Gambolati, Pini: Complex Projection Methods for Nonequilibrium Transport Equations

The finite element Laplace transform integration of nonequilibrium or nonideal contaminant transport in porous media yields complex unsymmetric equations which are to be repeatedly solved for several (typically ) complex-valued Laplace parameters in the Laplace domain. The solutions may be obtained with the aid of iterative projection methods such as Bi-CGSTAB (BiConjugate Gradient STABilized) and TFQMR (Transpose Free Quasi Minimal Residual), effectively preconditioned in the complex space. To be of any help a preconditioner should eliminate the almost totality of the original complex eigenvalues and compact the preconditioned eigenspectrum around unity. Inexpensive preconditioners based on the incomplete triangular factorization with zero fill-in (ILU(0)) are used. The results show that both complex Bi-CGSTAB and TFQMR perform very satisfactorily also in difficult advection dominated problems and converge to the desired solution in a few iterations for a problem size of the order of thousands, irrespective of the Laplace parameter whose complex part gradually increases from zero (p0) to its maximum value (p2M+1). The condition number of the preconditioned equations is very close to one with the asymptotic convergence significantly faster than that of the equivalent real projection solver.