A mixed finite elements approximation of inverse source problems for the wave equation with variable coefficients using observability

TL;DR

This paper introduces a mixed finite element method to approximate inverse source problems for the wave equation with variable coefficients, leveraging a novel boundary observability property for stability and convergence.

Contribution

The paper proposes a new mixed finite element approach for inverse wave problems with variable coefficients, establishing stability and convergence through a uniform boundary observability property.

Findings

Method achieves stable approximations of the inverse source problem.

Convergence is proven based on the new observability property.

Numerical results demonstrate the effectiveness of the approach.

Abstract

We consider an inverse problem for the linear one-dimensional wave equation with variable coefficients consisting in determining an unknown source term from a boundary observation. A method to obtain approximations of this inverse problem using a space discretization based on a mixed finite element method is proposed and analyzed. Its stability and convergence relay on a new uniform boundary observability property with respect to the discretization parameter.