Parallel flow evaluation and preconditioning of gradient eigensolvers

Developing parallel codes for computing the nonlinear flow in multiaquifer porous systems is an important task both for improving model efficiency and for performing large real-life simulations. When highly compressible multiaquifer systems are considered, the flow inside some layers is governed by nonlinear equations. An effective Finite Element (FE) procedure for solving these equations was developed, relying upon the partition of the solution procedure into layer-wise steps. Such procedure was implemented and tested on a multi--processor computer. A satisfactory degree of parallelization was achieved when computing the flow in a realistic nonlinear multiaquifer system. While studying numerical models, we analyzed the topic of computing a small number of the leftmost eigenpairs in the generalized problem Ax=l Bx, where A and B are large, sparse, symmetric positive definite FE matrices, arising from the numrical integration of our models. The eigenpairs are needed both for evaluating relevant characteristics of our systems, and for developing parallel PDE solvers. Our optimization method for evaluating a number of the smallest eigenpairs, called DACG (Deflation-Accelerated Conjugate Gradient), was parallelized. When effectively preconditioned, the efficiency of DACG well compares with that of established packages. Some approximate inverse preconditioners, as accelerators for our parallel DACG, were studied. Their performance and their potential in performing parallel computations were analyzed.