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

Chunmei Wang, Ludmil Zikatanov

TL;DR

This work develops a primal-dual weak Galerkin method for convection-diffusion equations under low regularity, coupling a primal variable with a dual/adjoint to enhance stability on general polyhedral meshes. The PDWG formulation uses discrete weak differential operators and a stabilizing bilinear form to achieve a symmetric, well-posed discrete problem, with rigorous a priori error estimates in $H^{\epsilon}$ for $\epsilon\in[0,1/2)$. Under mild regularity and coefficient assumptions, the method attains optimal convergence, and numerical tests confirm theoretical rates even in convection-dominated regimes, often with $\gamma=0$ when $s=k-1$. The approach enables accurate approximations on polyshedral meshes and accommodates low regularity solutions, with potential for fast solvers and broader PDE classes in future work.

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 $H^ε$-norm for $ε\in [0,\frac12)$. A series of numerical tests that validate the theory and are presented as well.

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

TL;DR

This work develops a primal-dual weak Galerkin method for convection-diffusion equations under low regularity, coupling a primal variable with a dual/adjoint to enhance stability on general polyhedral meshes. The PDWG formulation uses discrete weak differential operators and a stabilizing bilinear form to achieve a symmetric, well-posed discrete problem, with rigorous a priori error estimates in for . Under mild regularity and coefficient assumptions, the method attains optimal convergence, and numerical tests confirm theoretical rates even in convection-dominated regimes, often with when . The approach enables accurate approximations on polyshedral meshes and accommodates low regularity solutions, with potential for fast solvers and broader PDE classes in future work.

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 -norm for . A series of numerical tests that validate the theory and are presented as well.

Paper Structure

This paper contains 9 sections, 8 theorems, 83 equations, 5 figures, 20 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Discrete Weak Differential Operators
  4. Primal-Dual Weak Galerkin Formulation
  5. Solution Existence, Uniqueness, and Stability
  6. Error Equations
  7. Error Estimates
  8. Numerical Results
  9. Conclusions

Key Result

Lemma 4.1

\newlabelLemma5.1wy3655 The $L^2$ projection operators $Q_h$, ${\@fontswitch{}{\mathcal{}} Q}^{k-1}_h$ and ${\@fontswitch{}{\mathcal{}} Q}^{s}_h$ satisfy the following commutative diagram:

Figures (5)

  • Figure 8.1: Surface plot of $u_h$ on the unit square domain $\Omega_1$: left for the $C^0$-$P_2(T)/P_1({\partial T})/P_1(T)$ element, right for the $C^0$- $P_2(T)/P_1({\partial T})/P_0(T)$ element.
  • Figure 8.2: Surface plots for the primal variable $u_h$ on the unit square domain $\Omega_1$ with the $C^0$- $P_2(T)/P_1({\partial T})/P_0(T)$ element: left for the diffusion tensor $a=[10^{-1}, 0; 0, 10^{-1}]$, middle for the diffusion tensor $a=[10^{-3}, 0; 0, 10^{-3}]$, right for the diffusion tensor $a=[10^{-6}, 0; 0, 10^{-6}]$.
  • Figure 8.3: Contour plots for primal variable $u_h$: load function $f=0$ (left), load function $f=1$ (right). The square domain $\Omega_3$ and the $C^0$- $P_2(T)/P_1({\partial T})/P_1(T)$ element.
  • Figure 8.4: Contour plots for primal variable $u_h$: load function $f=0$ (left), load function $f=1$ (right). Cracked square domain $\Omega_4$ and the $C^0$- $P_2(T)/P_1({\partial T})/P_1(T)$ element.
  • Figure 8.5: Contour plots for the primal variable $u_h$: load function $f=0$ (left), load function $f=1$ (right). L-shaped domain $\Omega_5$ with the $C^0$- $P_2(T)/P_1({\partial T})/P_1(T)$ element.

Theorems & Definitions (16)

  • Remark 2.1
  • Lemma 4.1
  • proof
  • Lemma 5.1
  • proof
  • Remark 5.1
  • Theorem 5.2
  • proof
  • Lemma 6.1
  • proof
  • ...and 6 more