A Primal-Dual Discontinuous Galerkin Finite Element Method for Ill-Posed Elliptic Cauchy Problems

被引：0
作者：
Chen, Yanli [1 ]
Zhang, Tie [1 ,2 ]
Sheng, Ying [1 ]
机构：
[1] Northeastern Univ, Dept Math, Shenyang 110004, Liaoning, Peoples R China
[2] Northeastern Univ, State Key Lab Synthet Automat Proc Ind, Shenyang 110004, Liaoning, Peoples R China
来源：
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS | 2024年 / 16卷 / 04期
基金：
欧盟地平线“2020”;
关键词：
The ill-posed elliptic problem; discontinuous Galerkin method; primal-dual scheme; optimal error estimate; QUASI-REVERSIBILITY; DATA ASSIMILATION; INVERSE PROBLEMS; SOLVE; PART;
D O I：
10.4208/aamm.OA-2022-0108
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
We present a primal -dual discontinuous Galerkin finite element method for a type of ill-posed elliptic Cauchy problem. It is shown that the discrete problem attains a unique solution, if the solution of the ill-posed elliptic Cauchy problems is unique. An optimal error estimate is obtained in a H 1 -like norm. Numerical experiments are provided to demonstrate the efficiency of the proposed method.
引用
收藏
页码：860 / 877
页数：18
相关论文
共 28 条
[1]   STABLE DETERMINATION OF A CRACK FROM BOUNDARY MEASUREMENTS [J].
ALESSANDRINI, G .
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, 1993, 123 :497-516
[2]   Solving Cauchy problems by minimizing an energy-like functional [J].
Andrieux, S ;
Baranger, TN ;
Ben Abda, A .
INVERSE PROBLEMS, 2006, 22 (01) :115-133
[3]   AN INTERIOR PENALTY FINITE-ELEMENT METHOD WITH DISCONTINUOUS ELEMENTS [J].
ARNOLD, DN .
SIAM JOURNAL ON NUMERICAL ANALYSIS, 1982, 19 (04) :742-760
[4]   On Cauchy's problem:: II.: Completion, regularization and approximation [J].
Azaiez, Mejdi ;
Ben Belgacem, Faker ;
El Fekih, Henda .
INVERSE PROBLEMS, 2006, 22 (04) :1307-1336
[5]   Why is the Cauchy problem severely ill-posed? [J].
Ben Belgacem, Faker .
INVERSE PROBLEMS, 2007, 23 (02) :823-836
[6]   Local Convergence of the Lavrentiev Method for the Cauchy Problem via a Carleman Inequality [J].
Ben Belgacem, Faker ;
Duc Thang Du ;
Jelassi, Faten .
JOURNAL OF SCIENTIFIC COMPUTING, 2012, 53 (02) :320-341
[7]   Extended-domain-Lavrentiev's regularization for the Cauchy problem [J].
Ben Belgacem, Faker ;
Du, Duc Thang ;
Jelassi, Faten .
INVERSE PROBLEMS, 2011, 27 (04)
[8]  
Blum J.., 1989, Numerical Simulation and Optimal Control in Plasma Physics
[9]   Electrical impedance tomography [J].
Borcea, L .
INVERSE PROBLEMS, 2002, 18 (06) :R99-R136
[10]   A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation [J].
Bourgeois, L .
INVERSE PROBLEMS, 2005, 21 (03) :1087-1104
123