  A new primal-dual weak Galerkin (PD-WG) finite element method was developed
and analyzed in this article for first-order linear convection equations in
non-divergence form. The PD-WG method results in a symmetric discrete system
involving not only the original equation for the primal variable, but also the
dual/adjoint equation for the dual variable (also known as Lagrangian
multiplier). Optimal order of error estimates in various discrete Sobolev norms
are derived for the numerical solutions arising from the PD-WG scheme.
Numerical results are produced and reported to illustrate the accuracy and
effectiveness of the new PD-WG method.
