Open AccessNumerical method for recovering the piecewise constant right-hand side function of an elliptic equation from a boundary overdetermination data
Author(s) -
D Kh Ivanov,
Aleksandr E. Kolesov,
Petr N. Vabishchevich
Publication year - 2019
Publication title -
journal of physics. conference series
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.21
H-Index - 85
eISSN - 1742-6596
pISSN - 1742-6588
DOI - 10.1088/1742-6596/1392/1/012081
Subject(s) - mathematics , piecewise , mathematical analysis , boundary value problem , inverse problem , boundary (topology) , heaviside step function , free boundary problem , constant (computer programming) , elliptic boundary value problem , dirichlet boundary condition , computer science , programming language
An elliptic problem in an open set Ω is considered with Dirichlet boundary condition on piecewise smooth boundary ∂Ω. The inverse problem is to recover piecewise-constant source term, which means an identification function of an unknown subset D ⊂ Ω. An additional information is taken as Neumann boundary condition, which leads us to consider the inverse problem under Cauchy boundary data. In this work a new computational algorithm for recovery the unknown source term is proposed. The main idea is that the identification function of subset D is replaced by a Heaviside function over the solution of an auxiliary elliptic equation. Then we formulate a minimization problem of a residual for overdetermination data, when a control is taken as the right-hand side of the auxiliary equation. Numerical implementation is based on the finite element method applying the open-source computing platform FEniCS and its package dolfin-adjoint. The capabilities of the given computational algorithm are shown by results of numerical solutions of 2D test problems.