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Abstract

In this thesis the exactly divergence-free finite element method developed by Cockburn, Scheotzau and Kanschat in [13] and [14] is studied. This method is first reviewed in the context of Stokes problem. An interior penalty discontinuous Galerkin approach is formulated and analysed in the unified framework established in [13], [14] and [37]. Then we extend the method to non-isothermal flow problems, in particular, to a generalised Boussinesq equation. Following the work by Ricardo, Scheotzau and Qin in [34], the method is formulated and the numerical analysis is reviewed. Numerical examples are implemented and presented, which verify the theoretical error estimates and the exactly divergence-free property.