<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 S. Chandrasekhar. Hydrodynamic and Hydromagnetic Stability . Dover Publications, Newburyport, 2013. MR:0128226 . Zbl:0142.44103 http://zbmath.org/?q=an:0142.44103>http://www.ams.org/mathscinet-getitem?mr=MR0128226 . Zbl:0142.44103 http://zbmath.org/?q=an:0142.44103 . D. Farmer and D. J. Ivers. Aspects of a hybrid hp-finite element/spectral method for the linearised magnetohydrodynamic equations in spherical geometries. In W. McLean and A. J. Roberts editors, Proceedings of the 15th Biennial Computational Techniques and Applications Conference, CTAC-2010 , volume 52 of ANZIAM J. , pages C379–C394, July 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3950 [July 20, 2011] G. H. Golub and J. H. Welsch. Calculation of Gauss Quadrature Rules. Mathematics of Computation , 23:221–230, 1969. doi:10.2307/2004418 R. Hide and P. H. Roberts. Hydromagnetic flow due to an oscillating plane. Reviews of Modern Physics , 32(4):799–806, 1960. doi:10.1103/revmodphys.32.799 D. J. Ivers and C. G. Phillips. A vector spherical harmonic spectral code for linearised magnetohydrodynamics. In K. Burrage and Roger B. Sidje editors, Proceedings of the 10th Biennial Computational Techniques and Applications Conference, CTAC-2001 , volume 44 of ANZIAM J. , pages C423–C442, April 2003. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/689 [April 1, 2003] D. J. Ivers and C. G. Phillips. Scalar and vector spherical harmonic spectral equations of rotating magnetohydrodynamics. Geophysical Journal International , 175:955–974, 2008. doi:10.1111/j.1365-246x.2008.03944.x L. Kamenski, W. Huang, and H. Xu. Conditioning of finite element equations with arbitrary anisotropic meshes. Mathematics of computation , 83(289):2187–2211, 2014. doi:10.1090/s0025-5718-2014-02822-6 G. E. Karniadakis and S. Sherwin. Spectral hp Element Methods for Computational Fluid Dynamics, Second Edition . Oxford University Press, Oxford, 2005. doi:10.1093/acprof:oso/9780198528692.001.0001 D. E. Loper. General solution for the linearized Ekman–Hartmann layer on a spherical boundary. The Physics of Fluids , 13(12):2995–2998, 1970. doi:10.1063/1.1692891 D. E. Loper. Steady hydromagnetic boundary layer near a rotating, electrically conducting plate. The Physics of Fluids , 13(12):2999–3002, 1970. doi:10.1063/1.1692892 C. Schwab. p- and hp- Finite Element Methods, Theory and Applications in Solid and Fluid Mechanics . Oxford University Press, Oxford, 1998. MR:1695813 . Zbl:0910.73003 http://zbmath.org/?q=an:0910.73003>http://www.ams.org/mathscinet-getitem?mr=MR1695813 . Zbl:0910.73003 http://zbmath.org/?q=an:0910.73003 A. M. Soward and R. Hollerbach. Non-axisymmetric magnetohydrodynamic shear layers in a rotating spherical shell. Journal of Fluid Mechanics , 408:239–274, 2000. doi:10.1017/s0022112099007776