Body-fitting Meshes for the Discontinuous Galerkin Method

Abstract In this dissertation, a scheme capable of generating highly accurate, body fitting meshes and its application with the Discontinuous Galerkin Method is introduced. Unlike most other mesh generators, this scheme generates meshes consisting of quadrilateral or hexahedral elements exclusively. The high approximation quality is achieved by means of high order curved elements. The resulting meshes are highly suited for the application with high order methods, specifically the Discontinuous Galerkin Method, as large portions of the mesh are fully structured while matching the high order of the numerical method with a high order geometry representation at object and domain boundaries. The mesh scheme works in two steps. It chooses an interior volume that can be represented using a fully structured Cartesian mesh and connects it to the embedded objects and domain boundaries with a so called buffer-layer in a second step. Inside the layer, high order curved elements are applied for yielding high geometric representation accuracy. The resulting meshes are ideal for the application of high order methods. In the interior part the Cartesian structure can be exploited for obtaining high efficiency of the numerical method while the accuracy potential can be realized also in the presence of curved objects and boundaries. After introducing the mesh scheme, the Discontinuous Galerkin Method is described and applied for solving Maxwell’s equations. As a high order method it achieves exponential convergence under p-refinement. It is shown that using meshes produced by the novel scheme this property is achieved for curved domains as well. As an example, optimal convergence rates are demonstrated in a cylindrical cavity problem. In another example, the abilities of the method to produce correct spectral properties of closed resonator problems are investigated. To this end, a time-domain signal is recorded during the transient analysis. After applying the Fourier transform accurate frequency spectra are observed, which are free of spurious modes