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Efficient parallel solution of PDEs

The modeling of physical systems often leads to partial differential equations (PDEs). Usually, the equations or the domain where the equations are posed are so complicated that the analytic solution cannot be found. Thus, the equations must be solved using numerical methods. In order to do this, the PDEs are first discretized using the finite element method (FEM) or the finite difference method (FDM), for example. These methods often lead to the solution of large systems of (non)linear equations which can be very time andmemory consuming even in modern computers. Therefore, efficient parallel numerical solution methods for PDEs are needed in order to make the cost of simulations reasonable. In this report, we briefly describe the main ideas of fictitious domain, domain decomposition and fast direct methods. Then, we present their applications to acoustic scattering and fluid flow problems, which have quite a different nature. This is a continuation of our earlier research efforts reported in [4].