The numerical simulation of complex physical, biological, or chemical systems typically requires the efficient solution of huge linear systems of equations which consumes the majority (70% - 90%) of the overall simulation time. In this project, we will develop, analyse and implement a novel iterative solver based on so-called hierarchical matrix techniques. We will 1) adapt hierarchical matrices to provide an efficient and robust solver for convection-dominant problems, 2) analyse theoretical properties, 3) implement the proposed algorithms and perform numerical tests in comparison with other state-of-art techniques, and 4) study possible further applications for hierarchical matrices.
This novel approach of hierarchical matrices is of significant importance within its own field of numerical analysis and also with respect to practical large-scale computing challenges that scientists are currently facing. Examples for applications include models for magnetic fusion, electrochemical processes, the growth of ceramic nanostructures, or groundwater modeling. Hierarchical matrices have first been introduced in 1998, and the encouraging results that have been obtained for well-conditioned problems motivate the application of hierarchical matrix techniques in efficient solvers for these notoriously hard to solve systems of equations that occur in Computational Fluid Dynamics.
