Spectral / hp Methods For Elliptic Problems on Hybrid Grids

We review the basic algorithms of spectral/hp element methods on tetrahedral grids and present newer developments on hybrid grids consisting of tetrahedra, hexahedra, prisms, and pyramids. A unified tensor-product trial basis is developed for all elements in terms of non-symmetric Jacobi polynomials. We present in some detail the patching procedure to ensure C continuity and appropriate solution techniques including a multi-level Schur complement algorithm. In standard low-order methods the quality of the numerical solution of an elliptic problem depends critically on the grid used, especially in three-dimensions. Moreover, the efficiency to obtain this solution depends also on the grid, not only because grid generation may be the most computationally intensive stage of the solution process but also because it may dictate the efficiency of the parallel solver to invert the corresponding algebraic system. It is desirable to employ grids which can handle arbitrary geometric complexity and exploit existing symmetry and structure of the solution and the overall domain. Tetrahedral grids provide great flexibility in complex geometries but because of their unstructured nature they require more memory compared with structured grids consisting of hexahedra. This extra memory is used to store connectivity information as well as the larger number of tetraheda required to fill a specific domain, i.e. five to six times more tetrahedra than hexahedra. From the parallel solver point of view, large aspect ratio tetrahedra can lead to substantial degradation of convergence rate in iterative solvers, and certain topological constraints need to be imposed to maintain a balanced parallel computation. The methods we discuss in this paper address both of the aforementioned issues. First, we develop high-order hierarchical expansions with exponential convergence for smooth solutions, which are substantially less sensitive to grid distortions. Second, we employ hybrid grids consisting of tetrahedra, hexahedra, prisms, and pyramids that facilitate great discretisation flexibility and lead to substantial memory savings. An example of the advantage of hybrid grids was reported in [5] where only 170K tetrahedra in combination with prisms were employed to construct a hybrid grid around the high-speed-civil-transport aircraft instead of an estimated