Edge Element Approximation for the Spherical Interface Dynamo System

TL;DR

This paper develops an edge element approximation for a spherical interface dynamo system, providing theoretical analysis, stability, convergence, and simulation results that align with existing findings in magnetic field modeling.

Contribution

It introduces an edge element method tailored for the spherical dynamo system with quasi-vacuum boundary conditions, including theoretical proofs and numerical validation.

Findings

Existence, uniqueness, and stability of the solution established.

Convergence of the numerical scheme demonstrated.

Simulated magnetic field characteristics match existing models.

Abstract

Exploring the origin and properties of magnetic fields is crucial to the development of many fields such as physics, astronomy and meteorology. We focus on the edge element approximation and theoretical analysis of celestial dynamo system with quasi-vacuum boundary conditions. The system not only ensures that the magnetic field on the spherical shell is generated from the dynamo model, but also provides convenience for the application of the edge element method. We demonstrate the existence, uniqueness and stability of the solution to the system by the fixed point theorem. Then, we approximate the system using the edge element method, which is more efficient in dealing with electromagnetic field problems. Moreover, we also discuss the stability of the corresponding discrete scheme. And the convergence is demonstrated by later numerical tests. Finally, we simulate the three-dimensional time evolution of the spherical interface dynamo model, and the characteristics of the simulated magnetic field are consistent with existing work.