Treatment of the magnetic field for geodynamo simulations using the finite element method

We propose a scheme for calculating the magnetic field in a spherical shell, based on Earth's outer core, using the finite element method (FEM). The two most difficult problems for magnetohydrodynamics (MHD) simulations in a rotating spherical shell with FEM are solving the magnetic field outside the fluid shell, and connecting the magnetic field in the fluid shell to the exterior potential field at the boundary. To solve these problems, we extend the finite element mesh beyond the fluid shell and compute the vector potential of the magnetic field. To verify the present scheme, we consider three test case. First, we compare the FEM model with an analytical solution of Laplace's equation outside the fluid. Second, we evaluate free decay of a dipole field and compare the results with a spectral solution. Finally, compare the results of a simple kinematic dynamo problem with a spectral solution. The results suggest that the accuracy of the dipole field depends on the radius of the simulation domain, and that this error becomes sufficiently small if the radius of the outer region is approximately 6 times larger than the radius of the fluid shell.