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

  • Dinh Nho Hào EMAIL logo , Le Thi Thu Giang , Sergey Kabanikhin and Maxim Shishlenin

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.

MSC 2010: 65N20; 65N60; 35J15

Funding statement: The work by Dinh Nho Hào was partially supported by Vietnam Academy of Science and Technology (VAST) under the Grant VAST.HTQT.NGA.09/17-18, the work by S. I. Kabanikhin and M. A. Shishlenin was partially supported by RFBR (Grant 17-51-540004, 16-29-15120, 16-01-00755). Part of this work has been done during Dinh Nho Hào’s stay at Vietnam Institute for Advanced Study in Mathematics.

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Received: 2018-07-04
Revised: 2018-09-13
Accepted: 2018-09-13
Published Online: 2018-10-11
Published in Print: 2018-12-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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