Local Discontinuous Galerkin Methods with Novel Basis for Fractional Difusion Equations with Non-smooth Solutions

doi:10.1007/s42967-020-00104-3

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (1): 227-249.doi: 10.1007/s42967-020-00104-3

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Local Discontinuous Galerkin Methods with Novel Basis for Fractional Difusion Equations with Non-smooth Solutions

Liyao Lyu^1,2, Zheng Chen³

1 School of Mathematical Sciences, Soochow University, Suzhou 215006, Jiangsu Province, China;
2 Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA;
3 Department of Mathematics, University of Massachusetts, Dartmouth, MA 02747, USA

收稿日期:2020-08-28 修回日期:2020-11-11 出版日期:2022-03-20 发布日期:2022-03-01
通讯作者: Zheng Chen, Liyao Lyu E-mail:zchen2@umassd.edu;lyuliyao@msu.edu
基金资助:
We acknowledge the fnancial support from the Center for Scientifc Computing and Visualization Research at the University of Massachusetts Dartmouth for Mr. Liyao Lyu's research internship. This work was supported, in part, by a grant from the College of Arts and Sciences at the University of Massachusetts Dartmouth.

Local Discontinuous Galerkin Methods with Novel Basis for Fractional Difusion Equations with Non-smooth Solutions

Liyao Lyu^1,2, Zheng Chen³

1 School of Mathematical Sciences, Soochow University, Suzhou 215006, Jiangsu Province, China;
2 Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA;
3 Department of Mathematics, University of Massachusetts, Dartmouth, MA 02747, USA

Received:2020-08-28 Revised:2020-11-11 Online:2022-03-20 Published:2022-03-01
Contact: Zheng Chen, Liyao Lyu E-mail:zchen2@umassd.edu;lyuliyao@msu.edu
Supported by:
We acknowledge the fnancial support from the Center for Scientifc Computing and Visualization Research at the University of Massachusetts Dartmouth for Mr. Liyao Lyu's research internship. This work was supported, in part, by a grant from the College of Arts and Sciences at the University of Massachusetts Dartmouth.

摘要/Abstract

摘要： In this paper, we develop novel local discontinuous Galerkin (LDG) methods for fractional difusion equations with non-smooth solutions. We consider such problems, for which the solutions are not smooth at boundary, and therefore the traditional LDG methods with piecewise polynomial solutions sufer accuracy degeneracy. The novel LDG methods utilize a solution information enriched basis, simulate the problem on a paired special mesh, and achieve optimal order of accuracy. We analyze the L² stability and optimal error estimate in L²-norm. Finally, numerical examples are presented for validating the theoretical conclusions.

关键词: Local discontinuous Galerkin methods, Fractional difusion equations, Nonsmooth solutions, Novel basis, Optimal order of accuracy

Abstract: In this paper, we develop novel local discontinuous Galerkin (LDG) methods for fractional difusion equations with non-smooth solutions. We consider such problems, for which the solutions are not smooth at boundary, and therefore the traditional LDG methods with piecewise polynomial solutions sufer accuracy degeneracy. The novel LDG methods utilize a solution information enriched basis, simulate the problem on a paired special mesh, and achieve optimal order of accuracy. We analyze the L² stability and optimal error estimate in L²-norm. Finally, numerical examples are presented for validating the theoretical conclusions.

Key words: Local discontinuous Galerkin methods, Fractional difusion equations, Nonsmooth solutions, Novel basis, Optimal order of accuracy

中图分类号:

65M60

Liyao Lyu, Zheng Chen. Local Discontinuous Galerkin Methods with Novel Basis for Fractional Difusion Equations with Non-smooth Solutions[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 227-249.

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Local Discontinuous Galerkin Methods with Novel Basis for Fractional Difusion Equations with Non-smooth Solutions

Local Discontinuous Galerkin Methods with Novel Basis for Fractional Difusion Equations with Non-smooth Solutions

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