Computation of the interior transmission eigenvalues for elastic scattering in an inhomogeneous medium containing an obstacle

In this work, we study the interior transmission eigenvalues for elastic scattering in an inhomogeneous medium containing an obstacle. This problem is related to the reconstruction of the support of the inhomogeneity without the knowledge of the embedded obstacle by the far-field data or the invisibility cloaking of an obstacle. Our goal is to provide an efficient numerical algorithm to compute as many positive interior transmission eigenvalues as possible. We consider two cases of medium jumps: Case 1, where $$\mathbf{C }_0=\mathbf{C }_1$$ , $$\rho _0\ne \rho _1$$ , and Case 2, where $$\mathbf{C }_0\ne \mathbf{C }_1$$ , $$\rho _0=\rho _1$$ with either Dirichlet or Neumann boundary conditions on the boundary of the embedded obstacle. The partial differential equation problem is reduced to a generalized eigenvalue problem (GEP) for matrices by the finite element method. We will apply the Jacobi–Davidson (JD) algorithm to solve the GEP. Case 1 requires special attention because of the large number of zero eigenvalues, which depends on the discretization size. To compute the positive eigenvalues effectively, it is necessary to deflate the zeros to infinity at the beginning of the algorithm.