Superconvergence of UWLDG Method for One-Dimensional Linear Sixth-Order Equations

doi:10.1007/s42967-024-00390-1

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (2): 771-795.doi: 10.1007/s42967-024-00390-1

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Superconvergence of UWLDG Method for One-Dimensional Linear Sixth-Order Equations

Mengfei Wang¹, Yan Xu¹

School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, China

收稿日期:2023-09-23 修回日期:2024-01-29 接受日期:2024-02-17 出版日期:2025-06-20 发布日期:2025-04-21
通讯作者: Yan Xu,yxu@ustc.edu.cn;Mengfei Wang,wmf231306@mail.ustc.edu.cn E-mail:yxu@ustc.edu.cn;wmf231306@mail.ustc.edu.cn

Superconvergence of UWLDG Method for One-Dimensional Linear Sixth-Order Equations

Mengfei Wang¹, Yan Xu¹

School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, China

Received:2023-09-23 Revised:2024-01-29 Accepted:2024-02-17 Online:2025-06-20 Published:2025-04-21

摘要/Abstract

摘要： This paper concerns the superconvergence property of the ultraweak-local discontinuous Galerkin (UWLDG) method for one-dimensional linear sixth-order equations. The crucial technique is the construction of a special projection. We will discuss in three different situations according to the remainder of k, the highest degree of polynomials in the function space, divided by 3. We can prove the (2k - 1) th-order superconvergence for the cell averages when k ≡ 0 or 2 (mod 3). But if k ≡1 (mod 3), we can only prove a (2k - 2) th-order superconvergence. The same superconvergence orders can also be gained for the errors of numerical fluxes. We will also prove the superconvergence of order k + 2 at some special quadrature points. Some numerical examples are given at the end of this paper.

关键词: Linear sixth order equations, Ultraweak-local discontinuous Galerkin (UWLDG) method, Superconvergence

Abstract: This paper concerns the superconvergence property of the ultraweak-local discontinuous Galerkin (UWLDG) method for one-dimensional linear sixth-order equations. The crucial technique is the construction of a special projection. We will discuss in three different situations according to the remainder of k, the highest degree of polynomials in the function space, divided by 3. We can prove the (2k - 1) th-order superconvergence for the cell averages when k ≡ 0 or 2 (mod 3). But if k ≡1 (mod 3), we can only prove a (2k - 2) th-order superconvergence. The same superconvergence orders can also be gained for the errors of numerical fluxes. We will also prove the superconvergence of order k + 2 at some special quadrature points. Some numerical examples are given at the end of this paper.

Key words: Linear sixth order equations, Ultraweak-local discontinuous Galerkin (UWLDG) method, Superconvergence

中图分类号:

Mengfei Wang, Yan Xu. Superconvergence of UWLDG Method for One-Dimensional Linear Sixth-Order Equations[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 771-795.

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Superconvergence of UWLDG Method for One-Dimensional Linear Sixth-Order Equations

Superconvergence of UWLDG Method for One-Dimensional Linear Sixth-Order Equations

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