| In this dissertation, the author made mainly original contributions of new formulation of functionals and physical spline finite element method.; A unified and concise formulation of functionals was proposed and verified by typical examples: a scattering problem and several conducting structures with imperfectly conducting walls. This formulation is no longer “from the sky”; it can be considered as natural conclusions of Maxwell's equations. In the meantime, the new formulation can explain the spurious mode problem in a natural way since the divergence equations are not employed either explicitly or implicitly. Obviously, the same technique can be extended to more cases, for example time-domain problems.; An amazing technique-physical spline interpolation-was first developed. Physical differential equations were incorporated into interpolations of basic elements in finite element methods. This was named physical spline finite element method (PSFEM) by this author. Theoretically, the physical spline interpolation introduces many new features. First, physics equations can be used in the interpolations to make the interpolation problem-associated. The algorithm converges much faster than any general interpolations. This keeps the simplicity of the first order Lagrange interpolation. Secondly, the concept of basis functions may need to be re-examined. Thirdly, basis functions could be complex without simple geometric explanations.; The applications to one-dimensional and two-dimensional electromagnetic problems show the novel improvements of the newly developed PSFEM on accuracy, convergence and stability. |
