Complex fluid systems are commonly described by coupled equation systems. In this thesis, we discussed the details of deriving the finite element formula for solving coupled equation systems, especially for the complex fluid systems. We provided solutions to both the numerical scheme to each equation set and the coupling between the two physical systems. We adopted the four-step fractional method which separates the pressure equation from the momentum equation thus reduces the degree of freedoms for solving the flow equations. The SUPG term is incorporated in all the convection dominant equations to eliminate the spurious oscillations of velocity in the solutions. The coupled equation sets are treated in a semi-coupled way, where each equation set is solved solely and sequentially, and then the solution is updated in an iterative pattern at each time step until convergence of all unknown variables has been reached.

We applied this finite element numerical scheme on investigating the behaviors of two complex fluid systems. For the problem of flow past a heated cylinder, we numerically verified the existence of three vortex shedding patterns behind a heated cylinder. We further studied the relation between the vortex shedding pattern and the cylinder aspect ratio, and found that the velocity gradient along the cylinder axis direction plays important role on affecting the formation of vortex shedding patterns behind the heated cylinder. We further demonstrated the capability of this numerical scheme on solving coupled equation systems by showcasing its application on simulating the multiphase flows. We verified that there should be at least four elements through the interface to guarantee the accuracy of the numerical solution to the interface motion. We also proved the capability of the numerical framework on solving 3D flows on complicated geometries.

We also introduced the design and implementation of a fault-tolerant framework for handling high throughput simulation tasks, especially its usage on the stochastic analysis on complex systems. The key idea of the design of this framework is separating the scientific solver apart from the input parameters generator (the allocator) as an independent process, which enables the scientific solver to use multiple processors for large scale problems. This design avoids the risk in the MPI based design where the failure of a single process will cause the termination of the whole project. By incorporating the monitoring unit into the framework, we are able to recover the failed scientific solvers. The challenge of large data set management is resolved by utilizing the HDF5 database and its managing library, which provides us efficient data storing and retrieving functionality. The performance of this fault tolerant framework was tested by solving stochastic problems with very high stochastic dimensions over thousands of processors.