Structure-conserving numerical methods that aim at preserving certain structures of the PDEs at the discrete level have been an interesting research topic for many decades. The mimetic spectral element method, a recently developed arbitrary order structure-preserving method on orthogonal or curvilinear meshes, has also been drawing increasingly amount of attention. This dissertation is devoted to promoting the application and development of the mimetic spectral element method.
In this dissertation, we first give a comprehensive introduction on the mimetic spectral element method with applications to the Poisson problem, which is followed by a new development of the mimetic spectral element method for the Navier-Stokes equations. This new development is on a conservative dual-field discretization that conserves mass, kinetic energy and helicity for the 3D incompressible Navier-Stokes equations in the absence of dissipative terms. And when there are dissipative terms, the method correctly predicts the decay rates of the kinetic energy and helicity. It is a dual-field method in the sense that two evolution equations are employed and weak solutions are sought for each physical variable in two different finite dimensional function spaces. This novel method and the promising results reveal its potential in multiple research fields like turbulence modeling, sub-grid methods and large eddy simulation.
Despite the mimetic spectral element method possesses preferable properties due to its feature of structure-preserving, its demand of high computational power is a major limitation. To address this drawback, two techniques, hybridization and dual basis functions, are employed for the mimetic spectral element method, which leads to an extension that decreases the computational cost not only by reducing the size and lowering the condition number of the global linear system, but also by improving the feasibility for parallel computing.
A special component, the Complement, is embedded in this thesis. It aims to provide a more friendly introduction for the readers, especially those who are new to this specific area of numerical methods. In these web-based additions, there are instructors and well-documented scripts which allow readers to learn in an interactive way, thus to get some hands-on experience and eventually to obtain a deeper understanding of the method. This component can help the readers to more quickly and efficiently implement their own new ideas, which will in return contribute to the development of this method.
Overall, we conclude that this dissertation fulfilled the goal to promote the application and development of the mimetic spectral element method.
For a Dutch version of the summary, click on Samenvatting.
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