Optimal error estimates of ultra-weak discontinuous Galerkin methods with generalized numerical fluxes for multi-dimensional convection-diffusion and biharmonic equations
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- by Yuan Chen and Yulong Xing;
- Math. Comp. 93 (2024), 2135-2183
- DOI: https://doi.org/10.1090/mcom/3927
- Published electronically: December 6, 2023
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Abstract:
In this paper, we study ultra-weak discontinuous Galerkin methods with generalized numerical fluxes for multi-dimensional high order partial differential equations on both unstructured simplex and Cartesian meshes. The equations we consider as examples are the nonlinear convection-diffusion equation and the biharmonic equation. Optimal error estimates are obtained for both equations under certain conditions, and the key step is to carefully design global projections to eliminate numerical errors on the cell interface terms of ultra-weak schemes on general dimensions. The well-posedness and approximation capability of these global projections are obtained for arbitrary order polynomial space based on a wide class of generalized numerical fluxes on regular meshes. These projections can serve as general analytical tools to be naturally applied to a wide class of high order equations. Numerical experiments are conducted to demonstrate these theoretical results.References
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Bibliographic Information
- Yuan Chen
- Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
- Email: chen.11050@osu.edu
- Yulong Xing
- Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
- MR Author ID: 761305
- Email: xing.205@osu.edu
- Received by editor(s): March 3, 2023
- Received by editor(s) in revised form: September 12, 2023, September 21, 2023, and October 11, 2023
- Published electronically: December 6, 2023
- Additional Notes: The work of the second author was partially supported by the NSF grant DMS-1753581 and DMS-2309590.
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 2135-2183
- MSC (2020): Primary 65M12, 65M15, 65M60; Secondary 35G25
- DOI: https://doi.org/10.1090/mcom/3927
- MathSciNet review: 4759372