UBC Theses and Dissertations
Preconditioners for incompressible magnetohydrodynamics Wathen, Michael P.
The main goal of this thesis is to design efficient numerical solutions to incompressible magnetohydrodynamics (MHD) problems, with focus on the solution of the large and sparse linear systems that arise. The MHD model couples the Navier-Stokes equations that govern fluid dynamics and Maxwell's equations which govern the electromagnetic effects. We consider a mixed finite element discretization of an MHD model problem. Upon discretization and linearization, a large block 4-by-4 nonsymmetric linear system needs to be (repeatedly) solved. One of the principal challenges is the presence of a skew-symmetric term that couples the fluid velocity with the magnetic field. We propose two distinct preconditioning techniques. The first approach relies on utilizing and combining effective solvers for the mixed Maxwell and the Navier-Stokes sub-problems. The second approach is based on algebraic approximations of the inverse of the matrix of the linear system. Both approaches exploit the block structure of the discretized MHD problem. We perform a spectral analysis for ideal versions of the proposed preconditioners, and develop and test practical versions. Large-scale numerical results for linear systems of dimensions up to 10⁷ in two and three dimensions validate the effectiveness of our techniques. We also explore the use of the Conjugate Gradient (CG) method for saddle-point problems with an algebraic structure similar to the time-harmonic Maxwell problem. Specifically, we show that for a nonsingular saddle-point matrix with a maximally rank-deficient leading block, there are two sufficient conditions that allow for CG to be used. An important part of the contributions of this thesis is the development of numerical software that utilizes state-of-the-art software packages. This software is highly modular, robust, and flexible.
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