A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Difusion Problems

doi:10.1007/s42967-019-00024-x

Communications on Applied Mathematics and Computation ›› 2020, Vol. 2 ›› Issue (1): 31-55.doi: 10.1007/s42967-019-00024-x

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A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Difusion Problems

Qiang Du¹, Lili Ju², Jianfang Lu³, Xiaochuan Tian⁴

1 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA;
2 Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA;
3 South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou 510631, China;
4 Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA

收稿日期:2018-09-30 修回日期:2019-02-26 出版日期:2020-03-20 发布日期:2020-02-19
通讯作者: Jianfang Lu, Qiang Du, Lili Ju, Xiaochuan Tian E-mail:jfu@m.scnu.edu.cn;qd2125@columbia.edu;ju@math.sc.edu;xtian@math.utexas.edu
基金资助:
Q. Du's research is partially supported by US National Science Foundation Grant DMS-1719699, US AFOSR MURI Center for Material Failure Prediction Through Peridynamics, and US Army Research Ofce MURI Grant W911NF-15-1-0562. L. Ju's research is partially supported by US National Science Foundation Grant DMS-1818438. J. Lu's research is partially supported by Postdoctoral Science Foundation of China Grant 2017M610749. X. Tian's research is partially supported by US National Science Foundation Grant DMS-1819233.

A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Difusion Problems

Qiang Du¹, Lili Ju², Jianfang Lu³, Xiaochuan Tian⁴

1 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA;
2 Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA;
3 South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou 510631, China;
4 Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA

Received:2018-09-30 Revised:2019-02-26 Online:2020-03-20 Published:2020-02-19
Contact: Jianfang Lu, Qiang Du, Lili Ju, Xiaochuan Tian E-mail:jfu@m.scnu.edu.cn;qd2125@columbia.edu;ju@math.sc.edu;xtian@math.utexas.edu
Supported by:
Q. Du's research is partially supported by US National Science Foundation Grant DMS-1719699, US AFOSR MURI Center for Material Failure Prediction Through Peridynamics, and US Army Research Ofce MURI Grant W911NF-15-1-0562. L. Ju's research is partially supported by US National Science Foundation Grant DMS-1818438. J. Lu's research is partially supported by Postdoctoral Science Foundation of China Grant 2017M610749. X. Tian's research is partially supported by US National Science Foundation Grant DMS-1819233.

摘要/Abstract

摘要： There have been many theoretical studies and numerical investigations of nonlocal diffusion (ND) problems in recent years. In this paper, we propose and analyze a new discontinuous Galerkin method for solving one-dimensional steady-state and time-dependent ND problems, based on a formulation that directly penalizes the jumps across the element interfaces in the nonlocal sense. We show that the proposed discontinuous Galerkin scheme is stable and convergent. Moreover, the local limit of such DG scheme recovers classical DG scheme for the corresponding local difusion problem, which is a distinct feature of the new formulation and assures the asymptotic compatibility of the discretization. Numerical tests are also presented to demonstrate the efectiveness and the robustness of the proposed method.

关键词: Nonlocal difusion, Discontinuous Galerkin method, Interior penalty, Asymptotic compatibility, Strong stability preserving

Abstract: There have been many theoretical studies and numerical investigations of nonlocal diffusion (ND) problems in recent years. In this paper, we propose and analyze a new discontinuous Galerkin method for solving one-dimensional steady-state and time-dependent ND problems, based on a formulation that directly penalizes the jumps across the element interfaces in the nonlocal sense. We show that the proposed discontinuous Galerkin scheme is stable and convergent. Moreover, the local limit of such DG scheme recovers classical DG scheme for the corresponding local difusion problem, which is a distinct feature of the new formulation and assures the asymptotic compatibility of the discretization. Numerical tests are also presented to demonstrate the efectiveness and the robustness of the proposed method.

Key words: Nonlocal difusion, Discontinuous Galerkin method, Interior penalty, Asymptotic compatibility, Strong stability preserving

中图分类号:

Qiang Du, Lili Ju, Jianfang Lu, Xiaochuan Tian. A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Difusion Problems[J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 31-55.

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A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Difusion Problems

A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Difusion Problems

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