Sci. Comput. 33(1):75–113, 2007; Adjerid and Baccouch, J. Sci. Comput. 38(1):15–49, 2008; Adjerid and Baccouch Comput. Methods Appl. Mech. Eng. 200:162–177, 2011) does not apply near discontinuities and shocks and lead to inaccurate estimates under uniform mesh refinement. Here, we present several computational results obtained from adaptive refinement computations that suggest that even in the presence of shocks our error estimates converge to the true error under adaptive mesh refinement. We also show the performance of several adaptive strategies for hyperbolic problems with discontinuous solutions.
KeywordsAdaptive discontinuous Galerkin method Hyperbolic problems A posteriori error estimation Unstructured meshes
The authors are grateful to Bryan Johnson (undergraduate student at the University of Nebraska at Omaha) for applying the adaptive algorithms to the contact problem to generate the results for Example 3.
The work of the Slimane Adjerid author was supported in part by NSF grant DMS-0809262. The work of the Mahboub Baccouch author was supported by the NASA Nebraska Space Grant Program and UCRCA at the University of Nebraska at Omaha.
- S. Adjerid and T. Weinhart. Asymptotically exact discontinuous Galerkin error estimates for linear symmetric hyperbolic systems. Applied Numerical Mathematics, in press, 2011.Google Scholar
- B. Cockburn, G. E. Karniadakis, and C. W. Shu, editors. Discontinuous Galerkin Methods Theory, Computation and Applications, Lectures Notes in Computational Science and Engineering, volume 11. Springer, Berlin, 2000.Google Scholar
- K. Ericksson and C. Johnson. Adaptive finite element methods for parabolic problems I: A linear model problem. SIAM Journal on Numerical Analysis, 28: 12–23, 1991.Google Scholar
- W. H. Reed and T. R. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, 1973.Google Scholar