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A robust solver for H(curl) convection-diffusion and its local Fourier analysis

Jindong Wang, Shuonan Wu

TL;DR

This work addresses robustly solving 2D $H({\rm curl})$ convection-diffusion problems by combining a simplex-averaged finite element (SAFE) discretization with exponential-fitting techniques. A downwind, lexicographic Gauss-Seidel smoother and a kernel-corrected hybrid smoother are integrated into a geometric multigrid framework, with local Fourier analysis providing quantitative insights into smoothing behavior and two-level convergence. The key contributions include explicit discrete flux operators, a detailed stencil-based representation of SAFE, a principled kernel-correction strategy tied to $H({\rm grad})$ convection-diffusion, and rigorous LFA-backed evidence of robustness across convection- and diffusion-dominated regimes, including jump diffusion. The results demonstrate a practical, scalable solver for vector convection-diffusion in $H({\rm curl})$, offering potential extensions to 3D problems and connections to auxiliary-space preconditioners like the HX approach for Maxwell-type systems.

Abstract

In this paper, we present a robust and efficient multigrid solver based on an exponential-fitting discretization for 2D H(curl) convection-diffusion problems. By leveraging an exponential identity, we characterize the kernel of H(curl) convection-diffusion problems and design a suitable hybrid smoother. This smoother incorporates a lexicographic Gauss-Seidel smoother within a downwind type and smoothing over an auxiliary problem, corresponding to H(grad) convection-diffusion problems for kernel correction. We analyze the convergence properties of the smoothers and the two-level method using local Fourier analysis (LFA). The performance of the algorithms demonstrates robustness in both convection-dominated and diffusion-dominated cases.

A robust solver for H(curl) convection-diffusion and its local Fourier analysis

TL;DR

This work addresses robustly solving 2D convection-diffusion problems by combining a simplex-averaged finite element (SAFE) discretization with exponential-fitting techniques. A downwind, lexicographic Gauss-Seidel smoother and a kernel-corrected hybrid smoother are integrated into a geometric multigrid framework, with local Fourier analysis providing quantitative insights into smoothing behavior and two-level convergence. The key contributions include explicit discrete flux operators, a detailed stencil-based representation of SAFE, a principled kernel-correction strategy tied to convection-diffusion, and rigorous LFA-backed evidence of robustness across convection- and diffusion-dominated regimes, including jump diffusion. The results demonstrate a practical, scalable solver for vector convection-diffusion in , offering potential extensions to 3D problems and connections to auxiliary-space preconditioners like the HX approach for Maxwell-type systems.

Abstract

In this paper, we present a robust and efficient multigrid solver based on an exponential-fitting discretization for 2D H(curl) convection-diffusion problems. By leveraging an exponential identity, we characterize the kernel of H(curl) convection-diffusion problems and design a suitable hybrid smoother. This smoother incorporates a lexicographic Gauss-Seidel smoother within a downwind type and smoothing over an auxiliary problem, corresponding to H(grad) convection-diffusion problems for kernel correction. We analyze the convergence properties of the smoothers and the two-level method using local Fourier analysis (LFA). The performance of the algorithms demonstrates robustness in both convection-dominated and diffusion-dominated cases.
Paper Structure (27 sections, 2 theorems, 68 equations, 6 figures, 5 tables)

This paper contains 27 sections, 2 theorems, 68 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Simplex-averaged finite element (SAFE) on rectangle grids
  3. SAFE in variational form
  4. Discrete de Rham complex on structured grids
  5. Identities of exponentially-fitted type
  6. Local stiffness matrix
  7. Grid operator form and stencil representation
  8. Index sets
  9. Grid functions
  10. Grid operators
  11. Stencil representation of SAFE scheme
  12. A robust solver for SAFE
  13. Downwind lexicographic Gauss-Seidel smoother
  14. Hybrid smoother with kernel correction
  15. Multigrid method
  16. ...and 12 more sections

Key Result

Lemma 4.1

\newlabellm:Helmholtz-splitting0 For all $\bm{\theta}\in \bm{\Theta}$, it holds that Specifically, if we denote $\Psi(\bm{\theta}):=[\psi_J(\bm{\theta})_E\quad \psi_s(\bm{\theta})_E]$, it has the following transformation relationship with eq:standard-basis: where

Figures (6)

  • Figure 1: DOFs and basis of $\mathcal{N}_{[0]}$ on $T = (0,h)^2$.
  • Figure 1: Updating order for the (lexicographic) Gauss-Seidel smoother for the convection in the first quadrant ($b_1,b_2>0$).
  • Figure 1: Contour plots of $\rho^{\rm dw}$ (top) and $\rho^{\rm hybrid}$ (bottom) as a function of $\bm{\theta}$ with $h=1/32$, $\bm{\beta}=({1\over 2},{\sqrt{3}\over 2})^T$ and various $\varepsilon=1,10^{-2},10^{-4}$.
  • Figure 2: Updating order for the (lexicographic) Gauss-Seidel smoother of auxiliary problem for the convection in the first quadrant ($b_1,b_2>0$).
  • Figure 2: Contour plots of $S_{\Psi}^{\rm dw}(\bm{\theta})_{1,1}$ and $S_{\Psi}^{\rm dw}(\bm{\theta})_{2,1}$ as a function of $\bm{\theta}$ with $h=1/32$, $\bm{\beta}=({1\over 2},{\sqrt{3}\over 2})^T$ and various $\varepsilon=1$ (left) and $\varepsilon=10^{-4}$ (right).
  • ...and 1 more figures

Theorems & Definitions (11)

  • Definition 2.1: discrete flux operator
  • Remark 2.2: well-posedness of discrete flux operator
  • Remark 2.3: SAFE scheme on 3D cubic grid
  • Remark 2.4: stencil on 3D cubic grids
  • Remark 3.3: smoothers for auxiliary problem
  • Lemma 4.1: Fourier-Helmholtz splitting
  • Proof 1
  • Remark 4.2: discrete Helmholtz-type decomposition
  • Theorem 4.3: Fourier representation of auxiliary problem
  • Proof 2
  • ...and 1 more