# A primal-dual finite element method for first-order transport problems

## TL;DR

This paper introduces a primal-dual weak Galerkin finite element method for first-order transport problems, providing a stable, mass-conserving, and accurate numerical solution under low regularity conditions.

## Contribution

It develops a novel PD-WG finite element method that handles low regularity solutions and ensures local mass conservation for first-order transport equations.

## Key findings

- Achieves optimal error estimates in various norms.
- Provides numerical results demonstrating accuracy and stability.
- Ensures local mass conservation on each element.

## Abstract

This article devises a new numerical method for first-order transport problems by using the primal-dual weak Galerkin (PD-WG) finite element method recently developed in scientific computing. The PD-WG method is based on a variational formulation of the modeling equation for which the differential operator is applied to the test function so that low regularity for the exact solution of the original equation is sufficient for computation. The PD-WG finite element method indeed yields a symmetric system involving both the original equation for the primal variable and its dual for the dual variable (also known as Lagrangian multiplier). For the linear transport problem, it is shown that the PD-WG method offers numerical solutions that conserve mass locally on each element. Optimal order error estimates in various norms are derived for the numerical solutions arising from the PD-WG method with weak regularity assumptions on the modelling equations. A variety of numerical results are presented to demonstrate the accuracy and stability of the new method.