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Brian Cornille, University of Wisconsin-Madison
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In multi-timescale extended magnetohydrodynamics (MHD) simulation, representing Hall term effects is particularly challenging. When substituted into Faraday's Law, the Hall term introduces an advection-like term, which hampers mathematical coercivity in Galerkin finite-element methods. The first-order systems least squares (FOSLS) approach can be used to make the system coercive and stabilize the advection-like term in implicit computations.^{1} To better understand this numerical challenge, we consider a limit of the extended MHD system known as electron MHD (EMHD), which largely isolates the Hall term. In EMHD the FOSLS approach presents many choices for independent variable pairs. Here we study two options, vector fields {B, J} in the function space {H1, H1} and {B, E} in {H1, H(curl)}. We also consider two Galerkin formulations with B in H1 and H(curl). These formulations are being implemented using the MFEM library. Numerical properties of all four formulations are compared and contrasted. The multiscale nature of the EMHD tearing mode in a slab geometry provides a challenging test case that exposes strengths and weaknesses of the different formulations and permits validation with analytical theory.
^{1}C.A. Leibs, T.A. Manteuffel, Nested Iteration and First-Order Systems Least Squares for a Two-Fluid Electromagnetic Darwin Model. *SIAM Journal on Scientific Computing.* 37, S314-S333 (2015).
Supported by DOE CSGF under grant DE-FG02-97ER25308 and by DOE grant DE-SC0018001.

**Abstract Author(s): **B.S. Cornille, C.R. Sovinec