In the realm of scientific computing, solving a linear system of equations is often the main bottleneck of the calculations. We extend ideas from the fast multipole method and propose a novel fast linear solver for sparse and dense matrices. The proposed algorithm is fully algebraic and has numerically proved linear complexity with the problem size. Our method relies on the low-rank compression of the new fill-in blocks generated during the elimination process. The compressed fill-ins are computed and stored in a hierarchical tree structure. The proposed solver can be used as a stand-alone direct solver with tunable accuracy determined a priori, or can be employed as a preconditioner in conjunction with an iterative method.

In addition, we present our high performance computational framework developed to simulate heated particle-laden flows. We present various results on the effects of particle preferential concentration. Simulation of the heated particle-laden flows involves solving a variable coefficient Poisson equation. We use this case, as well as many other applications, to benchmark our proposed fast linear solver.