Low-Energy Basis Preconditioning for Elliptic Substructured Solvers Based on Unstructured Spectral/hp Element Discretization

The development and application of three-dimensional unstructured hierarchical spectral/hp element algorithms has highlighted the need for efficient preconditioning for elliptic solvers. Building on the work of Bica (Ph.D. thesis, Courant Institute, New York University, 1997) we have developed an efficient preconditioning strategy for substructured solvers based on a transformation of the expansion basis to a low-energy basis. In this numerically derived basis the strong coupling between expansion modes in the original basis is reduced thus making it amenable to block diagonal preconditioning. The efficiency of the algorithm is maintained by developing the new basis on a symmetric reference element and ignoring, in the preconditioning step, the role of the Jacobian of the mapping from the reference to the global element. By applying an additive Schwarz block preconditioner to the low-energy basis combined with a coarse space linear vertex solver we have observed reductions in execution time of up to three times for tetrahedral elements and 10 times for prismatic elements when compared to a standard diagonal preconditioner. Full details of the implementation and validation of the tetrahedral and prismatic element preconditioning strategy are set out below.