We present a finite element methodology tailored for the simulation of pulsatile flow in the full aorta and sinus of Valsalva interacting with highly deformable thin leaflets. We describe an extension of the so-called "Resistive Immersed Surface" method. To circumvent stability issues resulting from the bad conditioning of the linear system, especially when flow and geometry become complex after the inclusion of the aorta, we use a Lagrange multiplier technique that couples the dynamics of valve and flow. A banded level set variant allows to address the singularity of the resulting linear system while featuring, in addition to the parallel implementation, higher accuracy and an affordable computational burden. High-fidelity computational geometries are built and simulations are performed under physiological conditions. Several numerical experiments illustrate the ability of the model to capture the basic fluidic phenomena in both healthy and pathological configurations. We finally examine numerically the flow dynamics in the sinus of Valsalva after Transcatheter Aortic Valve Implantation. We show numerically that flow may be subject to stagnation in the lower part of the sinuses. We highlight the far-reaching consequences of this phenomenon and we hope inciting adequate studies to further investigate this potential clinical consequences.