Numerical studies of higher dimensional localized black holes and holographic Weyl semimetals In our everyday lives we experience three spatial dimensions and a fourth dimension of time. Neverthe less several intricate problems of modern physics may be mastered with the introduction of additional dimensions including the hierarchy problem and the unification of the fundamental forces. Furthermore dualities between certain strongly coupled quantum field theories and particular gravitational theories in higher dimensions were conjectured based on string theory which generically comes with the premise of extra dimensions. Specifically the AdS CFT correspondence or rather gauge gravity duality motivated the study of a wide variety of higher dimensional gravitational theories with additional matter. The first part of this thesis covers the numerical construction of localized black holes in five six and ten dimensional Kaluza Klein theories. We focus on static asymptotically flat vacuum solutions of Einstein’s field equations with one periodic compact dimension. Our study concentrates on the critical regime where the poles of the localized black holes are about to merge. We utilize a well adapted multi domain pseudo spectral scheme for obtaining high accuracy results and investigate the phase diagram of the localized solutions far beyond previous results. A spiral phase space structure is found for the five and six dimensional setups which matches to the results that were recently obtained for non uniform black strings. On the contrary the phase space structure of the ten dimensional configuration exhibits no spiraling behavior of the thermodynamical quantities. These critical exponents were extracted from the numerical data of the aforementioned configurations and show an excellent agreement with the theoretical predictions. In the second part of this thesis the AdS CFT correspondence is employed for studying strongly coupled Weyl semimetals. More concretely we numerically investigate the effects of inhomogenities within a holographic Weyl semimetal by using a pseudo spectral scheme including interfaces of Weyl semimetals and the impact of time independent disorder. When studying interfaces between differentWeyl semimetal phases we observe the appearance of an electric current that is restricted to the interface in the presence of an electric chemical potential. The related integrated current is universal in the sense that it only depends on the topology of the phases. These results may shed some light on anomalous transport for inhomogeneous magnetic fields. As another point we study the effects of time independent one dimensional disorder on the holographic Weyl semimetal quantum phase transition (QPT) with a particular focus on the quantum critical region. We observe the smearing of the sharp QPT linked to the appearance of rare regions where the order parameter is locally non zero. We discuss the role of the disorder correlation and we compare our results to expectations from condensed matter theory at weak coupling. We also analyze the interplay of finite temperature and disorder.