Pseudospectral matrix element methods for flow in complex geometry

A pseudospectral matrix element (PSME) method, which extended the global pseudospectral method to a multi-element scheme, has been applied to the solution of the incompressible, primitive variable, Navier-Stokes equations for complex geometries with rectilinear or curvilinear boundaries. For a simple complex geometry, a direct solution for pressure Poisson equation is feasible, while in a much more complex geometry the pressure solution is accomplished by a new implementation of domain decomposition approach. According to this approach, the computational domain can be divided into a number of overlapping subdomains where the grid points inside the overlapping area may or may not be located at the same place. Each subdomain can be mapped onto a square domain by an algebraic (or isoparametric) mapping, of simpler geometry with patched elements, in which the pressure solution is more easily obtained by an eigenfunction expansion technique for cartesian-type geometries or a direct solver for noncartesian-type geometries with rectilinear (or curvilinear) boundaries. With an iterative Schwarz alternating procedure (SAP) between subdomains, the complete solution is found. The novel feature of this approach are (i) the continuity equation is satisfied everywhere, in the interior (including the inter-element points) and on the boundary; (ii) reducing the global storage size to local (subdomain) storage locations for which parallel computation is easily implemented; (iii) producing the desired grid points without solving any grid-generating equations is easy; and (iv) consistent mass conservation holds at geometrical singular points despite their discontinuous slope (i.e., singular vorticity). Numerical examples of flow over a triangular and parabolic bump as well as flow in a bifurcation with a daughter branch entering the main channel at angles 45° and 90° are presented in this paper.