An $L^p$- Primal-Dual Weak Galerkin Method for Convection-Diffusion Equations

TL;DR

This paper introduces an $L^p$-primal-dual weak Galerkin method for convection-diffusion equations, providing theoretical error estimates and demonstrating its effectiveness through numerical experiments.

Contribution

The paper develops a new $L^p$-PDWG method for convection-diffusion equations, including existence, uniqueness, and optimal error estimates in various norms.

Findings

Optimal-order error estimates in $L^q$-norm for primal variables

Error bounds for dual variable in $W^{m,p}$ norm

Numerical results confirm efficiency and accuracy

Abstract

In this article, the authors present a new $L^p$- primal-dual weak Galerkin method ($L^p$-PDWG) for convection-diffusion equations with $p>1$. The existence and uniqueness of the numerical solution is discussed, and an optimal-order error estimate is derived in the $L^q$-norm for the primal variable, where $\frac 1p+\frac 1q=1$. Furthermore, error estimates are established for the numerical approximation of the dual variable in the standard $W^{m,p}$ norm, $0\le m\le 2$. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed $L^p$-PDWG method.