Low regularity primal–dual weak Galerkin finite element methods for convection–diffusion equations

Abstract

We consider finite element discretizations for convection–diffusion problems under low regularity assumptions. The derivation and analysis use the primal–dual weak Galerkin (PDWG) finite element framework. The Euler–Lagrange formulation resulting from the PDWG scheme yields a system of equations involving not only the equation for the primal variable but also its adjoint for the dual variable. We show that the proposed PDWG method is stable and convergent. We also derive a priori error estimates for the primal variable in the Hϵ-norm for ϵ∈[0,12). A series of numerical tests that validate the theory is presented as well.