Determination of the right-hand side in elliptic equations

ABSTRACT

The problem of determining a term in the right-hand side of elliptic equations from an observation on a part of the boundary is investigated. The inverse problem is formulated as an operator equation and then stabilized by Tikhonov regularization method. The regularized problem is discretized based on Hinze's variational discretization concept and the regularization parameter is chosen guaranteeing that when noise level and the discretization mesh size tend to zero, the solution of the discretized regularized problem converges to the $f^{\ast }$-minimum norm solution of the continuous inverse problem. Some numerical examples are presented for illustrating the performance of the proposed method.

KEYWORDS:

• Inverse source problem
• elliptic equations
• ill-posedness
• regularization
• finite element method

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35J15
• 35R30
• 65J20
• 65N21

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is partially supported by Vietnam Institute for Advanced Study in Mathematics (VIASM), and the International Center for Research and Postgraduate Training in Mathematics, Institute of Mathematics, VAST under the grant ICRTM02-2020.03 (Dinh Nho Hào and Le Thi Thu Giang).