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A streamline upwind/Petrov-Galerkin method for the magnetic advection-diffusion problem

Haochen Li, Yangfan Luo, Jindong Wang, Shuonan Wu

TL;DR

The paper develops a streamline upwind/Petrov-Galerkin method for the magnetic advection-diffusion problem in H(curl) conforming settings. By constructing a discrete magnetic advection operator via a lifting-based stabilization and introducing two stabilization terms, the authors obtain a stable, conforming scheme with optimal a priori error estimates. They prove discrete coercivity under mild assumptions and validate the theory with extensive 2D and 3D numerical experiments, demonstrating strong stability, reduced oscillations, and expected convergence rates, even in advection-dominated and layer-rich regimes. This approach extends SUPG concepts to vector-valued curl-curl problems and offers a robust framework for MHD-related simulations.

Abstract

This paper presents the development and analysis of a streamline upwind/Petrov-Galerkin (SUPG) method for the magnetic advection-diffusion problem. A key feature of the method is an SUPG-type stabilization term based on the residuals and weighted advection terms of the test function. By introducing a lifting operator to characterize the jumps of finite element functions across element interfaces, we define a discrete magnetic advection operator, which subsequently enables the formulation of the desired SUPG method. Under mild assumptions, we establish the stability of the scheme and derive optimal error estimates. Numerical examples in both two and three dimensions are provided to demonstrate the theoretical convergence and stabilization properties of the proposed method.

A streamline upwind/Petrov-Galerkin method for the magnetic advection-diffusion problem

TL;DR

The paper develops a streamline upwind/Petrov-Galerkin method for the magnetic advection-diffusion problem in H(curl) conforming settings. By constructing a discrete magnetic advection operator via a lifting-based stabilization and introducing two stabilization terms, the authors obtain a stable, conforming scheme with optimal a priori error estimates. They prove discrete coercivity under mild assumptions and validate the theory with extensive 2D and 3D numerical experiments, demonstrating strong stability, reduced oscillations, and expected convergence rates, even in advection-dominated and layer-rich regimes. This approach extends SUPG concepts to vector-valued curl-curl problems and offers a robust framework for MHD-related simulations.

Abstract

This paper presents the development and analysis of a streamline upwind/Petrov-Galerkin (SUPG) method for the magnetic advection-diffusion problem. A key feature of the method is an SUPG-type stabilization term based on the residuals and weighted advection terms of the test function. By introducing a lifting operator to characterize the jumps of finite element functions across element interfaces, we define a discrete magnetic advection operator, which subsequently enables the formulation of the desired SUPG method. Under mild assumptions, we establish the stability of the scheme and derive optimal error estimates. Numerical examples in both two and three dimensions are provided to demonstrate the theoretical convergence and stabilization properties of the proposed method.

Paper Structure

This paper contains 22 sections, 8 theorems, 113 equations, 23 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. SUPG method
  4. Stability analysis
  5. A priori error estimate
  6. Numerical experiment
  7. Example 1: 3D convergence tests
  8. Example 2: 2D convergence tests
  9. Example 3: Clarification of stabilization terms
  10. Example 4: Convergence with respect to degree of polynomial
  11. Example 5: Numerical performance in complex regions
  12. Example 6: Boundary layers
  13. Example 7: Internal layers
  14. Example 8: Manual refined meshes for boundary layers
  15. Example 9: Manual refined meshes for internal layers
  16. ...and 7 more sections

Key Result

Lemma 1

For any two vectors $\bm{v}_1, \bm{v}_2$, the following identities hold: Here $[\alpha]:=\alpha^+-\alpha^-$.

Figures (23)

  • Figure 3.1: Severe spurious oscillations in the standard Galerkin method \ref{['standard_Galerkin_method']}. The computational domain is $(0,1)^2$ with parameters $\varepsilon = 10^{-6}$, $\bm{\beta} = [1, 2]^T$, $\gamma = 0$, $\bm{f} = [1, 1]^T$, and $\bm{g} \equiv [0, 0]^T$. The solution is computed on a uniform mesh of $2 \times 16^2$ triangles using first-order Nédélec elements of the second kind.
  • Figure 6.1: 2D and 3D meshes
  • Figure 6.2: Example 4: Dependence of error of the SUPG scheme on $N$ and polynomial degree $k$ with $\delta_T = 0.4/N$.
  • Figure 6.3: Example 5: Complex regions
  • Figure 6.4: Example 6. Numerical solution obtained by the $S_h^1$ only method ($k=1$).
  • ...and 18 more figures

Theorems & Definitions (16)

  • Remark 1
  • Lemma 1: weighted average identities
  • proof
  • Lemma 2: inverse inequality
  • Lemma 3: trace inequality
  • Lemma 4: finite element trace inequality
  • Remark 2
  • Remark 3
  • Corollary 1: inverse inequality for $\nabla\times$ operator
  • Lemma 5: discrete coercivity
  • ...and 6 more