A Primal-Dual Discontinuous Galerkin Finite Element Method for Ill-Posed Elliptic Cauchy Problems

We present a primal-dual discontinuous Galerkin finite element method for a type of ill-posed elliptic Cauchy problem. It is shown that the discrete problem attains a unique solution, if the solution of the ill-posed elliptic Cauchy problems is unique. An optimal error estimate is obtained in a H1-like norm. Numerical experiments are provided to demonstrate the efficiency of the proposed method.