Create driver for incompressible resistive MHD using mixed FE

[jesusbonilla]

We'll follow [1,2] and references therein (ref's to be completed) to implement the FE approximation of incompressible resistive MHD equations. As a first step we will focus on low Re & Ha numbers to avoid introducing stabilization terms. We'll follow [1] to implement mixed FE.

Following the formulation in [1], the discrete problem reads: find and such that for any and it holds

where .

Discretizing above equations with implicit Euler results in exact satisfaction of the Gauss low for every time step, this follows from [Thm 1, 1]. In [Sect. 3.4, 1] explicit Picard and Newton formulations can be found.

Tasks

• Create a Julia project/driver for incompressible MHD equations.
• Implement above formulation using Gridap library.
• Implement Shercliff's benchmark test (See [Sec. 5.A, 2] for analytical solution).
• Implement Hunt's benchmark test (See [Sec. 5.B, 2] for analytical solution).
• Implement 2D island coalescence problem (a.k.a. reconnection problem?).

References

[1] K. Hu, Y. Ma & J. Xu Stable Finite Element Methods Preserving ∇ · B = 0 Exactly for MHD Models Numerische Mathematik, 135, 371–396 (2017)
[2] R. Planas STABILIZED FINITE ELEMENT FORMULATIONS FOR SOLVING INCOMPRESSIBLE
MAGNETOHYDRODYNAMICS PhD Thesis (2013)