A Robust Finite Element Method for Linearized Magnetohydrodynamics on General Domains

Area of Science:

• Computational physics
• Numerical analysis
• Fluid dynamics

Background:

• The finite element method (FEM) is a powerful numerical technique for solving partial differential equations.
• Linearized Magnetohydrodynamics (MHD) describes the behavior of electrically conducting fluids interacting with magnetic fields.
• Existing FEM for MHD often struggles with non-convex domains and less regular solutions.

Purpose of the Study:

• To generalize and improve the existing finite element method for linearized Magnetohydrodynamics.
• To develop a scheme capable of handling non-convex domains and less regular solutions.
• To prove the robustness of the proposed method with respect to fluid and magnetic Reynolds numbers.

Main Methods:

• Generalization of a previously introduced finite element method for linearized MHD.
• Development of a numerical scheme with enhanced capabilities for domain geometry and solution regularity.
• Theoretical analysis to establish pressure robustness and quasi-robustness properties.

Main Results:

• The proposed FEM scheme successfully handles non-convex domains.
• The method is effective for less regular solutions in MHD problems.
• The scheme is proven to be pressure robust and quasi-robust concerning fluid and magnetic Reynolds numbers.

Conclusions:

• The enhanced FEM provides a more versatile and robust approach for solving linearized MHD problems.
• The method's ability to handle complex scenarios expands its applicability in various scientific and engineering fields.
• Numerical tests validate the theoretical findings, confirming the method's efficacy.