This thesis introduces a higher-order optimization method for solving non-smooth variational problems on Riemannian manifolds. In this work, we apply the Riemannian Semismooth Newton (RSSN) method to a non-smooth non-linear optimality system derived in recent advances in manifold duality theory. In particular we will show a novel local convergence result for an inexact version of the Riemannian Semismooth Newton method and show state-of-the-art performance in numerical experiments by solving several `2-TV-like problems on manifolds with positive and negative curvature.

We have explored the concept of mobile computational fluid dynamics (CFD) solving. Using the computational
power of a smartphone we tried to generate real time simulations of relatively simple transient natural
convection problems in the laminar regime. The focus lies on finding a compromise between accuracy, speed
and stability: we want to make a real timemobile CFD solver that is as accurate as possible.
A JAVA application has been created that runs on Android. The SIMPLE algorithm has been implemented
in order to solve for the flow and heat transfer. The SIMPLE algorithm on the application was tested for
runtime performance and accuracy.
For a test problem, a real time simulation on a Nexus 5X has been realized with an accuracy of 5.4% on a
21x21 grid. Running the algorithm on a desktop gave a simulation 10 times faster than real time. Comparing
this algorithm to a version converted to MATLAB, gave similar solving speed. We concluded that the bottleneck
in the algorithm, solving matrix equations, could not be easily improved, because MATLAB solvers
perform quite optimal: a significantly faster CFD solver implementing the SIMPLE algorithm probably does
not exist.
Nevertheless, when trying to use the obtained results on water and air, we could not obtain any real time
solutions. It is up to further research to define restictions that can guarantee real time solutions for fluids.