  The authors propose and analyze a well-posed numerical scheme for a type of
ill-posed elliptic Cauchy problem by using a constrained minimization approach
combined with the weak Galerkin finite element method. The resulting
Euler-Lagrange formulation yields a system of equations involving the original
equation for the primal variable and its adjoint for the dual variable, and is
thus an example of the primal-dual weak Galerkin finite element method. This
new primal-dual weak Galerkin algorithm is consistent in the sense that the
system is symmetric, well-posed, and is satisfied by the exact solution. A
certain stability and error estimates were derived in discrete Sobolev norms,
including one in a weak $L^2$ topology. Some numerical results are reported to
illustrate and validate the theory developed in the paper.
