Global magnetohydrodynamic stability with COOL finite elements

The COOL finite element scheme relies on the construction of basis functions using variable order Legendre polynomials [1]. We have implemented this approach in the global linear ideal magnetohydrodynamic code TERPSICHORE [2]. The standard version of this code uses a hybrid method that combines piecewise constant and piecewise linear basis elements. The COOL method with Legendre polynomial order $p=1$ exactly recovers the original formulation. So far, we find that the optimal polynomial order lies around $p=3$ to $4$ (cubic to quartic). At higher order, numerical problems develop in the regions within half-interval mesh points of the magnetic axis and the edge of the plasma because extrapolation of poorly resolved equilibrium quantities at the Gauss-points of the Legendre polynomial can drive very local fictitious near-axis and/or edge mode structures.\\[4pt] [1] A.~Ahusborde, R.~Gruber, M.~Azaiez, M.~L.~Sawley, Phys.~Rev.~E \textbf{75} (2007) 056704.\\[0pt] [2] D.~V.~Anderson, W.~A.~Cooper, R.~Gruber, S.~Merazzi, U.~Schwenn, Int.~J.~Supercomp.~Appl.~\textbf{4} (1990) 34-47.