The Navier-Stokes equations govern the flow of viscous fluids such as air or water. Since no general solution is known, computer simulations are used to obtain approximate solutions. As computers are unable to handle continuous representations of fields, finite-dimensional projec
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The Navier-Stokes equations govern the flow of viscous fluids such as air or water. Since no general solution is known, computer simulations are used to obtain approximate solutions. As computers are unable to handle continuous representations of fields, finite-dimensional projection of fields is required, resulting in a loss of information. A difficulty in constructing these discrete solutions, is to obtain discrete solutions that have the same conservation properties as continuous solutions. Other challenges include the efficient handling of the nonlinear term in the equations. In this thesis we aim to construct a numerical method whose solutions satisfy the conservation of mass, kinetic energy and helicity exactly. To achieve this, a structure preserving (mimetic) spectral element method is employed. Such methods aim to represent properties of the continuous problem exactly in the discrete problem. These methods have their theoretic roots in differential geometry, where there is a clear connection between fields and the geometric objects they are associated with. From this association we note that the fields in the Navier-Stokes equations show a dual nature, where each field can be associated to two types of geometric objects. In [1] this dual nature is used to construct a mass, kinetic energy and helicity conserving method that handles the nonlinear term efficiently by using a leapfrog scheme for problems on periodic domains. In this thesis this work is extended to handle Dirichlet boundary conditions. When considering periodic boundary conditions, the constructed numerical method exhibits the same conservation properties of energy and helicity as the Navier-Stokes equations. When considering Dirichlet boundary conditions, some additional contributions to the dissipation rates of energy and helicity are noted in the cases of inflow into the domain and nonzero normal vorticity at the boundary respectively. These additional contributions cannot be meaningfully quantified within the employed approach, and are a result of strongly enforced boundary conditions on velocity. Furthermore, the numerical method constructs a pointwise divergence free velocity field at every other time step. The mimetic spectral element method allows for high order spatial approximation of fields. Optimal spatial convergence orders are observed for all fields. Temporal integration was done via a combination of the trapezoidal and midpoint rule and the expected second order temporal convergence is observed.