This thesis was concerned with the approximate solution of partial differential equations via discretization with composite finite elements. It deals with the application of this approach to problems where the computational domain is given through image data and thus possibly is of complex shape. We summarize the method's mathematical background, namely the Galerkin method and classical finite elements, and describe in detail the theory of composite finite elements as introduced by Hackbusch and Sauter. Based on simple elliptic model problems we further present an efficient strategy for calculating arbitrarily coarse approximations which nevertheless resolve the problem's computational domain and preserve characteristic properties of the exact solution. As part of the work, a program realizing the described method for Poisson-like problems in up to three dimensions was implemented in C++ using the DUNE framework. We use this program to practically verify the theoretical convergence properties of composite finite element discretizations and compare the potential of those discretizations with the potential of the classical finite element method.

@MastersThesis{Wes11,
  author       = {Westerheide, S.},
  title        = {Bildbasierte Loesung von Partiellen Differentialgleichungen mit Composite Finite Elements},
  school       = {Institute for Computational and Applied Mathematics, University of Muenster},
  month        = {march},
  year         = {2011},
  type         = {Diploma Thesis},
  keywords     = {composite finite elements, CFE, image based computing, complex geometry, coarse approximation},
  url          = \{/2011/Wes11},
}