Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing $N$ for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on $N\times \sqrt{N}$, $\sqrt{N}\times N$ and $\sqrt{N}\times \sqrt{N}$ meshes. It is shown that the combination FEM yields (up to a factor $lnN$) the same order of accuracy in the associated...

Efficient iterative solution of large linear systems on grid computers is a complex problem. The induced heterogeneity and volatile nature of the aggregated computational resources present numerous algorithmic challenges. This paper describes a case study regarding iterative solution of large sparse linear systems on grid computers within the software constraints of the grid middleware GridSolve and within the algorithmic constraints of preconditioned Conjugate Gradient (CG) type methods. We identify...