A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation

Dinh Nho Hào; Le Thi Thu Giang; Sergey Kabanikhin; Maxim Shishlenin

doi:10.1515/jiip-2018-0060

A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation

Dinh Nho Hào, Le Thi Thu Giang, Sergey Kabanikhin, Maxim Shishlenin

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

Original language	English
Pages (from-to)	835-857
Number of pages	23
Journal	Journal of Inverse and Ill-Posed Problems
Volume	26
Issue number	6
DOIs	https://doi.org/10.1515/jiip-2018-0060
Publication status	Published - 1 Dec 2018

Keywords

Cauchy problem
elliptic equation
finite difference scheme
ill-posed problems
non-local boundary value problems
regularization
very weak solution
INVERSE

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10.1515/jiip-2018-0060

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abstract = "We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.",

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AU - Hào, Dinh Nho

AU - Thu Giang, Le Thi

AU - Kabanikhin, Sergey

AU - Shishlenin, Maxim

PY - 2018/12/1

Y1 - 2018/12/1

N2 - We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

AB - We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

KW - Cauchy problem

KW - elliptic equation

KW - finite difference scheme

KW - ill-posed problems

KW - non-local boundary value problems

KW - regularization

KW - very weak solution

KW - INVERSE

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DO - 10.1515/jiip-2018-0060

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SP - 835

EP - 857

JO - Journal of Inverse and Ill-Posed Problems

JF - Journal of Inverse and Ill-Posed Problems

SN - 0928-0219

IS - 6

ER -

A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation

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Keywords

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