Multigrid methods with application to reservoir simulation

This dissertation is concerned with the study of multigrid methods for the solution of elliptic partial differential equations. The primary focus is on parallel multigrid methods and the application of multigrid methods to reservoir simulation. Multicolor Fourier analysis is used to analyze the behavior of standard multigrid methods for problems in one and two dimensions. The relationship between multicolor Fourier analysis and standard Fourier analysis is established. Multiple coarse grid methods for solving certain model problems in one and two dimensions are considered. For such methods, at each coarse grid level we use more than one coarse grid to improve convergence. For the application of multiple coarse grid methods to a given Dirichlet problem it is convenient to first construct a related extended problem. For solving an extended problem with a multiple coarse grid method, a "purification" procedure can be used to obtain Moore-Penrose solutions of the singular systems which are encountered. For solving anisotropic equations, semicoarsening and line smoothing techniques are used with multiple coarse grid methods to improve convergence. The two-level convergence factors of the multiple coarse grid methods are estimated by using a multicolor Fourier analysis. In a special case where each of the operators has the same stencil on each of the grid points on one level, the exact multilevel convergence factors of the multiple coarse grid methods can be obtained. For solving partial differential equations with discontinuous coefficients, the interpolation and restriction operators should include information about the coefficients of the equations. Matrix-dependent interpolation and restriction operators based on the Schur complement can be used in nonsymmetric cases. A semicoarsening multigrid solver with matrix-dependent interpolation and restriction operators is used in UTCOMP, a three-dimensional, multiphase, multicomponent, compositional reservoir simulator developed at The University of Texas at Austin. The numerical experiments are carried out on different computing systems. The results obtained from the analysis and the numerical experiments indicate that the multigrid methods are promising.