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

A Primal-dual weak Galerkin finite element method for linear convection equations in non-divergence form