Comparison of some parallel Krylov solvers for large scale groundwater contaminant transport simulations

Some popular iterative solvers for non-symmetric systems arising from the finite-element discretization of three-dimensional groundwater contaminant transport problem are implemented and compared on distributed memory parallel platforms. This paper attempts to determine which solvers are most suitable for the contaminant transport problem under varied conditions for large scale simulations on distributed parallel platforms. The original parallel implementation was targeted for the 1024 node Intel paragon platform using explicit message passing with the NX library. This code was then ported to SGI Power Challenge Array, Convex Exemplar, and Origin 2000 machines using an MPI implementation. The performance of these solvers is studied for increasing problem size, roughness of the coefficients, and selected problem scenarios. These conditions affect the properties of the matrix and hence the difficulty level of the solution process. Performance is analyzed in terms of convergence behavior, overall time, parallel efficiency, and scalability. The solvers that are presented are BiCGSTAB, GMRES, ORTHOMIN, and CGS. A simple diagonal preconditioner is used in this parallel implementation for all the methods. The results indicate that all methods are comparable in performance with BiCGSTAB slightly outperforming the other methods for most problems. The authors achieved very good scalability in all the methods up to 1024 processors of the Intel Paragon XPS/150. They demonstrate scalability by solving 100 time steps of a 40 million element problem in about 5 minutes using either BiCGSTAB or GMRES

Non-negative mixed finite element formulations for a tensorial diffusion equation

We consider the tensorial diffusion equation, and address the discrete maximum-minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. The discrete maximum-minimum principle is the discrete version of the maximum-minimum principle. In this paper we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques (in particular, quadratic programming). These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative constraint to the variational statement of the Raviart-Thomas formulation. The second non-negative formulation based on the variational multiscale formulation. For the former formulation we comment on the affect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems.Comment: 40 pages using amsart style file, and 15 figure

Upscaling models of solute transport in porous media through genetic programming

10.2166/hydro.2007.028Journal of Hydroinformatics94251-26