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Local projection stabilization methods for $\boldsymbol{H}({\rm curl})$ and $\boldsymbol{H}({\rm div})$ advection problems

Yangfan Luo, Jindong Wang, Shuonan Wu

TL;DR

This work develops local projection stabilization (LPS) methods for advection problems in the vector spaces $H(\mathrm{curl})$ and $H(\mathrm{div})$, using conforming finite elements of arbitrary order within a unified framework. Stability and accuracy are achieved by enriching the approximation spaces with $H(\mathrm{d})$ bubble functions ($d=\mathrm{curl}$ or $\mathrm{div}$), establishing a local inf-sup condition and a modified interpolation operator $j_h$ that yields optimal a priori error estimates in the energy norm, with a convergence rate of $O(h^{r+\frac12})$ (and $O(h^{r+1})$ in the $L^2$ norm) under quasi-uniform meshes. The analysis decomposes the error into stability, consistency, and projection components, and the method is validated through comprehensive 2D and 3D numerical experiments that demonstrate the necessity of both bubble enrichment and the stabilization terms $S_h^1$ and $S_h^2$. The results indicate that the proposed LPS framework provides stable, accurate discretizations for convection-dominated vector transport in applications such as magnetohydrodynamics (MHD).

Abstract

We devise local projection stabilization (LPS) methods for advection problems in the $\boldsymbol{H}$(curl) and $\boldsymbol{H}$(div) spaces, employing conforming finite element spaces of arbitrary order within a unified framework. The key ingredient is a local inf-sup condition, enabled by enriching the approximation space with appropriate $\boldsymbol{H}$(d) bubble functions (with d = curl or div). This enrichment allows for the construction of modified interpolation operators, which are crucial for establishing optimal a priori error estimates in the energy norm. Numerical examples are presented to verify both the theoretical results and the stabilization properties of the proposed method.

Local projection stabilization methods for $\boldsymbol{H}({\rm curl})$ and $\boldsymbol{H}({\rm div})$ advection problems

TL;DR

This work develops local projection stabilization (LPS) methods for advection problems in the vector spaces and , using conforming finite elements of arbitrary order within a unified framework. Stability and accuracy are achieved by enriching the approximation spaces with bubble functions ( or ), establishing a local inf-sup condition and a modified interpolation operator that yields optimal a priori error estimates in the energy norm, with a convergence rate of (and in the norm) under quasi-uniform meshes. The analysis decomposes the error into stability, consistency, and projection components, and the method is validated through comprehensive 2D and 3D numerical experiments that demonstrate the necessity of both bubble enrichment and the stabilization terms and . The results indicate that the proposed LPS framework provides stable, accurate discretizations for convection-dominated vector transport in applications such as magnetohydrodynamics (MHD).

Abstract

We devise local projection stabilization (LPS) methods for advection problems in the (curl) and (div) spaces, employing conforming finite element spaces of arbitrary order within a unified framework. The key ingredient is a local inf-sup condition, enabled by enriching the approximation space with appropriate (d) bubble functions (with d = curl or div). This enrichment allows for the construction of modified interpolation operators, which are crucial for establishing optimal a priori error estimates in the energy norm. Numerical examples are presented to verify both the theoretical results and the stabilization properties of the proposed method.

Paper Structure

This paper contains 16 sections, 9 theorems, 92 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local projection stabilization (LPS) methods
  4. Bubble space and modified interpolation operator
  5. Convergence analysis
  6. Stability
  7. Consistency error
  8. A priori error estimate
  9. Numerical experiments
  10. Example 1: 2D convergence tests
  11. Example 2: 3D convergence tests
  12. Example 3: Clarification of stabilization terms
  13. Example 4: Clarification of bubble enrichment
  14. Example 5: 2D circular layer test
  15. Example 6: 2D interior layer test
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $f$ be a face of $K \in \mathcal{T}_h$. For $v|_K \in H^1(K)$ the following estimate holds where C is a constant independent of $h_K$.

Figures (3)

  • Figure 4.1: Normal and tangent vectors associated with the $H(\mathrm{d})$ bubble functions.
  • Figure 6.1: Example 5: First component of the LPS solution featuring a circular interior layer.
  • Figure 6.2: Example 6: First component of the LPS solution featuring an interior layer.

Theorems & Definitions (19)

  • Lemma 2.1: trace inequality
  • Lemma 2.2: inverse inequality
  • Lemma 4.1: curl local inf-sup
  • proof
  • Lemma 4.2: div local inf-sup
  • proof
  • Remark 4.1
  • Theorem 4.1: modified interpolation operator
  • proof
  • Lemma 5.1: coercivity
  • ...and 9 more