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

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 properties relay on a new uniform boundary observability inequality with respect to the discretization parameter. This fundamental uniformity property of the observation is not verified in the case of more usual discretization schemes, such as centered finite differences or classical finite elements. For the mixed finite elements it is proved by combining lateral energy estimates with Fourier techniques.