Numerical methods for solving problems with a large contrast in the coefficients are investigated in this report. These types of problems typically appear in basin modeling. Specifically, the deflation and restricted additive Schwarz (RAS) methods are compared for their effectiveness in solving this type of problem in combination with the conjugate gradient method, both in terms of iterations and computation time. It is shown that the RAS method converges to the correct solution in a small amount of iterations. However, the deflation method, in combination with another preconditioner, performs better in terms of computation time. An important observation is that the relative residual can only be used as a reliable stopping criterion when the deflation method is used. The methods can be combined into the DRASCG method, which converges in an extremely small number of iterations. In a parallel environment, the speedup of the RASCG method is limited by a load imbalance in the amount of work required for the application of the preconditioner for each subdomain. The deflation method obtains good speedup, that is close to the ideal speedup. The methods are compared when the number of subdomains is increased. It is shown that a classic data distribution is not effective for this type of problem. The deflation method is shown to be robust for a physics based domain decomposition.

In this thesis the beta log-gas probability density function is discussed. It is shown that there is a strong link between this density function and Jacobi matrices. A change of variables exercise shows that the distribution of eigenvalues is exactly like the quadratic beta log-gas. The change of variables gives the normalization constant for the quadratic beta log-gas. Finally, it is made likely that the Jacobi matrix adheres to Wigners semicircle law, and that the beta log-gas is limited by the semicircle distribution.