  A new numerical method is devised and analyzed for a type of ill-posed
elliptic Cauchy problems by using the primal-dual weak Galerkin finite element
method. This new primal-dual weak Galerkin algorithm is robust and efficient in
the sense that the system arising from the scheme is symmetric, well-posed, and
is satisfied by the exact solution (if it exists). An error estimate of optimal
order is established for the corresponding numerical solutions in a scaled
residual norm. In addition, a mathematical convergence is established in a weak
$L^2$ topology for the new numerical method. Numerical results are reported to
demonstrate the efficiency of the primal-dual weak Galerkin method as well as
the accuracy of the numerical approximations.
