The purpose of this paper is to analyze some features of contaminant flow passing through cracked porous medium, such as the influence of fracture network on the advection and diffusion of contaminant species, the impact of adsorption on the overall transport of contaminant wastes. In order to precisely describe the whole process, we firstly build the mathematical model to simulate this problem numerically. Taking into consideration of the characteristics of contaminant flow, we employ two partial differential equations to formulate the whole problem. One is flow equation; the other is reactive transport equation. The first equation is used to describe the total flow of contaminant wastes, which is based on Darcy law. The second one will characterize the adsorption, diffusion and convection behavior of contaminant species, which describes most features of contaminant flow we are interested in. After the construction of numerical model, we apply locally conservative and compatible algorithms to solve this mathematical model. Specifically, we apply Mixed Finite Element (MFE) method to the flow equation and Discontinuous Galerkin (DG) method for the transport equation. MFE has a good convergence rate and numerical accuracy for Darcy velocity. DG is more flexible and can be used to deal with irregular meshes, as well as little numerical diffusion. With these two numerical means, we investigate the sensitivity analysis of different features of contaminant flow in our model, such as diffusion, permeability and fracture density. In particular, we study Kd values which represent the distribution of contaminant wastes between the solid and liquid phases. We also make omparisons of two different schemes and discuss the advantages of both methods.

This paper describes a procedure for the construction of monopoles on three-dimensional Euclideanspace, starting from their rational maps. A companion paper, ‘Euclidean monopoles and rationalmaps’, to appear in the same journal, describes the assignment to a monopole of a rational map, from$\Bbb{CP}^1$ to a suitable flag manifold. In describing the reverse direction, this paper completes the proofof the main theorem therein.

A construction of monopoles from solutions to Nahm's equations (a systemof ordinary differential equations) has been well-known for certain gauge groups for some time. Thesesolutions are hard to construct however, and the equations themselves become increasingly unwieldy when thegauge group is not $\mbox{SU}(2).$

Here, in contrast, a rational map is the only initial data. Butwhereas one can be reasonably explicit in moving from Nahm data to a monopole, here the monopole is onlyobtained from the rational map after solving a partial differential equation.

A non-linear flowequation, essentially just the path of steepest descent down the Yang-Mills-Higgs functional, is set up. Itis shown that, starting from an ‘approximate monopole’ - constructed explicitly from the rationalmap - a solution to the flow must exist, and converge to an exact monopole having the desired rationalmap.

1991 Mathematics Subject Classification: 53C07, 53C80, 58D27, 58E15, 58G11.