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We consider the incompressible magnetohydrodynamic equations, linearised about a steady basic state in spherical geometries. The angular dependence is discretised using a Galerkin method based on spherical harmonics. This produces a coupled system of ordinary differential equations in radius, which may contain up to fourth order derivatives, after separation of the time dependence. In applications small magnetic, viscous or thermal diffusion may lead to boundary layers of second or higher order. We investigate key aspects of the discretisation of the radial equations using one dimensional $hp$-finite element methods through two model problems: reduction of order versus higher order elements for an ordinary differential equation of fourth order; and fourth order boundary layers. Results indicate that higher order elements are to be favoured over a reduction of order, and that robust exponential convergence may be achieved for the symmetric two layer problem investigated.

Aspects of a hybrid hp-finite element/spectral method for the linearised magnetohydrodynamic equations in spherical geometries