%0 Generic
%A Engwer, Christian
%D 2009
%F heidok:9990
%K Discontinuous Galerkin Method , Finite Elements , Higher Order , Complex Domain , Unfitted Finite Elements , Structured Grids , Numerical Upscaling
%R 10.11588/heidok.00009990
%T An Unfitted Discontinuous Galerkin Scheme for Micro-scale Simulations and Numerical Upscaling
%U https://archiv.ub.uni-heidelberg.de/volltextserver/9990/
%X The aim of this thesis is the development of a new discretization method for solving partial differential equations on complex shaped domains.  Many biological, physical, and chemical applications involve processes on such domains and the numerical treatment of such processes is a challenging task. The proposed method offers a higher-order discretization where the mesh is not required to resolve the complex shaped boundary. The method combines the Unfitted Finite Element method with a Discontinuous Galerkin discretization. Trial and test functions are defined on a structured grid and their support is restricted according to the domain boundary. Essential boundary conditions are imposed weakly via the Discontinuous Galerkin formulation. Thus, the mesh is not required to resolve the domain boundary but higher-order ansatz functions can still be used. Hence it is possible to vary the size of the ansatz space independently of the geometry. For an elliptic test problem, stability and convergence properties of the method are analyzed numerically. Even though some assumptions of the underlying Discontinuous Galerkin method regarding the finite element mesh cannot be guaranteed, the method is stable in all tests and converges optimally. The control over the size of the approximation space is especially attractive for applications like numerical upscaling and multi-scale simulations. In this thesis the method is successfully applied to numerical upscaling of a stationary flow problem and to a time-dependent transport problem, where the complex domains used are artificially generated as well as experimentally measured structures, obtained from micro X-ray CT scans.