# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## An ultraweak-local discontinuous Galerkin method for PDEs with high order spatial derivativesHTML articles powered by AMS MathViewer

by Qi Tao, Yan Xu and Chi-Wang Shu
Math. Comp. 89 (2020), 2753-2783 Request permission

## 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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• 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
• 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.