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UBC Theses and Dissertations

Iterative solution of a mixed finite element discretisation of an incompressible magnetohydrodynamics problem Wathen, Michael


The aim of this thesis is to develop and numerically test a large scale preconditioned finite element implementation of an incompressible magnetohydrodynamics (MHD) model. To accomplish this, a broad-scope code has been generated using the finite element software package FEniCS and the linear algebra software PETSc. The code is modular, extremely flexible, and allows for implementing and testing different discretisations and linear algebra solvers with relatively modest effort. It can handle two- and three-dimensional problems in excess of 20 million degrees of freedom. Incompressible MHD describes the interaction between an incompressible electrically charged fluid governed by the incompressible Navier-Stokes equations coupled with electromagnetic effects from Maxwell’s equations in mixed form. We introduce a model problem and a mixed finite element discretisation based on using Taylor-Hood elements for the fluid variables and on a mixed N ́ed ́elec pair for the magnetic unknowns. We introduce three iteration strategies to handle the non-linearities present in the model, ranging from Picard iterations to completely decoupled schemes. Adapting and extending ideas introduced in [Dan Li, Numerical Solution of the Time-Harmonic Maxwell Equations and Incompressible Magnetohydrodynamics Problems, Ph.D. Dissertation, The University of British ii Columbia, 2010], we implement a preconditioning approach motivated by the block structure of the underlying linear systems in conjunction with state of the art preconditioners for the mixed Maxwell and Navier-Stokes subproblems. For the Picard iteration scheme we implement an inner-outer preconditioner. The numerical results presented in this thesis demonstrate the efficient performance of our preconditioned solution techniques and show good scalability with respect to the discretisation parameters.

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