Partial eigensolution for transient groundwater flow equations

Spectral methods offer a worthwhile alternative to traditional time stepping schemes for the integration in time of large sets of finite elements (FE) sub-surface equations. The spectral (or reduction) approach requires the solution of a generalized symmetric eigenproblem involving the matrix pencil H,P, H and P being the stiffness and capacity matrix, respectively. The transient FE response is primarily described by the leftmost eigenpairs whose evaluation represents the basic computational burden of the spectral method. A new technique for finding several of the smallest eigenpairs has been developed relying on a deflation-accelerated conjugate gradient (DACG) scheme where a sequence of objective functions are successively optimized in deflated subspaces of decreasing size. Convergence at each deflation step is controlled by the relative separation of two distinct adjacent characteristic values and asymptotic rate of convergence appears to be inversely proportional to the spectral condition number of the Hessian of the Rayleigh quotient in the current restricted subspace. Preliminary results show that preconditioning of the conjugate gradient (CG) method with H-1 greatly improves convergence which may prove to be slow only toward repeated or very close eigenvalues.