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Discontinuous Galerkin error estimation for hyperbolic problems on unstructured triangular meshes |
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| Authors: | Mahboub Baccouch Slimane Adjerid |
| Affiliation: | 1. Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182, United States;2. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, United States;1. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, PR China;2. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, Canada;1. Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing, 100124, China;2. Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA;3. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, 411105, China;1. Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada;2. School of Mathematical Sciences, Ocean University of China, Qingdao, Shandong, 266100, China;3. School of Mathematics, Shandong University, Jinan, Shandong, 250100, China |
| Abstract: | We extend the error analysis of Adjerid and Baccouch [1], [2] for the discontinuous Galerkin discretization error to variable-coefficient linear hyperbolic problems as well as nonlinear hyperbolic problems on unstructured meshes. We further extend this analysis to transient hyperbolic problems and prove that the local superconvergence results presented in [1] still hold for both steady and transient variable-coefficient linear and nonlinear problems. This local error analysis allows us to construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on each element of general unstructured meshes. We illustrate the superconvergence and the efficiency of our a posteriori error estimates by showing computational results for several linear and nonlinear numerical examples. |
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