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Mixed discontinuous Galerkin finite element methods for incompressible magnetohydrodynamics

University of British Columbia

We develop and analyze mixed discontinuous Galerkin finite element methods for the numerical approximation of incompressible magnetohydrodynamics problems. Incompressible magnetohydrodynamics is the area of physics that is concerned with the behaviour of electrically conducting, resistive, incompressible and viscous fluids in the presence of electromagnetic fields. It is modelled by a system of nonlinear partial differential equations, which couples the Navier-Stokes equations with the Maxwell equations. In the first part of this thesis, we introduce an interior penalty discontinuous Galerkin method for the numerical approximation of a linearized incompressible magnetohydrodynamics problem. The fluid unknowns are discretized with the discontinuous ℘k-℘k-1 element pair, whereas the magnetic variables are approximated by discontinuous ℘k-℘k+1 elements. Under minimal regularity assumptions, we carry out a complete a priori error analysis and prove that the energy norm error is optimally convergent in the mesh size in general polyhedral domains, thus guaranteeing the numerical resolution of the strongest magnetic singularities in non-convex domains. In the second part of this thesis, we propose and analyze a new mixed discontinuous Galerkin finite element method for the approximation of a fully nonlinear incompressible magnetohydrodynamics model. The velocity field is now discretized by divergence-conforming Brezzi-Douglas-Marini elements, and the magnetic field by curl-conforming Nédélec elements. In addition to correctly capturing magnetic singularities, the method yields exactly divergence-free velocity approximations, and is thus energy-stable. We show that the energy norm error is convergent in the mesh size in possibly non-convex polyhedra, and derive slightly suboptimal a priori error estimates under minimal regularity and small data assumptions. Finally, in the third part of this thesis, we present two extensions of our discretization techniques to time-dependent incompressible magnetohydrodynamics problems and to Stokes problems with nonstandard boundary conditions. All our discretizations and theoretical results are computationally validated through comprehensive sets of numerical experiments.

Text

2011-05-25

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/34789

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/