This thesis is concerned with the analysis and implementation of the hp-version interior penalty discontinuous Galerkin finite element method (DGFEM) on computational meshes consisting of general polygonal/polyhedral (polytopic) elements. Two model problems are considered: general advection-diffusion-reaction boundary value problems and time dependent parabolic problems. New hp-version a priori error bounds are derived based on a specific choice of the interior penalty parameter which allows for edge/face-degeneration as well as an arbitrary number of faces and hanging nodes per element. The proposed method employs elemental polynomial bases of total degree p (Pp- bases) defined in the physical coordinate system, without requiring mapping from a given reference or canonical frame. A series of numerical experiments highlighting the performance of the proposed DGFEM are presented. In particular, we study the competitiveness of the p-version DGFEM employing a Pp-basis on both polytopic and tensor-product elements with a (standard) DGFEM and FEM employing a (mapped) Qp-basis. Moreover, a careful theoretical analysis of optimal convergence rate in p for Pp-basis is derived for several commonly used projectors, which leads to sharp bounds of exponential convergence with respect to degrees of freedom (dof) for the Pp-basis.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:702689 |
| Date | January 2017 |
| Creators | Dong, Zhaonan |
| Contributors | Georgoulis, Emmanuil |
| Publisher | University of Leicester |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/2381/39140 |
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