High-order numerical methods and algorithms: Eigen-based spectral element approach

This thesis focuses on the construction of the eigen-based high-order expansion bases for spectral elements. In high-order approaches with spectral elements or p-finite elements, basis functions are very important. They are used to discretize the partial differential equations (approximate the solution functions). Different bases lead us to different system matrices in the final algebraic system of equations. The properties of these matrices in turn influence the number of iterations to covergence needed by iterative solvers. We want to find a set of basis functions that can lead us to system matrices with high numerical efficiency, which can be solved by as few number of iterations with iterative solvers as possible. Therefore, we can save computation time. We construct eigen-based bases from an existing basis for structured and unstructured elements. Then we construct macro eigen-based bases from the eigen-based bases. System matrices given by eigen-based bases exhibit superior numerical efficiency, which give us system matrices of low condition number, sparse matrix pattern, clustered distribution of eigenvalues compared with the existing bases. As we expected, the number of conjugate gradient iterations to convergence obtained with the eigen-based bases are fewer than those with the existing expansion bases. Numerical examples are provided to verify the assumptions and to demonstrate the improved performance of the eigen bases and macro eigen bases