An application of domain decomposition methods with non-conforming spectral element/Fourier expansions for the incompressible Navier–Stokes equations

An application of domain decomposition methods is presented for the incompressible Navier–Stokes equations. Non-conforming spectral element/Fourier expansions in the separate domains are employed, and a simple iterative algorithm is used, based on the Dirichlet/Neumann method at the domain interface. Thus, a new element in the present approach is that patching of mixed algebraic/trigonometric polynomial spaces is applied at the domain interface, whereas usually in domain decomposition methods finite-order polynomial spaces have been employed in the separate domains. By applying a coupled scheme for the velocity and pressure fields in each domain, a stable algorithm is obtained. New numerical results are presented on a 3-D model problem of flow in a channel, where both bounding surfaces have corrugations, but of different orientation. Smooth solutions across the domain interface are obtained. Steady flow and the onset of flow instability is simulated and discussed. The results demonstrate that spectral element/Fourier expansions, which have been previously used to study flows in geometries with one homogeneous dimension, may be employed to tackle flow problems in relatively more complex geometries. Furthermore, the results suggest that decomposition into domains with 3-D elements and domains with 2-D elements/Fourier expansions, or domains handled by spectral methods, may be an attractive possibility. The advantage is due to the orthogonality and decoupling of the Fourier modes, which leads to a computational load increasing only linearly with resolution. A related attractive feature is that a natural way for parallel implementation is offered.