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.