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An ultraweak-local discontinuous Galerkin method for PDEs with high order spatial derivatives


Authors: Qi Tao, Yan Xu and Chi-Wang Shu
Journal: Math. Comp. 89 (2020), 2753-2783
MSC (2010): Primary 65M60; Secondary 35G25
DOI: https://doi.org/10.1090/mcom/3562
Published electronically: August 4, 2020
MathSciNet review: 4136546
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Abstract: In this paper, we develop a new discontinuous Galerkin method for solving several types of partial differential equations (PDEs) with high order spatial derivatives. We combine the advantages of a local discontinuous Galerkin (LDG) method and the ultraweak discontinuous Galerkin (UWDG) method. First, we rewrite the PDEs with high order spatial derivatives into a lower order system, then apply the UWDG method to the system. We first consider the fourth order and fifth order nonlinear PDEs in one space dimension, and then extend our method to general high order problems and two space dimensions. The main advantage of our method over the LDG method is that we have introduced fewer auxiliary variables, thereby reducing memory and computational costs. The main advantage of our method over the UWDG method is that no internal penalty terms are necessary in order to ensure stability for both even and odd order PDEs. We prove the stability of our method in the general nonlinear case and provide optimal error estimates for linear PDEs for the solution itself as well as for the auxiliary variables approximating its derivatives. A key ingredient in the proof of the error estimates is the construction of the relationship between the derivative and the element interface jump of the numerical solution and the auxiliary variable solution of the solution derivative. With this relationship, we can then use the discrete Sobolev and Poincaré inequalities to obtain the optimal error estimates. The theoretical findings are confirmed by numerical experiments.


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Additional Information

Qi Tao
Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
Email: taoq@mail.ustc.edu.cn

Yan Xu
Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
Email: yxu@ustc.edu.cn

Chi-Wang Shu
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
MR Author ID: 242268
Email: chi-wang_shu@brown.edu

Keywords: Discontinuous Galerkin (DG) method; high order equation; error estimate; discrete Sobolev and Poincaré inequalities.
Received by editor(s): September 2, 2019
Received by editor(s) in revised form: February 21, 2020
Published electronically: August 4, 2020
Additional Notes: The research of the first author was supported by the China Scholarship Council.
The research of the second author was supported by National Numerical Windtunnel grants NNW2019ZT4-B08, Science Challenge Project TZZT2019-A2.3, and NSFC grants 11722112.
The second author is the corresponding author.
The research of the third author was supported by NSF grant DMS-1719410.
Article copyright: © Copyright 2020 American Mathematical Society