Eine Linienmethode zur approximativen Lösung inverser Probleme für elliptische Differentialgleichungen

In this thesis we deal with the development, theoretical examination and numerical implementation of a method of lines for the Cauchy-problem for elliptic partial differential equations. We consider both the Laplace-equation and a more general elliptic equation containing a diffusion coefficient, which depends on one of the space variables. Our main results for both differential operators are the regularization of the illposed Cauchy-Problem and
the proof of error estimates leading to convergence results for the method of lines. We base them principally on two major foundations. The first one is a conditional stability result for the continuous Cauchy-problem, of which the proof in parts of the relevant literature was not carried out thoroughly enough. The second one consists of introducing certain finitedimensional spaces, onto which the possibly perturbed Cauchy-data is projected. For the more general PDE, comprehensive additional examinations are required, which reveal the convergence of the eigenvalues and eigenvectors of the discrete approximation of a Sturm-
Liouville eigenvalue problem. We finish the thesis with the presentation of the results of some of our numerical computations and discuss them referring to the knowledge, we have gained by our preceding theoretical work. The reader can find a more detailed overview beyond the scope of this abstract in Section 1.4.