A sequential computational procedure for the efficient and reliable prediction of combustion instabilities in future gas turbine generations is developed in the present thesis. Special focus is put on the accurate inclusion of damping mechanisms. Essentially, the procedure consists of the creation of a linear computational model, a subsequent linear stability analysis and finally an investigation of nonlinear saturation mechanisms. A particularly efficient linear computational approach is the combination of spatially resolved finite element method approaches based on the Helmholtz equation with one-dimensional network models to account for acoustic losses. To increase the accuracy of the Helmholtz approach, a methodology to include the advection of sound waves in arbitrary mean flow fields is developed. Similar to the regular Helmholtz equation, this approach requires the development of a transformation procedure for the combination with network models to avoid energetic errors at the coupling interfaces. Using this linear computational model for a modal stability analysis with established flame driving models allows to identify potentially unstable oscillation states. The corresponding modal results can subsequently be exploited to investigate nonlinear damping mechanisms by means of reduced order models. Therefore, a universal methodology coupled with nonlinear resonator models is developed. Finally, the efficiency of the approach and thus its applicability to industrial setups is demonstrated on the basis of a geometrically complex configuration representative for a commercial gas turbine combustor. This highlights the significance of nonlinear damping mechanisms for limit-cycle oscillations.