Low-energy basis preconditioning for elliptic substructured solvers based on unstructured spectral/hp element discretization

The development and application of three-dimensional unstructured hierarchi cal spectral/hp element algorithms has highlighted the need for efficient p reconditioning for elliptic solvers. Building on the work of Bica (Ph.D. th esis, Courant Institute, New York University. 1997) we have developed an ef ficient preconditioning strategy for substructured solvers based on a trans formation 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 e lement. By applying an additive Schwarz block preconditioner to the low-ene rgy basis combined with a coarse space linear vertex solver we have observe d reductions in execution time of up to three times for tetrahedral element s and 10 times for prismatic elements when compared to a standard diagonal preconditioner. Full details of the implementation and validation of the te trahedral and prismatic element preconditioning strategy are set out below. (C) 2001 Academic Press.