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Two-Level Finite Element Iterative Algorithm Based on Stabilized Method for the Stationary Incompressible

Qili Tang1, Min Hou1, Yajie Xiao1

  • 1Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China.

Entropy (Basel, Switzerland)
|July 8, 2023
PubMed
Summary

This study introduces a novel two-level stabilized finite element algorithm for solving incompressible magnetohydrodynamic (MHD) equations. The method enhances computational efficiency while maintaining accuracy, offering significant time savings in numerical simulations.

Keywords:
Oseen iterationfinite element methodstabilized methodstationary incompressible MHDtwo-level method

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Area of Science:

  • Computational fluid dynamics
  • Numerical analysis
  • Magnetohydrodynamics

Background:

  • Stationary incompressible magnetohydrodynamic (MHD) equations present numerical challenges.
  • Low regularity of magnetic fields requires specialized techniques.
  • Existing finite element methods may face restrictions like the inf-sup condition.

Purpose of the Study:

  • To develop and analyze a computationally efficient two-level stabilized finite element algorithm for solving stationary incompressible MHD equations.
  • To address the challenges posed by low regularity of magnetic fields.
  • To improve upon existing one-level methods in terms of computational cost.

Main Methods:

  • Combination of stabilization technique, Oseen iterative method, and two-level finite element algorithm.
  • Lagrange multiplier technique for the magnetic field sub-problem.
  • Stabilized method for the flow field sub-problem to bypass inf-sup condition restrictions.

Main Results:

  • Stability and convergence analysis for both one- and two-level algorithms are provided.
  • The two-level method achieves the same convergence order as the one-level method when h=O(H2).
  • Significant computational cost savings are demonstrated, with the two-level method being up to three times faster.

Conclusions:

  • The proposed two-level stabilized finite element method is effective for solving stationary incompressible MHD equations.
  • The method offers substantial computational advantages over one-level approaches, particularly with Nédélec elements.
  • This work contributes to efficient numerical solutions in magnetohydrodynamics.