Overlapping Schwarz Methods for Unstructured Spectral Elements

A parallel and scalable domain decomposition method for unstructured and hybrid spectral element discretizations of elliptic problems is introduced and studied. The spectral elements are affine images of the reference triangle or square in two dimensions and of the reference tetrahedron, pyramid, prism, or cube in three dimensions. The method is based on overlapping Schwarz techniques applied to the Schur complement of the discrete system and is implemented as a preconditioner for a Krylov space method. Numerical results in two and three dimensions show that the iteration counts of our method are bounded by a constant independent of the spectral degree and the number of subdomains. The resulting elliptic solver can be used in Navier–Stokes simulations using the spectral element code NekTar.