Low Regularity Primal-Dual Weak Galerkin Finite Element Methods for Convection-Diffusion Equations

TL;DR

This paper introduces a stable and convergent primal-dual weak Galerkin finite element method tailored for convection-diffusion equations with low regularity, providing theoretical error estimates and numerical validation.

Contribution

The paper develops a novel PDWG finite element method for convection-diffusion problems under low regularity, including stability analysis and a priori error estimates.

Findings

Method is stable and convergent

Provides error estimates in $H^{oldsymbol{ extepsilon}}$-norm

Numerical tests validate theoretical results

Abstract

We propose a numerical method for convection-diffusion problems under low regularity assumptions. We derive the method and analyze it using 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 $H^{\epsilon}$-norm for $\epsilon\in [0,\frac12)$. A series of numerical tests that validate the theory and are presented as well.