<em>hp</em>-finite element method for coupled linear systems of mixed-order differential equations with boundary layer behaviour

Abstract

Spectral methods for the horizontal dependence in linearised magnetohydrodynamics in spherical geometries and plane layers give rise to mixed-order (second and fourth) systems of equations in the radial or vertical dependence. The spherical case was covered by Ivers & Phillips [Geophys J.~Intl 175, 2008]. Previous work of Farmer & Ivers [ANZIAM J. 52, 2011] indicated how to extend results appearing by Schwab [1998] to multiple boundary layers retaining robust exponential convergence (in the small parameter appearing in the coefficient of the highest derivative terms). This article investigates the recovery of the interior solution for different model problems. The main result is that the robust convergence results reported by Farmer & Ivers [2011] do carry through for coupled mixed-order problems. It is also shown that for a coupled mixed-order system there is little difference between using a \(C^0\) expansion for the second-order variables and \(C^1\) expansion for the fourth-order, versus the algorithmically simpler choice of a \(C^1\) expansion for both. Calibration of the method to determine the boundary layer widths and the degree of approximation for the interior when the exact solution is unknown, is discussed. The results of this article can be applied to solving the linearised magnetohydrodynamics equations in spherical geometries and plane layers. References

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