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Description:
The present dissertation provides insight into the fields of phase decomposition in lipid membranes and the separation of chiral objects, so called enantiomers, using the C0-Interior Penalty Discontinuous Galerkin (C0-IPDG) method, the fictitious domain Lagrange multiplier (FDLM) method, and automated image processing tools. Phase decomposition in lipid membranes has been the subject of numerous investigations by both experiment and theoretical simulation. However quantitative comparison of the simulated data to the experimental results are rare. In this work, we present a novel way of comparing the temporal development of liquid-ordered domains obtained from numerically solving the Cahn-Hilliard equation and by experimentally inducing a phase transition in giant unilamellar vesicles. This relies on calculating the structure factor of the domain pattern. This way, we reveal three distinct regimes of decomposition in both, experiment and simulation. As a result, we design and investigate the convergence of adaptive C0-IPDG methods of any polynomial order for the biharmonic problem. Numerical results are presented which illustrate the performance of the adaptive C0-IPDG approach. The existence of chiral lipid molecules further leads to the question of enantiomer separation. Enantiomers differ by their orientation which possibly results in completely different properties. Since they mostly occur as so-called racemic compounds their cost-effective separation is an important issue in applications. This work investigates a new cost-effective idea for separation using surface acoustic wave generated vorticity patterns. For this purpose, we simulate transport of rigid chiral particles in a fluidic environment by an adaption of the fictitious domain Lagrange multiplier method which has been designed to study the motion of rigid particles in carrier fluids. Numerical results are presented which illustrate the feasibility of enantiomer separation in flow fields consisting of pairwise counter-rotating vortices.