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Error estimates for a helicity-preserving finite element discretisation of an incompressible magnetohydrodynamics system

L. Beirao da Veiga, K. Hu, L. Mascotto

TL;DR

The paper addresses the problem of deriving rigorous error estimates for a seven-field, helicity- and energy-preserving finite element discretisation of incompressible MHD. It develops a structure-preserving FE method based on a de Rham sequence with commuting projections, a discrete curl operator, and divergence-free constraints to ensure discrete conservation of energy and magnetic and cross helicities. The main result is an optimal-order a priori error estimate for the semi-discrete scheme, showing convergence of the velocity and magnetic field (and associated variables) with rate $h^{k+1}$ under suitable regularity, while preserving key physical invariants. This work provides theoretical guarantees for the reliability and physical fidelity of helicity-preserving MHD simulations and lays a foundation for fully discrete analyses with time integrators that maintain quadratic invariants.

Abstract

We derive error estimates of a finite element method for the approximation of solutions to a seven-fields formulation of a magnetohydrodynamics model, which preserves the energy of the system, and the magnetic and cross helicities on the discrete level.

Error estimates for a helicity-preserving finite element discretisation of an incompressible magnetohydrodynamics system

TL;DR

The paper addresses the problem of deriving rigorous error estimates for a seven-field, helicity- and energy-preserving finite element discretisation of incompressible MHD. It develops a structure-preserving FE method based on a de Rham sequence with commuting projections, a discrete curl operator, and divergence-free constraints to ensure discrete conservation of energy and magnetic and cross helicities. The main result is an optimal-order a priori error estimate for the semi-discrete scheme, showing convergence of the velocity and magnetic field (and associated variables) with rate under suitable regularity, while preserving key physical invariants. This work provides theoretical guarantees for the reliability and physical fidelity of helicity-preserving MHD simulations and lays a foundation for fully discrete analyses with time integrators that maintain quadratic invariants.

Abstract

We derive error estimates of a finite element method for the approximation of solutions to a seven-fields formulation of a magnetohydrodynamics model, which preserves the energy of the system, and the magnetic and cross helicities on the discrete level.
Paper Structure (8 sections, 12 theorems, 120 equations)

This paper contains 8 sections, 12 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. The model problem
  3. Preservation of the energy, and the magnetic and cross helicities
  4. The method
  5. Discrete spaces, discrete operators and various approximants
  6. The discrete problem, preservation properties and well-posedness
  7. Error estimates for the semi-discrete scheme
  8. Proof of Theorem \ref{['theorem:helicity-discrete']}

Key Result

Theorem 2.1

The following identity holds true:

Theorems & Definitions (19)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 9 more