Towards a fictitious domain method with optimally smooth solutions

The main focus of this thesis is the smoothness of the solutions provided by fictitious domain methods for elliptic boundary value problems, and the construction of a new fictitious domain method which does not introduce artificial singularities, yielding better performance. After settling the notation and the theoretical background, the fairly popular fictitious domain / Lagrange multiplier (FDLM) method is analyzed. It is shown that the singularity introduced by the Lagrange multiplier significantly degrades the performance of the method even when the solution to the original problem is smooth. This is not only documented for the non-adaptive case, but also, using wavelet techniques, for the adaptive case, deriving upper bounds for the convergence rates that are independent of the order of the approximation method. In the second part of the thesis we consider the construction of a new fictitious domain method that does not introduce any singularities. For this, a special least squares formulation is introduced and discretized. Several results on the smoothness preserving property are proven. Finally, implementation techniques are discussed and a prototype is successfully tested

Towards a fictitious domain method with optimally smooth solutions

The main focus of this thesis is the smoothness of the solutions provided by fictitious domain methods for elliptic boundary value problems, and the construction of a new fictitious domain method which does not introduce artificial singularities, yielding better performance. After settling the notation and the theoretical background, the fairly popular fictitious domain / Lagrange multiplier (FDLM) method is analyzed. It is shown that the singularity introduced by the Lagrange multiplier significantly degrades the performance of the method even when the solution to the original problem is smooth. This is not only documented for the non-adaptive case, but also, using wavelet techniques, for the adaptive case, deriving upper bounds for the convergence rates that are independent of the order of the approximation method. In the second part of the thesis we consider the construction of a new fictitious domain method that does not introduce any singularities. For this, a special least squares formulation is introduced and discretized. Several results on the smoothness preserving property are proven. Finally, implementation techniques are discussed and a prototype is successfully tested