Computation of interior elastic transmission eigenvalues using a conforming finite element and the secant method

The interior elastic transmission eigenvalue problem, arising from the inverse scattering theory of non-homogeneous elastic media, is nonlinear, non-self-adjoint and of fourth order. This paper proposes a numerical method to compute real elastic transmission eigenvalues. To avoid treating the non-self-adjoint operator, an auxiliary nonlinear function is constructed. The values of the function are generalized eigenvalues of a series of self-adjoint fourth order problems. The roots of the function are the transmission eigenvalues. The self-adjoint fourth order problems are computed using the H²-conforming Argyris element. The secant method is employed to search the roots of the nonlinear function. The convergence of the proposed method is proved.