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

TL;DR

This paper introduces a primal-dual weak Galerkin finite element method for linear convection equations in non-divergence form, providing a symmetric system with optimal error estimates and demonstrating high accuracy through numerical tests.

Contribution

The paper develops a novel PD-WG finite element method that handles non-divergence form convection equations with proven optimal error bounds and symmetric discretization.

Findings

Optimal error estimates in various norms

Symmetric discrete system involving primal and dual variables

Numerical results confirm accuracy and effectiveness

Abstract

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.