An $L^p$-primal-dual finite element method for first-order transport problems

TL;DR

This paper introduces an $L^p$-primal-dual weak Galerkin method for first-order transport problems, ensuring local mass conservation, with proven existence, uniqueness, and optimal error estimates, validated by numerical experiments.

Contribution

It develops a novel $L^p$-PDWG method for transport problems that guarantees local mass conservation and provides rigorous error analysis.

Findings

Method achieves optimal error estimates.

Numerical results confirm efficiency and accuracy.

Mass conservation is maintained locally on each element.

Abstract

A new $L^p$-primal-dual weak Galerkin method ($L^p$-PDWG) with $p>1$ is proposed for the first-order transport problems. The existence and uniqueness of the $L^p$-PDWG numerical solutions is established. In addition, the $L^p$-PDWG method offers a numerical solution which retains mass conservation locally on each element. An optimal order error estimate is established for the primal variable. A series of numerical results are presented to verify the efficiency and accuracy of the proposed $L^p$-PDWG scheme.